Cambridge General Mathematics Year 11

9781107627291 Powers  Cambridge Preliminary Mathematics General Second Edition CMYK

Second

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Edition

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G K Powers Cambridge Preliminary Mathematics General

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Powers

Cambridge Preliminary Mathematics General Second Edition has been completely revised for the Stage 6 Mathematics General syllabus implemented from 2013, to prepare you for the HSC General 1 or General 2 course. Designed to cater for a wide range of learning styles and abilities, this student-friendly text reinforces the skills you need to manage your personal finances and effectively participate in an increasingly complex society. It enhances statistical literacy and encourages the application of relevant technologies, while developing other essential employability skills. This Second Edition closely follows the syllabus to cover all required strands and the two focus studies: Mathematics and Driving, and Mathematics and Communication. These focus studies are presented in two discrete chapters to provide teachers with the flexibility to integrate focusstudy content across the strands in a way that suits the teaching and learning approaches, abilities and knowledge in their classroom. A suggested program for integrating the focus studies is provided for teachers via Cambridge GO, but the options are many. Features: • Syllabus topics and their content are listed at the start of each chapter to outline the concepts to be covered. • Precise step-by-step worked solutions encourage independent learning. • Graded exercises focus first on basic skills and understanding, building on these through Development questions that encourage you to apply your understanding to contextualised problem solving. Challenge questions in each exercise extend and enrich students, and extra challenge questions are available in the Teacher Resource Package. • Essential rules, formulae and important concepts are highlighted throughout, while a comprehensive glossary and HSC formula sheet are ideal for quick reference and revision. • Applications of relevant technologies are incorporated, such as graphics calculator and spreadsheet activities, examples and questions. • End-of-chapter summaries and multiple-choice and short-response questions provide opportunities to consolidate and revise knowledge and understanding. • Two complete HSC Practice Papers with answers provide valuable preparation for the HSC course and exams.

Cambridge

Preliminary Mathematics General Second

Edition

n mo ry m Co mina li se pre our c or f l 1&2 a ner Ge

Cambridge

PRELIMINARY MATHEMATICS GENERAL Second Edition G K Powers

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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477 Williamstown Road, Port Melbourne, VIC 3207, Australia Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.edu.au Information on this title: www.cambridge.org/9781107627291 © The Powers Family Trust 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 under the title Cambridge Preliminary General Mathematics Second edition 2013 6th printing 2013, 2014 Cover design by Sylvia Witte Typeset by Aptara Corp. Printed in China by Print Plus Ltd A Cataloguing-in-Publication entry is available from the catalogue of the National Library of Australia at www.nla.gov.au ISBN 978-1-107-62729-1 Paperback Additional resources for this publication at www.cambridge.edu.au/GO Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this publication, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 Email: [email protected] Reproduction and communication for other purposes Except as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism or review) no part of this publication may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above. Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Contents Introduction vii Acknowledgements Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6

xii

Earning and managing money

1

FM1

Salary and wages 1 Overtime and special allowances 6 Annual leave loading and bonuses 11 Commission 15 Piecework, royalties and income from government Gross pay, net pay and deductions 22 Budgeting 27 Chapter summary 31 Sample HSC – Objective-response questions 32 Sample HSC – Short-answer questions 33 Algebraic manipulation

35

AM1

Adding and subtracting like terms 35 Multiplication and division of algebraic terms Expanding algebraic expressions 43 Factorising algebraic expressions 47 Substitution 50 Linear equations 53 Equations with fractions 60 Using formulas 63 Chapter summary 69 Sample HSC – Objective-response questions Sample HSC – Short-answer questions 71 Units of measurement and applications

18

39

70

73

MM1

Units of measurement 73 Measurement errors 79 Scientific notation and significant figures 83 Calculations with ratios 88 Rates and concentrations 92 Percentage change 96 Chapter summary 99 Sample HSC – Objective-response questions 100 Sample HSC – Short-answer questions 101

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iv

Contents

Chapter 4 Statistics and society, data collection and sampling     103 4.1 4.2 4.3 4.4

Statistical inquiry     103 Classification of data     108 Sample types     113 Designing a questionnaire     118 Chapter summary     121 Sample HSC – Objective-response questions     122 Sample HSC – Short-answer questions     123

Chapter 5 Interpreting linear relationships     125 5.1 5.2 5.3 5.4 5.5

AM2

Graphing linear functions     125 Gradient and intercept     130 Gradient-intercept formula     134 Simultaneous equations     138 Linear functions as models     142 Chapter summary     147 Sample HSC – Objective-response questions     148 Sample HSC – Short-answer questions     149

Chapter 6 Investing money     151 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

DS1

FM2

Simple interest     151 Simple interest graphs     156 Compound interest     160 Compound interest graphs     164 Using prepared tables     168 Financial institutions: costs     172 Appreciation and inflation     175 Shares and dividends     179 Chapter summary     183 Sample HSC – Objective-response questions     184 Sample HSC – Short-answer questions     185

HSC Practice Paper 1     187 Chapter 7 Displaying and interpreting single data sets     193 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

DS2

Frequency tables     193 Grouped frequency tables     197 Cumulative frequency     200 Range and interquartile range     204 Frequency and cumulative frequency graphs     209 Box-and-whisker plots     215 Sector and divided bar graphs     220 Radar charts     224 Dot plots and stem-and-leaf plots     227 Chapter summary     231 Sample HSC – Objective-response questions     232 Sample HSC – Short-answer questions     233

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v

Contents

Chapter 8 Applications of perimeter, area and volume     235 8.1 8.2 8.3 8.4 8.5 8.6

Pythagoras’ theorem     235 Perimeter     239 Area     244 Field diagrams     250 Volume of prisms and cylinders     254 Capacity     258 Chapter summary     261 Sample HSC – Objective-response questions     262 Sample HSC – Short-answer questions     263

Chapter 9 Relative frequency and probability     265 9.1 9.2 9.3 9.4 9.5 9.6

FM3

Allowable deductions     293 Taxable income     297 Medicare levy     301 Calculating tax     304 Calculating GST     309 Graphing tax rates     313 Chapter summary     317 Sample HSC – Objective-response questions     318 Sample HSC – Short-answer questions     319

Chapter 11 Summary statistics     321 11.1 11.2 11.3 11.4 11.5

PB1

Relative frequency     265 Multistage events     271 Systematic lists     274 Definition of probability     279 Range of probabilities     283 Complementary events     286 Chapter summary     289 Sample HSC – Objective-response questions     290 Sample HSC – Short-answer questions     291

Chapter 10 Taxation     293 10.1 10.2 10.3 10.4 10.5 10.6

MM2

DS3

The median     321 Mean and mode     325 The mean from larger data sets     330 Standard deviation     335 Comparison of summary statistics     339 Chapter summary     343 Sample HSC – Objective-response questions     344 Sample HSC – Short-answer questions     345

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vi

Contents

Chapter 12 Similarity and right-angled triangles     347 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

Similar figures and scale factors     347 Problems involving similar figures     352 Scale drawings     357 Trigonometric ratios     360 Using the calculator in trigonometry     365 Finding an unknown side     369 Finding an unknown angle     373 Applications of right-angled triangles     376 Angles of elevation and depression     380 Chapter summary     385 Sample HSC – Objective-response questions     386 Sample HSC – Short-answer questions     387

Chapter 13 Mathematics and communication     389 13.1 13.2 13.3 13.4 13.5

FSCo

Mobile phone plans     389 Phone usage tables and graphs     396 File storage     399 Digital downloads     403 Digital download statistics     406 Chapter summary     409 Sample HSC – Objective-response questions     410 Sample HSC – Short-answer questions     411

Chapter 14 Mathematics and driving     413 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9

MM3

FSDr

Cost of purchase     413 Insurance     418 Stamp duty     421 Running costs (fuel)     424 Straight-line depreciation     428 Declining balance depreciation     431 Safety     435 Blood alcohol content     441 Driving statistics     446 Chapter summary     451 Sample HSC – Objective-response questions     452 Sample HSC – Short-answer questions     453

HSC Practice Paper 2     455  SC formula sheet    461 H Glossary     463 Answers     469

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Introduction New syllabus and Focus studies Cambridge Preliminary Mathematics General Second Edition has been completely revised for the stage 6 Mathematics General syllabus to be implemented from 2013, and the HSC General 2 examination being implemented in 2014. The Preliminary course is a common preparation for both the General 1 and General 2 courses at HSC This resource closely follows the syllabus and is divided into strands, topics and focus studies. Each focus study is contained in a single chapter to provide easy access, and is designed to be integrated across the strands. Teachers can decide on the integration depending on the ability and knowledge of their students. The teaching program outlines one method of integration.

Additional new features in the second edition: • • • • • • •

Companion website on Cambridge GO (www.cambridge.edu.au/go) with a downloadable digital version and an online interactive version of the textbook. Extra resources have been added to the teacher resources and to GO – details given below. Teaching program for the new syllabus can be downloaded from GO. The more challenging questions are identified and extra ones have been added to the exercises and to the companion website. Extensive exercises divided into foundation, development and challenge questions cater for students at different levels, and facilitate differentiation into General 1 and General 2 courses. The sample HSC objective response questions can also be accessed via GO in a self-marking ‘Quiz Me’ format for web browsers and smartphones. Two complete HSC Practice Papers.

Existing features retained from the first edition: • • • • • •

Important concepts in boxes for easy reference. Excel spreadsheet activities integrated in the text. Graphics calculator explanations and problems integrated into the text. Chapter reviews containing a summary plus sample HSC objective-response (multiple-choice) and short-answer questions. Comprehensive glossary and HSC formula sheet. HOTmaths integrated program available (requires subscription).

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vii

viii

Introduction

Guide to the icons used in the textbook Identifies challenge questions in the exercises. Challenge questions 1

(placed at the end of each chapter)

A PowerPoint file or Word worksheet containing extra challenge questions is available on Cambridge GO.

1.1 1B 14.1

14A

Integrated HOTmaths course available (access by teacher account or student subscription). Spreadsheet file available on Cambridge GO. Used in Chapters 1–12 to indicate where the teaching program suggests that a Focus Study section be done next. An alternative worksheet format is available for the exercise on Cambridge GO. Study Guide 8

(placed on the Chapter Summary bar)

A PowerPoint file containing a study guide is available on Cambridge GO.

Additional Resources in the Teacher’s Resource package on Cambridge GO • • • • •

Lesson Notes – a new resource: PowerPoint files containing comprehensive lesson notes and additional examples which can be used in class or given to students as tutorials Chapter tests as worksheets, with answers Literacy worksheets – activities to help with mathematical terminology Spreadsheet skills worksheets – to use with spreadsheet files provided Copies of the teaching programs and scope and sequence charts

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Introduction

ix

About the author Greg Powers is currently the Head of Mathematics at Cabramatta High School and the coordinator of the Mathematics Head Teacher Western Network. He is an experienced classroom teacher, having taught for over 30 years in a range of different schools. Greg has been a senior marker for the HSC, educational consultant for the Metropolitan South West Region and presented at numerous MANSW inservices. He has also enjoyed several curriculum roles with the Department of Education and Training. Greg is an experienced author who has written numerous texts on mathematics and technology.

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Acknowledgements The author and publisher wish to thank the following sources for permission to reproduce material: Images: Wikimedia Commons, p.9 (bottom), p.175, p.176, p.179 (bottom); Australian Bureau of Statistics. Reprinted with permission. © Commonwealth of Australia, p.106; Privacy Commission, p.107; Picture by Steve Bowbrick, flickr.com/photos/bowbrick, p.178; Courtesy of the Commonwealth Bank of Australia, p.417; Courtesy of the NSW Office of State Revenue, p.421; Courtesy of the Department of Commerce Western Australia, p.424; andesign101 / Shutterstock.com, p.425 (top); EvrenKalinbacak / Shutterstock.com, p.429; zstock / Shutterstock.com, p.437; Andre Dobroskok / Shutterstock.com, p.439; ronfromyork / Shutterstock.com, p.449; Fedor Selivanov / Shutterstock.com, p.453; All other images 2012 used under license from Shutterstock.com. Every effort has been made to trace and acknowledge copyright. The publisher apologises for any accidental infringement and welcomes information that would redress this situation.

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C H A P T E R

1

Earning and managing money Syllabus topic — FM1 Earning and managing money Calculate payments from a salary Calculate wages using hourly rate, overtime rates and allowances Calculate annual leave loading and bonuses Calculate earnings based on commission, piecework and royalties Determine deductions and calculate net pay Evaluate a prepared budget

1.1 Salary and wages Salary 1.1

Salary is a payment for a year’s work which is then divided into equal monthly, fortnightly or weekly payments. People who are paid a salary include teachers and nurses. Advantages • Permanent employment • Superannuation, sick and holiday pay

Disadvantages • No overtime for extra work • Hours are fixed

Converting salary to weeks, fortnights and months 1 year = 52 weeks Example 1

1 year = 26 fortnights

1 year = 12 months

Calculating from a salary

Mitchell earns a salary of $65 208 per annum. He is paid fortnightly. How much does he receive each fortnight? Assume there are 52 weeks in the year. 1 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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2

Preliminary Mathematics General

Solution 1 2 3 4

Write the quantity to be found. Divide the salary by the number of fortnights in a year or 26. Evaluate and write using correct units. Write your answer in words.

Fortnightly pay = 65 208 ÷ 26 = $2508.00 Mitchell is paid $2508 per fortnight.

Wages Wage is a payment for work calculated on an hourly basis. People who are paid a wage include shop assistants, factory workers and mechanics. Advantages • Permanent employment • Superannuation, sick and holiday pay • Overtime payments for extra work Example 2

Disadvantages • No incentive to work hard each hour • Hours are fixed

Calculating a wage

Jasmine is paid at a rate of $1098 for a 40-hour week. a How much does Jasmine earn per hour? b What wages will Jasmine receive for a week where she works 38 hours? Solution 1 2 3 4 5 6 7

Write the quantity to be found. Divide the amount by the number of hours worked. Evaluate and give answer correct to two decimal places. Write the quantity to be found. Multiply the rate by the number of hours worked. Evaluate and write using correct units. Write your answer in words.

Salary A payment for a year’s work, which is then divided into equal monthly, fortnightly or weekly payments.

a

Wage per hour = 1098 ÷ 40 = $27.45 Jasmine earns $27.45 per hour.

b

Wage for 38 hours = 27.45 × 38 = $1043.10 Jasmine receives $1043.10 for the week.

Wage A payment for a week’s work and is calculated on an hourly basis.

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Chapter 1 — Earning and managing money

Exercise 1A 1

Emily earns a salary of $92 648. Write, to the nearest dollar, her salary as amounts per: a week. b fortnight. c month.

2

The annual salary for 4 people is shown in the table below. Calculate their weekly and fortnightly payments. (Answer correct to the nearest dollar.) Name

Salary

a

Abbey

$57 640

b

Blake

$78 484

c

Chloe

$107 800

d

David

$44 240

Week

Fortnight

3

What is Zachary’s fortnightly income if he earns a salary of $43 056?

4

Find the annual salary for the following people. a Amber earns $580 per week. b Tyler earns $1520 per fortnight. c Samuel earns $3268 per month. d Ava earns $2418 per week.

5

Harrison is a civil engineer and who earns a salary of $1500 per week. a How much does he receive per fortnight? b How much does he receive per year?

6

What is Yasmeen’s annual salary if her salary per fortnight is $1610?

7

Dylan receives a weekly salary payment of $1560. What is his annual salary?

8

Stephanie is paid $1898 per fortnight and Tahlia $3821 per month. Calculate each person’s equivalent annual income. Who earns the most per week and by how much?

9

Laura is paid $1235 per fortnight and Ebony $2459 per month. Which person receives the highest annual salary and by how much?

10

Tran is paid $1898 per week and Jake $8330 per month. Calculate each person’s equivalent annual income. What is the difference between their annual salaries?

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3

4

Preliminary Mathematics General

11

Joshua works as a labourer and is paid $25.50 an hour. How much does he earn for working the following hours? a 35 hours b 37 hours c 40 hours d 42 hours

12

Lily earns $29.75 an hour. If she works 6 hours each day during the week and 4 hours a day during the weekend, find her weekly wage.

13

Determine the wage for a 37-hour week for each of the following hourly rates. a $12.00 b $9.50 c $23.20 d $13.83

14

Determine the income for a year (52 weeks) for each of the following hourly rates. Assume 40 hours of work per week. a $7.59 b $15.25 c $18.78 d $11.89

15

Suchitra works at the local supermarket. She gets paid $22.50 per hour. Her time card is shown below. Day

In

Out

Monday

9.00 a.m. 5.00 p.m.

Tuesday

9.00 a.m. 6.00 p.m.

Wednesday 8.30 a.m. 5.30 p.m.

a b

Thursday

9.00 a.m. 4.30 p.m.

Friday

9.00 a.m. 4.00 p.m.

How many hours did Suchitra work this week? Find her weekly wage.

16

Grace earns $525 in a week. If her hourly rate of pay is $12.50, how many hours does she work in the week?

17

Zachary is a plumber who earned $477 for a day’s work. He is paid $53 per hour. How many hours did Zachary work on this day?

18

Lucy is a hairdresser who earns $24.20 per hour. She works an 8-hour day. a How much does Lucy earn per day? b How much does Lucy earn per week? Assume she works 5 days a week. c How much does Lucy earn per fortnight? d How much does Lucy earn per year? Assume 52 weeks in the year.

19

Alyssa is paid $36.90 per hour and Connor $320 per day. Alyssa works a 9-hour day. Who earns the most per day and by how much?

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Chapter 1 — Earning and managing money

5

Development 20

Feng is retiring and will receive 7.6 times the average of his salary over the past three years. In the past three years he was paid $84 640, $83 248 and $82 960. Find the amount of his payout.

21

Liam’s salary is currently $76 000. He will receive salary increases as follows: 5% increase from 1 July and then a 5% increase from 1 January. What will be his new salary from 1 January?

22

Create the spreadsheet below.

1A

a b

c d

Cell E5 has a formula that multiplies cells C5 to D5. Enter this formula. Enter the hours worked for the following employees: Liam – 20 Lily – 26 Tin – 38 Molly – 40 Noth – 37.5 Nathan – 42 Joshua – 38.5 Fill down the contents of E5 to E12. Edit the hourly pay rate of Olivia Cini to $16.50. Observe the change in E5.

23

Isabelle earns $85 324 per annum. Isabelle calculated her weekly salary by dividing her annual salary by 12 to determine her monthly payment and then divided this result by 4 to determine her weekly payment. What answer did Isabelle get, what is the correct answer, and what is wrong with Isabelle’s calculation?

24

Lucy earns $8 per hour and Ebony earns $9 per hour. Last week they both earned at least $150. What is the least number of hours that Lucy could have worked last week?

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6

Preliminary Mathematics General

1.2 Overtime and special allowances Overtime 1.2

Overtime rates apply when employees work beyond the normal working day. Payment for overtime is usually more than the normal pay rate. For example, a person whose normal pay rate is $10 an hour would receive $20 ($10 × 2) an hour if they were paid overtime at double time. Another common overtime rate is time-and-a-half. It is the normal pay rate multiplied by 1 12 or 1.5. Here a person would receive $15 ($10 × 1.5) an hour. Overtime rates Time-and-a-half rate Double time rate Example 3

– normal pay rate × 1.5 – normal pay rate × 2

Calculating wages involving overtime

John works for a building construction company. Find John’s wage during one week where he works 40 hours at the normal rate of $16 an hour, 3 hours at time-and-a-half rates and 1 hour at double time rates.

Solution 1 2 3

4 5

Write the quantity to be found. Wage = (40 × 16) normal pay Normal wage is 40 multiplied by $16. Payment for time-and-a-half is + (3 × 16 × 1.5) time-and-a-half pay 3 multiplied by $16 multiplied by 1.5. Payment for double time is 1 + (1 × 16 × 2) double time pay multiplied by $16 multiplied by 2. = $744.00 Evaluate and write your answer in words. John’s wage is $744.

Special allowances Employees receive an allowance if they work under difficult or dangerous conditions such as wet weather, extreme temperatures, confined spaces or isolated areas. Allowances are also paid when an employee has an expense related to their line of work such as uniform, meals, travel or tools. © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Chapter 1 — Earning and managing money

Casual work Casual work involves a set amount paid for each hour’s work. It can be paid weekly or fortnightly. Advantages • Working hours are flexible • Pay rate is often higher Example 4

Disadvantages • No superannuation, sick or holiday pay • May lose job when not needed

Calculating casual pay

Milan is employed on a casual basis for a fast-food company. His rate of pay is $15 per hour plus time-and-half on Saturday and double time on Sunday. Last week Milan worked from 10.30 a.m. until 2.30 p.m. on Thursday, from 9.30 a.m. until 2.00 p.m. on Saturday, and from 12 noon until 4 p.m. on Sunday. How much did Milan earn last week? Solution 1 2

3

4

5 6

Write the quantity to be found. Normal wage is 4 hours (Thursday 10.30 a.m. until 2.30 p.m.) multiplied by $15. Payment for time-and-a-half is 4.5 hours (Saturday 9.30 a.m. until 2.00 p.m.) multiplied by $15 multiplied by 1.5. Payment for double time is 4 hours (Sunday 12 noon until 4 p.m.) multiplied by $15 multiplied by 2. Evaluate and write using correct units. Write your answer in words.

Wage = (4 × 15)

normal pay

+ (4.5 × 15 × 1.5)

time-and-a-half pay

+ (4 × 15 × 2) double time pay

= $281.25 Milan’s wage is $281.25

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7

8

Preliminary Mathematics General

Exercise 1B 1

Calculate the payment for working 4 hours overtime at time-and-a half given the following normal pay rates. a $18.00 b $39.50 c $63.20 d $43.83

2

Calculate the payment for working 3 hours overtime at double time given the following normal pay rates. a $37.99 b $19.05 c $48.78 d $61.79

3

Andrew earns $32.50 an hour as a driver. He works 38 hours a week at normal time and 5 hours a week at double time. Find his weekly wage. Answer correct to the nearest cent.

4

Mei is a casual employee who worked 8 hours at normal time and 2 hours at time-and-ahalf. Her normal rate of pay is $12.30 per hour. What is her pay for the above time?

5

Oliver earns $23.80 an hour. He earns normal time during week days and time-and-a-half on weekends. Last week he worked 34 hours during the week and 6 hours during the weekend. Find his weekly wage.

6

George works in a take-away food store. He gets paid $18.60 per hour for a standard 35-hour week. Additional hours are paid at double time. His time card is shown below. Day

b

Out

Monday

8.30 a.m. 4.30 p.m.

Tuesday

9.00 a.m. 6.00 p.m.

Wednesday

8.45 a.m. 5.45 p.m.

Thursday

9.00 a.m. 6.30 p.m.

Friday a

In

10.00 a.m. 8.00 p.m.

How many hours did George work this week? Find his weekly wage.

7

Dave works for 5 hours at double time. He earns $98.00. Find his normal hourly rate.

8

Ella works 3 hours at time-and-a-half and earns $72.00. Find her normal hourly rate.

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9

9

Zahid is paid a set wage of $774.72 for a 36-hour week, plus time-and-a-half for overtime. In one particular week he worked 43 hours. What are Zahid’s earnings?

10

Samantha is paid a set wage of $962.50 for a 35-hour week, plus double time for overtime. In one particular week she worked 40 hours. What are Samantha’s earnings?

11

A window washer is paid $22.50 per hour and a height allowance of $55 per day. If he works 9 hours each week day, calculate the: a amount earned each week day b total weekly earnings for five days of work.

12

Anna works in a factory and is paid $18.54 per hour. If she operates the oven she is paid temperature allowance of $4.22 per hour in addition to her normal rate. Find her weekly pay if she works a total of 42 hours including 10 hours working the oven.

13

Scott is a painter who is paid a normal rate of $36.80 per hour plus a height allowance of $21 per day. If Scott works 9 hours per day for 5 days on a tall building, calculate his total earnings.

14

Kathy is a scientist who is working in a remote part of Australia. She earns a salary of $86 840 plus a weekly allowance of $124.80 for working under extreme and isolated conditions. Calculate Kathy’s fortnightly pay.

15

Chris is a soldier and is paid $27 per hour plus an additional allowance of $12.50 per hour for disarming explosives. What is his total weekly pay if he works from 6 a.m. to 2 p.m. for 7 days a week on explosives?

16

A miner earns a wage of $46.20 per hour plus an allowance of $28.20 per hour for working in cramped spaces. The miner worked a 10-hour day for 5 days in small shaft. What is his weekly pay?

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Preliminary Mathematics General

Development 17

Vien is employed on a casual basis. His rate of pay is shown below. Last week Vien worked from 11.30 a.m. until 3.30 p.m. on Thursday, from 8.30 a.m. till 2.00 p.m. on Saturday, and from 12 noon till 6 p.m. on Sunday. How much did Vien earn last week? Rate of pay Weekdays

$18.60 per hour

Saturday

Time-and-a-half

Sunday

Double time

18

A mechanic’s industrial award allows for normal rates for the first 7 hours on any day. It provides for overtime payment at the rate of time-and-a-half for the first 2 hours and double time thereafter. Find a mechanic’s wage for a 12-hour day if their normal pay rate is $42.50 an hour.

19

Abbey’s timesheet is shown Day In Out Meal break opposite. She gets paid $12.80 Monday 8.30 a.m. 5.30 p.m. 1 hour per hour during the week, time-and-a-half for Saturdays and Tuesday 8.30 a.m. 3.00 p.m. 1 hour double time for Sundays. Abbey Wednesday 8.30 a.m. 5.30 p.m. 1 hour is not paid for meal breaks. Thursday 8.30 a.m. 9.00 p.m. 2 hours a How much did Abbey earn at Friday 4.00 p.m. 7.00 p.m. No break the normal rate of pay during this week? Saturday 8.00 a.m. 4.00 p.m. No break b How much did Abbey earn Sunday 10.00 a.m. 3.00 p.m. 30 minutes from working at penalty rates during this week? c What percentage of her pay did Abbey earn by working at penalty rates?

20

Connor works a 35-hour week and is paid $18.25 per hour. Any overtime is paid at time-and-a-half. Connor wants to earn enough overtime to earn at least $800 each week. What is the minimum number of hours overtime that Connor will need to work?

21

Max works in a shop and earns $21.60 per hour at the normal rate. Each week he works 15 hours at the normal rate and 4 hours at time-and-a-half. a Calculate Max’s weekly wage. b Max aims to increase his weekly wage to $540 by working extra hours at the normal rate. How many extra hours must Max work? c Max’s rate of pay increased by 5%. What is his new hourly rate for normal hours? d What will be Max’s new weekly wage assuming he maintains the extra working hours?

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1.3 Annual leave loading and bonuses Annual leave loading 1.3

Annual leave loading is a payment calculated as a fixed percentage of the normal pay over a fixed number of weeks. It is usually paid at the beginning of the annual holidays to meet the increased expenses of a holiday. Annual leave loading Annual leave loading or holiday loading is usually at the rate of 17 12 %. Holiday loading = 17 12 % × Normal weekly pay × Number of weeks leave

Example 5

Finding the annual leave loading

Thomas works a 40-hour week at a rate of $18.50 per hour. He receives 17 12 % of 4 weeks normal pay as holiday loading. What is Thomas’s pay for the holiday? Solution 1 2

3 4 5 6 7 8 9 10

Write the quantity (4 weeks pay) to be found. Multiply the pay rate by the number of hours worked during the week by the number of weeks (4). Evaluate. Write the quantity (loading) to be found.

(

)

Multiply 0.175 17 12 % by 4 weeks pay (2960). Evaluate. Write the quantity (holiday pay) to be found. Add the 4 weeks pay (2960) and the loading (518). Evaluate. Write your answer in words.

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4 weeks pay = (18.50 × 40 × 4) = $2960

Loading = 17 12 % of $2960 = 0.175 × 2960 = $518 Holiday pay = $2960 + $518 = $3478

Thomas receives $3478 in holiday pay.

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Preliminary Mathematics General

Bonus A bonus is an extra payment or gift earned as reward for achieving a goal. It is paid in addition to the normal income. Bonuses are an incentive for employees to work harder. For example, an employee may receive a bonus of 5% of their annual salary or $1000. Bonus Bonus is an extra payment or gift earned as reward for achieving a goal.

Example 6

Calculating a bonus

Amber’s employer has decided to reward all their employees with a bonus. The bonus awarded is 5% of their annual salary. What is Amber’s bonus if her annual salary is $68 560? Solution 1 2 3 4

Write the quantity (bonus) to be found. Multiply the bonus percentage (5%) by the annual salary ($68 560). Evaluate. Write the answer in words.

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Bonus = 5% of $68560 = 0.05 × 68560 = $3428 Amber receives a bonus of $3428.

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13

Exercise 1C 1

A business pays 17 12 % holiday loading on 4 weeks normal pay. Calculate the amount of holiday loading for these employees. a Nicholas earns $6240 normal pay for 4 weeks. b Kumar earns $5130 normal pay for 4 weeks. c Samantha earns $5320 per fortnight. d Andrew earns $2760 per fortnight. e Bilal earns $1680 per week.

2

The local government pays its employees 17.5% holiday loading on 4 weeks normal pay. Calculate the amount of holiday loading for these employees. a Paige earns an annual salary of $105 560. (Assume 52 weeks in a year.) b Jack earns an annual salary of $58 760. (Assume 52 weeks in a year.) c Riley earns $32 per hour and works a 35-hour week. d A’ishah earns $41.50 per hour and works a 37-hour week.

3

Laura works a 37-hour week at a rate of $20.50 per hour. When she takes her 4 weeks annual leave, she is paid a loading of 17 12 %. What is Laura’s holiday pay when she takes her leave?

4

Ethan is paid $660 per week. He receives a holiday leave loading of 17.5% for three weeks holiday pay. What is his total holiday pay?

5

A bonus is awarded as a percentage of a person’s annual salary. The percentage awarded depends on the person’s achievements. Calculate the following bonuses. a 6% of $48 360 b 3% of $96 540 c 2% of $103 290 d

4.5% of $65 420

e

2 12 % of $88 580

f

1 14 % of $164 400

6

Grace received a bonus of 12% of her weekly wage. What was Grace’s bonus if her weekly wage is $1850?

7

Patrick’s boss has decided to reward all employees with a bonus. The bonus awarded is 7 3 % of their annual salary. What is Patrick’s bonus if his annual salary is $74 980? 4

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Preliminary Mathematics General

Development 8

Chen receives 17.5% of 4 weeks normal pay as leave loading. If Chen’s leave loading was $379.40, what was his normal weekly pay?

9

Create the spreadsheet below.

1C

a

b c d e f

10

Jim receives holiday loading of 17 12 % of 4 weeks pay. His loading was $996.80. a b

11

The formula for cell E5 is ‘=IF(AND(C5>10,D5>50),400,0)’. It is the formula that calculates a $400 bonus if the employer has more than 10 years of service and more than 50 hours of overtime. Fill down the contents of E6 to E10 using this formula. Edit the overtime amount for Sienna Humes to 52. Observe the changes in E7. Edit the years of service for Ava White to 10. Observe the changes in E10. Edit the overtime amount for Dylan Fraser to 60. Observe the changes in E6. Edit the years of service for Xay Sengmany to 20. Observe the changes in E9. Edit the overtime amount for Benjamin Huynh to 40. Observe the changes in E8.

Find his normal weekly pay. Find his normal hourly pay rate if he usually works a 40 hour week.

Chloe’s annual salary is $72 800. a Calculate her weekly wage. b Holiday loading is calculated at 17 12 % of four weeks pay. Calculate Chloe’s holiday loading. c Chloe’s employer has proposed to increase her annual salary by 1%. What is Chloe’s new annual salary? d The increase in Chloe’s annual salary is compensation for removing holiday loading. Explain why Chloe is worse off financially with the 1% increase.

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1.4 Commission Commission is usually a percentage of the value of the goods sold. People such as real estate agents and salespersons are paid a commission. 1.4

Advantages • Higher sales increase the income • May receive a small payment (retainer) plus the commission

Disadvantages • Income may vary each week • Competition for customers is usually high

Commission Commission = Percentage of the value of the goods sold

Example 7

Finding the commission

Zoë sold a house for $650 000. Find the commission from the sale if her rate of commission was 1.25%. Solution 1 2 3 4

Commission = 1.25% of $650 000 = 0.0125 × 650 000 = $8125 Commission earned is $8125.

Write the quantity (commission) to be found. Multiply 1.25% by $650 000. Evaluate and write using correct units. Write the answer in words.

Example 8

Finding the commission

An electrical goods salesman is paid $570.50 a week plus 4% commission on all sales over $5000 a week. Find his earnings in a week where his sales amounted to $6800. Solution 1 2 3 4 5

Commission on sales of over $5000 is $1800. Write the quantity (earnings) to be found. Add weekly payment and commission of 4% on $1800. Evaluate and write using correct units. Write the answer in words.

Sales = 6800 – 5000 = 1800 Earnings = 570.50 + (4% of $1800) = 570.50 + (0.04 × 1800) = $642.50 Earnings were $642.50.

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Preliminary Mathematics General

Exercise 1D 1

Jake earns a commission of 4% on the sales price. What is the commission on the following sales? a $8820 b $16 740 c $34 220

2

Michael Tran is a real estate agent. He earns 2% on all sales. Calculate Michael’s commission on these sales. a $456 000 b $420 000 c $285 500 d $590 700

3

Olivia sold a car valued at $54 000. Calculate Olivia’s commission from the sale if her rate of commission is 3%.

4

Sophie earns a weekly retainer of $355 plus a commission of 10% on sales. What are Sophie’s total earnings for each week if she made the following sales? a $760 b $2870 c $12 850

5

Chris earns $240 per week plus 25% commission on sales. Calculate Chris’s weekly earnings if he made sales of $2880.

6

Ella is a salesperson for a cosmetics company. She is paid $500 per week and a commission of 3% on sales in excess of $800. a What does Ella earn in a week when she makes sales of $1200? b What does Ella earn in a week when she makes sales of $600?

7

A real estate agent charges a commission of 5% for the first $20 000 of the sale price and 2.5% for the balance of the sale price. Copy and complete the following table. Sale price a

$150 000

b

$200 000

c

$250 000

d

$300 000

5% commission on $20 000

2.5% commission on balance

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Development 8

Jade is a real estate agent and is paid an annual salary of $18 000 plus a commission of 2.5% on all sales. She is also paid a car allowance of $50 per week. What was Jade’s total yearly income if she sold $1 200 000 worth of property?

9

The commission that a real estate agent charges for selling a property is based on the selling price and is shown below. Selling price

Commission

First $20 000

5%

Next $120 000

3%

Thereafter

1%

What is the commission charged on properties with the following selling prices? a $100 000 b $150 000 c $200 000 10

Harry is a sales person. He earns a basic wage of $300 per week and receives commission on all sales. Last week he sold $20 000 worth of goods and earned $700. What was Harry’s rate of commission?

11

Caitlin and her assistant, Holly, sell perfume. Caitlin earns 20% commission on her own sales, as well as 5% commission on Holly’s sales. What was Caitlin’s commission last month when she made sales of $1800 and Holly made sales of $2000?

12

A real estate agency charges a commission for selling a property based on the selling price below. Commission rates Up to $300 000

4%

$300 000 and over

5%

Bailey is paid by the real estate agent $180 per week plus 5% of the commission received by the real estate agency. This week, Bailey sold one property for $290 000 and one for $600 000. He sold no properties in the previous week. a What is the commission paid to the real estate agency for the property worth $290 000? b What is the commission paid to the real estate agency for both properties? c Calculate Bailey’s pay for this week. d What is Bailey’s average weekly income for the two-week period?

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Preliminary Mathematics General

1.5 Piecework, royalties and income from government Piecework Piecework is a fixed payment for work completed. People who are employed to complete a particular task, such as an electrician installing lights, are earning piecework. Advantages • Incentive to work hard. Income increases with more work completed • Often flexible hours and work place

Disadvantages • No permanent employment • No superannuation, sick or holiday pay

Piecework Piecework = Number of units of work × Amount paid per unit

Example 9

Calculating a piecework payment

Noah is a tiler and charges $47 per square metre to lay tiles. How much will he earn for laying tiles in a room whose area is 14 square metres? Solution 1 2 3 4

Write the quantity (earnings) to be found. Multiply number of square metres (14) by the charge ($47). Evaluate and write using correct units. Write the answer in words.

Earnings = 14 × $47 = $658

Noah earns $658.

Royalties A royalty is a payment for the use of intellectual property such as a book or song. It is calculated as a percentage of the revenue or profit received from its use. People such as creative artists and authors receive a royalty. Advantages • Incentive to work hard. Income increases with a better product • Flexible hours and work place

Disadvantages • Income varies according to sales • No superannuation, sick and holiday pay

Royalty Royalty = Percentage of the goods sold or profit received

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Chapter 1 — Earning and managing money

Example 10

19

Calculating a royalty

Andrew is an author and is paid a royalty of 12% of books sold. Find his royalties if there were 2480 books sold at $67.50 each. Solution 1 2 3 4

Write the quantity (royalty) to be found. Multiply 12% by the total sales or 2480 × $67.50 Evaluate and write using correct units. Write the answer in words.

Royalty = 12% of (2480 × $67.50) = 0.12 × 2480 × 67.50 = $20 088 Andrew earns $20 088 in royalties.

Incomes from the government Some people receive a pension, allowance or benefit from the government. For example, the age pension is payable for a person who has reached 65 years of age (male). The requirements for receiving these incomes may change according to the priorities of the current government.

Example 11

Calculating an income from the government

Youth allowance helps people studying, undertaking training or in an apprenticeship. Status

Allowance per fortnight

Under 18, at home

$194.50

Under 18, away from home

$355.40

18 and over, away from home

$355.40

18 and over, at home

$233.90

How much youth allowance does Ryan receive in a year if he is over 18 and living at home while studying? Solution 1 2 3 4

Write the quantity (allowance) to be found. Multiply allowance per fortnight ($233.90) by 26. Evaluate and write using correct units. Write the answer in words.

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Allowance = $233.90 × 26 = $6081.40 Youth allowance is $6081.40. Cambridge University Press

20

Preliminary Mathematics General

Exercise 1E 1

A dry cleaner charges $9 to clean a dress. How much do they earn by dry cleaning: a 250 dresses? b 430 dresses? c 320 dresses?

2

Abbey is an artist who makes $180 for each large portrait and $100 for each small portrait. How much will she earn if she sells 13 large and 28 small portraits?

3

Angus works part-time by addressing envelopes at home and is paid $23 per 100 envelopes completed, plus $40 to deliver them to the office. What is his pay for delivering 2000 addressed envelopes?

4

Emilio earns a royalty of 24% on net sales from writing a fiction book. There were $18 640 net sales in the last financial year. What is Emilio’s royalty payment?

5

Calculate the royalties on the following sales. a 3590 books sold at $45.60 with a 8% royalty payment b 18 432 DVDs sold at $20 with a 10% royalty payment c 4805 computer games sold at $65.40 with a 5% royalty payment

6

Austudy provides financial help for people aged 25 or older who are studying full-time. Status

a b

7

Fortnightly payment

Single, no children

$355.40

Single, with children

$465.60

Partnered, with children

$390.20

Partnered, no children

$355.40

How much does Madison receive in a year if she is single with a child and studying full-time? Madison is 29 years old. How much does Oscar receive in a year if he is partnered with no children and studying full-time? Oscar is 35 years old.

Child-care benefit is available to support parents in the workforce. The rate per fortnight is shown below. No. of children Fortnightly pay 1

$337.00

2

$704.34

3

$1099.26

Calculate the yearly payment for: a One child b Two children

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c

Three children

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21

Development 8

Tahlia receives $19.40 for delivering 200 brochures. She receives an additional $30 per day when delivering in wet weather. How much does she receive for delivering: a 600 brochures on a clear day? b 1000 brochures on a clear day? c 800 brochures on a wet day? d 1400 brochures on a wet day?

9

Mitchell works in a factory that makes key rings. Each key ring completed earns him $0.34. Mitchell also receives an additional $25 if he works on the weekend. How much does he earn for making: a 420 key rings on Friday? b 460 key rings on Wednesday? c 380 key rings on Saturday? d 230 key rings on Sunday?

10

A doctor charges each patient $33.50 for a consultation. He works for 6 hours a day and usually sees 5 patients per hour. a How much money does the doctor receive each day? b The doctor also has costs of $410 per day. What is the profit for the day?

11

Austudy is reduced by 50 cents for every dollar between $62 and $250 of fortnightly income. Tyler is 28 years of age, partnered and has one child. He is studying full-time but earning $126 per fortnight in a part-time job. What will be Tyler’s fortnightly payment from Austudy? Use the Austudy table on the previous page.

12

Anthony writes crime novels. He has just received his half-yearly statement of sales of his latest novel. He has been informed that 20 000 copies were printed and there are 8760 left in stock. Anthony receives 15% of the retail price as royalties. a How many copies of his latest novel were sold? b What is Anthony’s royalty if the retail price of his latest novel is $24.95? c What is Anthony’s royalty if the retail price of $24.95 was discounted by 10%?

13

The maximum youth allowance is reduced by $1 for every $4 that the youth’s parents’ income is over $31 400. By how much is Charlotte’s youth allowance reduced if her parents earn a combined income of $34 728?

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Preliminary Mathematics General

1.6 Gross pay, net pay and deductions

1.6

Employers must pay the minimum rate of an award or enterprise agreement. The rate will depend on the type of work and the actual times worked. Gross pay is the total of an employee’s pay including allowances, overtime pay, commissions and bonuses. It is the amount of money before any deductions are made. The amount remaining after deductions have been subtracted is called the net pay or ‘take-home pay’. Deductions are a regular amount of money subtracted from a person’s wage or salary. People have many different deductions subtracted from their gross pay such as: • Income tax – a charge that funds the government’s operations • Superannuation – an investment for retirement. An employer must contribute 9% of the employee’s wages into a superannuation fund. • Health insurance – private insurance to cover medical and dental costs • Union fee – payment for union membership. Gross pay, net pay and deductions Net pay = Gross pay − Deductions

Example 12

Calculating the net pay

Laura is a nurse who receives a gross weekly wage of $2345. She has the following deductions taken from her pay: • Income tax – $861 • Health fund payments – $48.25 • Superannuation – $67.95. What is Laura’s net pay?

Solution 1 2 3 4 5

Write the quantity (net pay) to be found. Write the formula for net pay. Substitute the values for gross pay and deductions. Evaluate and write using correct units. Write the answer in words.

Net pay = Gross pay − deductions = 2345 − (861 + 48.25 + 67.95) = $1367.80

Laura’s net pay is $1367.80

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Chapter 1 — Earning and managing money

Example 13

23

Reading information from a pay slip

Oscar received the following pay slip. What amount is received this pay for: a gross pay? b net pay? c superannuation? d PAYG tax?

Ordinary time Annual holiday Total Gross Earnings

Hours 26.00 0.0

Rate Amount This Pay Year to Date $25.000 $650.00 $25.994 $ 0.0 $650.00 $1 300.00

PAYG Tax

$100.00

$ 200.00

Social Club HECS Repayments Superannuation Less Post-tax deductions

$ $ $ $

2.00 13.00 35.00 50.00

$ 4.00 $ 26.00 $ 70.00 $ 100.00

Net Pay

$450.00

$ 900.00

$450.00

$ 900.00

Direct Credit to account: 00000000 Total Payments Solution 1 2 3 4

Read the value for gross earnings. Read the value for net pay. Read the value for superannuation. Read the value for PAYG tax.

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a b c d

Gross pay is $650.00. Net pay is $450.00. Superannuation is $35.00. PAYG tax is $100.00.

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24

Preliminary Mathematics General

Exercise 1F 1

Calculate the weekly net pay for these people. a Isabella receives a gross pay of $1386 and has total deductions of $875. b Kim-Ly receives a gross pay of $985 and has total deductions of $265. c Christopher receives a gross pay of $715 and has total deductions of $222.

2

Calculate the weekly net earnings for these people. a Daniel receives a gross weekly wage of $1056 and has deductions of $294.75 for income tax, $28.80 for superannuation and $325.05 for loan repayments. b Hannah receives a gross weekly wage of $3042 and has deductions of $1068 for income tax, health fund payments for $50.85, superannuation for $53.55 and savings for $450. c Kapil receives a gross weekly wage of $2274. He has deductions of $768 for income tax, $28.95 for health insurance, $49.02 for superannuation, $15.30 for life insurance and $450 for loan repayments.

3

Jack’s annual gross pay is $48 750. The deductions are $9150 for income tax, $1462 for health insurance and $5280 for superannuation. a What are Jack’s total deductions? b What is Jack’s annual net pay?

4

Calculate the weekly gross pay for these people. a Aaron receives a net weekly pay of $1245 and has deductions of $374.15 for income tax, $45.60 for superannuation and $25.20 for union membership. b Hannah receives a net weekly pay of $2645 and has deductions of $1068 for income tax, $53.95 for health fund payments and $83.75 for superannuation. c Ivan receives a net weekly pay of $2511. He has deductions of $913 for income tax, $31.95 for health insurance, $59.46 for superannuation, $18.20 for life insurance and $470 for loan repayments.

5

Harry’s net pay is $57 908. The deductions are $12 580 for income tax, $2087 for health insurance and $6910 for superannuation. What is Harry’s gross pay?

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25

6

Joshua is on a working holiday. He picked pieces of fruit on a farm as follows: Monday – 170 Tuesday – 130 Wednesday – 145 Thursday – 210 Friday – 190. a What is his gross salary at $0.55 per piece of fruit? b What is his net salary if he has total deductions of $121?

7

Charlotte owns an investment property that is rented out for $320 per week. She pays the real estate agent a fee of 3% for managing the property. a How much does she pay the real estate agent each week? b How much does Charlotte receive each week from the investment property? c What is the net income received by Charlotte from this property over the year?

8

Nicholas receives a yearly gross salary of $74 568. He pays 18% of his weekly gross salary in income tax. He contributes 9% of his weekly gross salary to his superannuation fund and has $155 in miscellaneous deductions each week. a What is his gross weekly pay? b How much income tax is deducted each week? c How much superannuation is he contributing each week? d What is the total amount of deductions made each week? e What is his net weekly pay this week?

9

Lakshmi receives a fortnightly pay of $2240. She pays 15% of her weekly gross salary in income tax. She contributes 9% of her weekly gross salary to her superannuation fund and has $95 in miscellaneous deductions each week. a What is her gross weekly pay? b How much income tax is deducted each week? c How much superannuation is she contributing each week? d What is the total amount of deductions made each week? e What is her net weekly pay this week?

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26

Preliminary Mathematics General

Development

13.1

10

Charlie is a building worker who receives $48.50 per hour for a 38-hour week. In addition he receives an allowance of $3.50 per hour for work on a multistorey development. Charlie is currently working on six-storey apartment block. Each week he has deducted from his pay a superannuation contribution of 9% of his gross pay and union fees of $28.45. Because he only started working late in the financial year, he doesn’t yet have to pay tax. a What is his gross weekly pay this week? b How much superannuation is he contributing each week? c What is his net weekly pay this week?

11

Emily receives a gross fortnightly salary of $2703 and has deductions of $891.75 for income tax, $54.30 for health fund payments, $753 for car loan payments and $14.55 for union subscription. a What is Emily’s net income each fortnight? b What percentage of her gross income is deducted for income tax? (Answer correct to one decimal place.)

12

Liam received a gross fortnightly salary of $3795. His pay deductions were $937.20, for income tax, $215.25 for superannuation, $21.45 for union fees and $201 for a home loan repayment. a What is his net income each fortnight? b What was his weekly net pay? c What percentage of his gross income was deducted for income tax? (Answer correct to one decimal place.) d If Liam’s loan repayment increased by 10%, what was his new fortnightly net pay?

13

Jane normally works 37 hours a week at $54 per hour. In one particular week she worked 42 hours and received overtime at the rate of time-and-a-half. Her deductions for the week were income tax $602.20, medical fund $49.60, superannuation $74.40 and motor vehicle repayment $417.40. a What was Jane’s gross weekly wage? b What was her net income for the week? c What percentage of her gross income is spent on a motor vehicle repayment? Answer correct to the nearest per cent.

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Chapter 1 — Earning and managing money

27

1.7 Budgeting Budgeting involves balancing of income and the expenses. It is planning how to manage your income. Budgets are created for a specified time such as weekly, monthly or yearly. Creating a budget

1.7 1 2 3 4

List all the income categories. List all the expense categories. Calculate the total of the income and expenses categories. Balance the budget by modifying the categories or by entering a balance category.

Example 14

Balancing a budget

Balance the following weekly budget. Income

Expenses

Salary

$1726.15

Clothing

$ 73.08

Bonus

$

Gifts and Christmas

$114.80

Investment

$ 156.78

Groceries

$467.31

Part-time work

$ 393.72

Insurance

$171.34

Loan repayments

$847.55

Motor vehicle costs

$105.96

Phone

$ 38.26

Power and heating

$ 51.82

Rates

$ 54.82

Recreation

$216.79

Work-related costs

$ 68.76

20.00

Balance Total

Total Solution 1

Add all the income.

2

Add the all the expenses excluding the ‘balance’. Subtract the total expenses from the total income. Write the result of step 3 as the balance.

3 4

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Income = 1726.15 + … + 393.72 = $2296.65 Expenses = 73.08 + … + 68.92 = $2210.49 Balance = Income − Expenses = 2296.6 − 2210.49 = $86.16 Cambridge University Press

28

Preliminary Mathematics General

Example 15

Creating a budget

Maya and Logan have a weekly net wage of $954. Their monthly expenses are home loan repayment $1032, car loan repayment $600, electricity $102, phone $66 and car maintenance $120. Their other expenses include insurance $2160 annually, rates $1800 annually, food $180 weekly, petrol $48 fortnightly and train fares $36 weekly. Maya and Logan allow $72 for miscellaneous items weekly and need to save $84 per week for a holiday next year. a Prepare a monthly budget for Maya and Logan. Assume there are four weeks in a month. b What is the balance? c How can Maya and Logan ensure they have their holiday next year? Solution a

Income Wage

Expenses $3816

Home loan repayment Car loan repayment

$600

Electricity

$102 $66

Phone Car maintenance

$120

Insurance

$180

Rates

$150

Food

$720

Petrol

$96

Train fares

$144

Miscellaneous

$288

Holiday

$336

Balance

−$18

$3816 1 2 3 4

5

$1032

$3816

List all the monthly income categories. List all the monthly expenses categories. Calculate the total income and expenses categories. Subtract the total expenses from the total income to calculate the balance.

a

Solution is shown above. Total income = $3816 Total expenses = $3834

b

The balance is −$18. A negative balance indicates an increase in their income or reduction in their expenses.

c

Balance = $3816 − $3834 = −$18 Maya and Logan need to increase income or reduce expenses by $18.

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Chapter 1 — Earning and managing money

Exercise 1G 1

Oscar and Jill are living in a unit. Part of their budget is shown below. Calculate the total amount paid over one year for: a

Electricity

b

Insurance

c

Food

d

Rent.

Item

When

Cost

Electricity

Quarterly

$ 384

Food

Weekly

$ 360

Insurance

Biannually $ 1275

Rent

Monthly

$ 1950

2

Sarah earns $67 365 annually. She has budgeted 20% of her salary for rent. How much should she expect to pay to rent an apartment for one year?

3

Adam has constructed a yearly budget as shown below. Income

Expenses

Wage

$60 786.22 Clothing

$ 4 634.42

Interest

$

$ 1 543.56

674.15 Council rates Electricity

$ 1 956.87

Entertainment

$ 4 987.80

Food

$17 543.90

Gifts and Christmas $ 5 861.20 Insurance

$ 2 348.12

Loan repayments

$16 789.34

Motor vehicle costs

$ 2 458.91

Telephone

$

832.98

Work-related costs

$

812.67

Balance Total a b c 4

Total

Calculate the total income. Calculate the total expenses. Balance the budget.

Dimitri had a total weekly income of $104 made up of a part-time job earning $74 and an allowance of $30. He decided to budget his expenses in the following way: sport – $24, movies – $22, school – $16 and food – $20. a Prepare a weekly budget showing income and expenses. b What is the balance?

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30

Preliminary Mathematics General

Development 5

Create the spreadsheet below.

1G

a b c d e

6

14.1

The formula for cell E5 is ‘=C5/$C$7’. It is the formula for relative percentage. Fill down the contents of E5 to E7 using this formula. Enter formulas in E9:E17 to calculate the relative percentages for expenses. Edit the amount spent per month on eating out from $200 to $240. Observe the changes. Edit the amount of savings per month from $300 to $360. Observe the changes. Edit the amount of car expenses per month from $100 to $150. Observe the changes.

Ava has a gross fortnightly pay of $1896. a Ava has a mortgage with an annual repayment of $13 676. Calculate the amount that Ava must budget each fortnight for her mortgage. b Ava has budgeted $180 per week for groceries, $60 per week for entertainment, $468 per year for medical expenses and $80 per week to run a car. Express these as fortnightly amounts and calculate their total. c Ava has an electricity bill of $130 per quarter, telephone bill of $91 per quarter and council rates of $1118 per annum. Express these amounts annually and convert to fortnightly amounts. What is the total of these fortnightly amounts? d Prepare a fortnightly budget showing income and expenses.

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Chapter 1 — Earning and managing money

Salary and wages

• •

Overtime and special allowances

• • • • •

Annual leave loading and bonuses

• • •

Commission

• •

Piecework, royalties and government

• • • •

Gross pay, net pay and deductions

• • •

Budgeting

•

Study guide 1

Salary – payment for a year’s work, which is then divided into equal monthly, fortnightly or weekly payments Wage – payment for work that is calculated on an hourly basis Overtime – work beyond the normal working day Casual rate – set amount paid for each hour’s work Time-and-a-half rate = normal rate × 1.5 Double time rate = normal rate × 2 Allowance – payment for difficult or dangerous conditions Annual leave loading – payment for going on holidays Holiday loading = 17 12 % × normal weekly pay × weeks leave Bonus – extra payment or gift earned as a reward Commission – percentage of the value of the goods sold Retainer – small payment in addition to the commission Piecework – payment for work completed Piecework = Number of units of work × Amount paid per unit Royalty – percentage of the goods sold or profit received Government income – pension, allowance or benefit Gross pay – total of the employee’s pay including allowances, overtime pay, commissions and bonuses Deductions – regular amount of money subtracted from a person’s wage or salary such as income tax Net pay = Gross pay – Deductions Budgeting – balancing of income and expenses. Budgets are created for a specified time such as weekly.

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Review

Chapter summary – Earning and managing money

31

Review

32

Preliminary Mathematics General

Chapter HSC Sample summary – Objective-response – Earning Moneyquestions 1

Alyssa receives a salary of $85 640. How much does she receive each fortnight? A $3293.84 B $3293.85 C $1646.92 D $1646.93

2

Christopher receives a normal hourly rate of $22.60 per hour. What is his pay when he works 8 hours at a normal rate and 3 hours at time-and-a-half? A $180.80 B $248.60 C $282.50 D $316.40

3

Rana works a 38-hour week at a rate of $26.00 per hour. She receives 17 1 % of 4 weeks 2 normal pay as holiday loading. What is Rana’s holiday loading? A $172.90 B $691.60 C $3952.00 D $4643.60

4

Taylah earns a weekly retainer of $425 plus a commission of 8% on sales. What are her weekly earnings when she made sales of $8620? A $34.00 B $459.00 C $689.60 D $1114.60

5

Ahmet is a carpet layer and charges $37.50 per square metre of carpet laid. How much will he earn for laying carpet in a room whose area is 9 square metres? A $37.50 B $46.50 C $337.50 D $675.00

6

Isabelle earns a royalty of 18% on net sales from writing her autobiography. There were $24 520 net sales in the last year. What is Isabelle’s royalty payment? A $4413.60 B $20 106.40 C $24 520.00 D $28 933.60

7

Angus’s net pay is $68 806. The deductions are $20 630 for income tax, $1051 for health insurance and $5487 for superannuation. What is his gross pay? A $27 168 B $41 638 C $47 125 D $95 974

8

Adam has the following bills: electricity $250 per quarter, phone $70 per month, rates $1200 per year and rent $300 per week. What is the total amount Adam should budget for the year? A $358 B $1553 C $1820 D $18 640

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Chapter 1 — Earning and managing money

33

1

Jake earns $96 470.40 per annum and works an average of 48 hours per week. a What is his average weekly wage? b Calculate Jake’s hourly rate of pay.

2

Alex works for a fast-food company and is paid $13.50 per hour for a 35-hour week. He gets time-and-a-half pay for overtime worked on the weekdays and double time for the weekends. Last week he worked a normal 35-hour week plus three hours of overtime during the week and four hours of overtime on the weekend. What was his wage last week?

3

Carlo’s employer has decided to reward all employees with a bonus. The bonus awarded is 6 14 % of their annual salary. What is Carlo’s bonus if his annual salary is $85 940?

4

The public service provides all employees with a 17 12 % holiday loading on four weeks normal wages. Lucy works a 37-hour week for the public service in Canberra. She is paid a normal hourly rate of $32.40. a How much will Lucy receive in holiday loading? b Calculate the total amount of pay that Lucy will receive for her holidays.

5

Chelsea is a real estate agent and charges the following commission for selling the property: 3% on the first $45 000, then 2% for the next $90 000 and 1 12 % thereafter. a What is Chelsea’s commission if she sold a property for $240 000? b How much would the owner of the property receive from the sale?

6

Patrick is a comedian who makes $120 for a short performance and $260 for a long performance. How much will he earn if he completes 11 short and 12 long performances? Challenge questions 1

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Review

Chapter HSC Sample summary – Short-answer – Earning Money questions

Review

34

Preliminary Mathematics General

7

Bailey is paid a royalty of 11.3% on the net sales of his book. The net sales of his book in the last financial year was $278 420. a What is Bailey’s royalty payment in the last financial year? b Net sales this financial year are expected to decrease by 15%. What is the expected royalty payment for this financial year?

8

The maximum youth allowance is reduced by $1 for every $4 that the youth’s parents’ income exceeds $31 400. By how much is Hannah’s youth allowance reduced if her parents earn a combined income of $35 624?

9

William works as a builder. His annual union fees are $278.20. William has his union fees deducted from his weekly pay. What is the size of William’s weekly union deduction?

10

Quan received a gross fortnightly salary of $2968. His pay deductions were $765.60 for income tax, $345.15 for superannuation and $23.40 for union fees. a What was his fortnightly net pay? b What percentage of his gross income was deducted for income tax? (Answer correct to one decimal place.)

11

Joel is a carpet layer and charges $16 per square metre to lay carpet. How much will he earn for laying carpet in a house whose area is 32 square metres?

12

Daniel has a gross monthly wage of $3640. He has the following deductions taken from his pay: $764 for income tax, $71.65 for superannuation and $23.23 for union membership. What is Daniel’s net pay?

13

Hannah has budgeted $210 per week for groceries, $70 per week for leisure, $23 per fortnight for medical expenses and $90 per week to run a car. Calculate the monthly expenses. Assume 4 weeks in a month.

14

Amelie earns $90 345 annually. She has budgeted 30% of her salary for a loan repayment. How much should she expect to pay for a loan repayment for one year? Challenge questions 1

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C H A P T E R

2

Algebraic manipulation Syllabus topic — AM1 Algebraic manipulation Add and subtract like terms Multiply and divide algebraic terms Expand and factorise algebraic expressions Evaluate the subject of the formula through substitution Solve linear equations involving up to 3 steps Solve equations following substitution

2.1 Adding and subtracting like terms A pronumeral (letter) represents a number. It may stand for an unknown value or series of values that change. For example, in the equation x + 5 = 8, x is a pronumeral that represents a value. Its value can be determined because we know 3 + 5 = 8, so x = 3. 2.1 When a term has a pronumeral and a number, the number is written before the pronumeral and is called the coefficient. For example, the term 3xy has a coefficient of 3 and its pronumerals are written after the coefficient in alphabetical order.

Like terms Terms that have exactly the same pronumerals such as 2a and 5a are called like terms. Only like terms can be added and subtracted. It involves adding and subtracting the coefficients. Adding and subtracting like terms simplifies the algebraic expression. It is often called collecting the like terms.

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36

Preliminary Mathematics General

Adding and subtracting like terms 1 2 3

Find the like terms or the terms that have exactly the same pronumerals. Only like terms can be added or subtracted; unlike terms cannot. Add or subtract the coefficients or the numbers before the pronumeral of the like terms.

Example 1

Adding and subtracting like terms

Simplify 2ab + 3 + 5ab − 7. Solution 1 2

Rewrite the expression by grouping the like terms. Add and subtract the coefficients.

Example 2

2ab + 3 + 5ab − 7 = (2ab + 5ab) + 3 − 7 = 7ab − 4

Adding and subtracting like terms

Simplify 4y + 6y2 − 3y − 5y2. Solution 1 2

Rewrite the expression by grouping the like terms. Add and subtract the coefficients.

4y + 6y2 − 3y − 5y2 = 6y2 − 5y2 + 4y − 3y = y2 + y

Adding and subtracting algebraic fractions To add and subtract algebraic fractions rewrite each fraction as an equivalent fraction with a common denominator, then add or subtract the numerators. A common denominator can always be found by multiplying the denominators of both fractions together. Example 3

Simplify

Adding and subtracting algebraic fractions

x x + . 6 4

Solution 1 2 3 4

Find a common denominator for 6 and 4. Both 6 and 4 divide into 12. Alternatively, multiply 6 by 4 and use 24. Multiply the first fraction by 2 (6 × 2 = 12) and the second fraction by 3 (4 × 3 = 12). Write the equivalent fractions. Add the numerators of the equivalent fractions.

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x x x×2 x×3 + = + 6 4 6×2 4×3 2 x 3x = + 12 12 5x = 12

Cambridge University Press

Chapter 2 — Algebraic manipulation

Exercise 2A 1

2

3

4

5

Choose the like terms out of each of the following. a 4r, 6p, r, 7 b 5x, 3xy, 2x c 2 d xy, 4xy, xy , 3yx e 3, 2m, mn, 9m f

-a, 5a, -8b, 7 c, cd, cde, dc, ce

Simplify by collecting like terms. a

4y + 3y

b

3 p + 1177 p

c

7h − 6 h

d

3xx − x

e

d + 4d

f

6 y − 1122 y

g

− d + ( −4 d )

h

33tt + ( −11t )

i

11 f − ( −5 f )

j

4 hg + 66gh

k

5ab + 22ba

l

xyz + 3 xyz

Simplify by collecting like terms. a

5c + 4 + 2 c

b

4 f +4 f −7

c

8 + 5r 5r + 12 r

d

6 x + 4 y − 3x

e

3b + 7a − 2 a

f

h + 2d − 6 h

g

4 de + ed − 22de

h

7a + b + 2a 2 a − 2b

i

xy + 2 yx + 3 xy

j

6ba − 2b 2b + ( − aabb )

k

7a + ( −b ) + 22aa − 2b

l

5 g + h + (− g ) + 8h

Simplify by collecting like terms. a

8 x 2 − 3x − x 2 + 4

b

4 a + a 2 − 33aa 2 + a

c

7t + 8t 2 − 6t − 7t 2

d

3m 2 + 8m − 4 m − m 2

e

e 2 + 2e + e 2 − e

f

d + d 2 − 5d + d 2

g

2w + w 2 + 5 + w

h

6 − v2 + v − 4

i

8r − 7 − 7r − 33r 2

Add or subtract the algebraic fractions. a

a a + 3 3

b

3x 2 x − 5 5

c

2 m 3m + 4 4

d

3x x − 7 7

e

d 2d + 11 11

f

6y 2y − 15 15

g

4 s 9s + 3 3

h

9f 4f − 8 8

i

y y + 2 4

j

7e e − 6 3

k

g 4g + 2 6

l

r 4r − 2 10

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38

Preliminary Mathematics General

Development 6

7

Which of the following is equivalent to m + m + m + n + n ? a

2 m + m + 22n

b

m3 + n2

c

5m − 4 m + 3n − n

d

7m − 4 m + 2 n − n

e

3n − n + m − −22m

f

−2 m + 5m − 2 n + 3n

Copy and add like terms where possible to complete the table. +

x

x+y

3x

3x 7y x-y 2y 8

Matteo has $y for shopping. He spent $x for a pair of jeans, $3x for a shirt and $2x for a belt. Write an expression in simplified form for how many dollars he has left.

9

The perimeter of a plane shape is the distance around the boundary of the shape. The plane shape opposite is a rectangle with a length l and a breadth b. Write an expression for the perimeter of this rectangle by collecting like terms.

10

The isosceles triangle opposite has three sides whose lengths are 3x + y, 3x + y and x + 2y. Write an expression in simplified form for the perimeter of this triangle by collecting like terms.

b l

3x + y

x + 2y 3x + y

11

Add or subtract the algebraic fractions. a d g j

w w + 4 3 z z − 3 5 u 4u + 10 15 3a a a − + 5 4 2

b e h k

a a − 4 5 3h h + 8 6 3e e − 4 10 7x x x − + 10 6 3

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c f i l

x 2x + 7 3 5r r − 12 8 w w w + + 2 4 6 d d d+ − 2 10

Cambridge University Press

Chapter 2 — Algebraic manipulation

39

2.2 Multiplication and division of algebraic terms

2.2

Algebraic terms are multiplied and divided to form a single algebraic expression. Terms usually contain a coefficient before a pronumeral. The multiplication sign between the coefficient and the pronumeral is omitted. For example, the algebraic term 4x can be written in expanded form as 4 × x. After an algebraic expression is written in expanded form, the coefficients can be multiplied or divided and the pronumerals can be multiplied or divided. Index notation should be used to write expressions in a shorter way such as a × a = a2. If the algebraic terms contain fractions it is easier to cancel any common factors in the numerator and denominator. This makes the calculations easier. Multiplication and division of algebraic terms 1 2 3 4 5 6

Write in expanded form. If the algebraic term is a fraction cancel any common factors. Multiply and divide the coefficients. Multiply and divide the pronumerals. Write the coefficient before the pronumerals. Write the pronumerals in alphabetical order and express in index notation.

Example 4

Multiplying algebraic terms

Simplify the following. a 2cd × (−3de )

b

x2 × 3x × 4 x

Solution 1 2 3 4 1 2 3 4 5

Write in expanded form. Multiply the coefficients (2 × −3 = −6). Write the pronumerals in alphabetical order. Express answer using index notation (d × d = d 2). Write in expanded form. Multiply the coefficients (1 × 3 × 4 = 12). Write the coefficient before the pronumerals. Write the pronumerals using index notation. Express the answer using index notation (x2 × x × x = x4).

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a

2 cd × (−3de ) = 2 × c × d × −3 × d × e = −6 × c × d × d × e = −6 cd 2 e

b

x 2 × 3 x × 4 x = 12 × x 2 × x × x = 12 × x 2 × x1 × x1 = 12 × x 4 = 12 x 4

Cambridge University Press

40

Preliminary Mathematics General

Dividing algebraic terms

Example 5

Simplify 18a 2b ÷ 6 a. Solution 1 2 3 4

5 6

Write in fraction form. Write in expanded form. Divide the coefficients (18 ÷ 6 = 3). Cancel the pronumeral a in both the numerator and denominator (common factor of a, aa = a ÷ a = 1). Write the coefficient before the pronumerals. Write the pronumerals in alphabetical order. Dividing algebraic terms with fractions

Example 6

Simplify

18a 2b 6a 18 × a × a × b = 6×a 3× a × a1 × b = a1 = 3× a ×b = 3ab

18a 2b ÷ 6a 6a =

x 8 xy × . 12 y 20

Solution 1 2

3

4

5 6 7 8

Write fractions in expanded form. Determine any common factors in the numerator and the denominator. Cancel out the common factors (4 is a common factor of 8 and 12, 2 is a common factor of 2 and 4). Cancel the pronumeral y in both the numerator and denominator as it is a common factor. Multiply the numerators together. Multiply the denominators together. Express the answer using index notation (x × x = x2). Write the coefficient before the pronumerals. However, it is acceptable to leave the answer as x .

x 8 xy x 8× x× y × = × 12 y 20 12 × y 20 4 1× 2 1 × x × y 1 x = × 41 × 3 × y1 42 × 5 x× x = 3× 2 × 5 x2  1 2 = or x  30  30 

2

30

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Cambridge University Press

Chapter 2 — Algebraic manipulation

Exercise 2B 1

Multiply these algebraic terms. a 6g × 4 b 7 × 5m 20 x × 5 x d a × 7a e

f

2 × 4d 4 ×7 5s × s × 6

i

−33 f × 1155 f

Multiply these algebraic terms. a mn × 4mn b 15 yz × 3 y

c

5 pq × (−2 pr )

g

2

3e × 7e × 2

h

4 w × (−4 (−4w )

d

3x 2 × 2 x

e

2 r 3 × 4r 2

f

24 q2 × 2 q2

g

3st 3 × 5 st

h

−de 2 × ( −5d 2 e )

i

z2 × 4z × 5z

c

6 mn 18

3

What is the product of 4uw2 and uw?

4

Simplify these algebraic terms. 12 y 16 a a b 6 4

5

6

c

d

14b 2b

e

12 m 3 12 m

f

2 xyz 26 x

g

8 x2 64 x

h

5r 2 15rs

i

10 pq 2q

c

27 h2 ÷ ( −3)

Write in fraction form and simplify. a 20 z ÷ 4 b 22 w 2 ÷ 2 d

33 y ÷ 11 y

e

12 ab ÷ 4 a

f

25kj 2 ÷ 5kj

g

4 x 3 ÷ 2288 x 2

h

2 s ÷ 8s 2

i

7m ÷ 2211m 3

2

Express 16 xy in simplest form. 12 x

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41

42

Preliminary Mathematics General

Development 7

Multiply the algebraic terms in the first column by each algebraic term given to complete the table. ×

p

2p2

pq

5p 6pq 2q2

8

9

Simplify the following. c

5w × 2w 9

2 3a × a 6

f

5w 2 w × 7 9

m 6 mn × 9n 15

i

d c × e 3d

a

8y × 4y 3

b

6d ×

d

5 x 12 × 4 x

e

g

10a 3b × b 4

h

2d 5

Divide the algebraic terms in the first column by each algebraic term given to complete the table. ÷

2r

r

4rt

8r 12rt 4r2 10

11

12

5x 2

The plane shape opposite is a rectangle with a length of 52x and a breadth of 3x2. Write an expression in simplest form for the area of this rectangle.

3x2

3 2 Express 8 x y in simplest form. 16 x 2 y 3

Simplify the following. a

15h2 3k 3 × 3k 4

b

21v 2 5u 3 × 15uv 7

c

18a 2 6b 3 a × × 12b 9a 2

d

9m 3 × 6 mn 2 3m 2 n

e

3( m + 1) 4m3 × 8m 2( m + 1)

f

7 y4 5( y − 2 ) × 10( y − 2 ) 21 y 6

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Chapter 2 — Algebraic manipulation

43

2.3 Expanding algebraic expressions Grouping symbols in algebraic expressions indicate the order of operations. The two most commonly used grouping symbols are parentheses ( ) and brackets [ ]. They are removed by using the distributive law or a × (b + c) = ab + ac. This is illustrated below. 2.3

Using order of operations 2 × ( 3 + 1) = 2 × 4 =8

Using distributive law 2 × ( 3 + 1) = 2 × 3 + 2 × 1 =6+2 =8

To expand an algebraic expression using the distributive law, multiply the number or terms inside the grouping symbols by the number or term outside the grouping symbols. The resulting algebraic expression is simplified by collecting the like terms. Make sure you remember to multiply all the terms inside the grouping symbol by the number or term outside the grouping symbols. Expanding algebraic expressions 1

2

Multiply the number or term outside the grouping symbol by the a first term inside the grouping symbol. b second term inside the grouping symbol. Simplify and collect like terms if required. a(b + c ) = a × b + a × c a(b − c ) = a × b − a × c = ab + ac = ab − ac

Example 7

Expanding algebraic expressions

Expand 5( 2 y − 3). Solution 1 2 3

Multiply the first term inside the parenthesis (2y) by the number outside the parenthesis (5). Multiply the second term inside the parenthesis (-3) by the number outside the parenthesis (5). Write in simplest form.

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5( 2 y − 3) = 5 × 2 y − 5 × 3 = 10 y − 15

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Preliminary Mathematics General

Example 8

Expanding algebraic expressions

Expand -(m - 5). Solution 1 2 3

Multiply the first term inside the parenthesis (m) by the number outside the parenthesis (-1). Multiply the second term inside the parenthesis (-5) by the number outside the parenthesis (-1). Write in simplest form.

Example 9

−( m − 5) = −1 × ( m − 5) = −1 × m − 1 × −5 = −m + 5

Expanding and simplify algebraic expressions

Remove the grouping symbols for 2(3x + 4) + 3(x - 1) and simplify if possible. Solution 1 2 3 4

Multiply the first term inside the parenthesis (3x) by the number outside the parenthesis (2). Multiply the second term inside the parenthesis (+4) by the number outside the parenthesis (2). Repeat the first two steps for the second parenthesis. Simplify by collecting the like terms.

Example 10

2( 3 x + 4 ) + 3( 3( x − 1) = 2 × 3 x + 2 × 4 + 3 × x + 3 × −1 = 6 x + 8 + 3x − 3 = 9x + 5

Expanding and simplifying algebraic expressions

Expand a(3a + 2) - a(a - 1) and simplify if possible. Solution 1 2

3 4

Multiply the first term inside the parenthesis (3a) by the term outside the parenthesis (a). Multiply the second term inside the parenthesis (+2) by the term outside the parenthesis (a). Repeat the first two steps for the second parenthesis. Simplify by collecting the like terms.

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a( 3a + 2 ) − a( a − 1) = a × ( 3a + 2 ) − a × ( a − 1) = a × 3a + a × 2 − a × a − a × −1 = 3a 2 + 2 a − a 2 + a = 2 a 2 + 3a

Cambridge University Press

Chapter 2 — Algebraic manipulation

Exercise 2C 1

Ryan was required to remove the grouping symbols. This was his solution. 3( 3 x − 2 ) = 9 x − 2 Where is the error in Ryan’s working?

2

3

4

Expand each of the following. a

3( a + 2 )

b

2( d + 1)

c

d

2( 3 x + 4 )

e

2(5 x − 7)

f

7(b − 2 ) 4( 9b + 1)

g

4(5 + 2t )

h

6(1 − 2 w )

i

5( 3 + 9d )

j

8(5e − 2 d )

k

5( 4 a + 9b )

l

7( 2 h + 8 g )

Expand each of the following. −3( y + 5) a −4( x + 3) b

c

− (b + 8)

d

−7( k − 2 )

e

−6( w − 1)

f

−2( x − 13)

g

−2( 4 + 2 q)

h

−5( 3 − 4 r )

i

−7(8 − 2 s )

c

n( n + 10)

f

d (6 d − 2 )

i

c(d + 4e )

Expand each of the following. b

d

y( y + 1) x( 2 x − 3)

e

v(v + 4) e( 3e + 5)

g

z (7e + 33ff )

h

a( 2b − 3c )

a

5

6

Remove the grouping symbols and simplify if possible. a

2( g + 1) + 4 g

b

7( s + 2 ) + s

c

3( y − 9) − 2 y

d

5 x − 4( 4( x − 2 )

e

6 z + 2( 2( z − 1)

f

3q − 7( 7( q − 5)

Remove the grouping symbols and simplify if possible. 4( x − 1) + 2 x + 5 b 7( 3 y − 2 ) + 4 y − 2 a c

2(5b + 2 ) − b − 8

d

4 r + 17 17 + 5( 5( r − 3)

e

2 n − 8 + 3( n + 2 )

f

5q + 2 − ( q + 9)

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46

Preliminary Mathematics General

Development 7

8

Simplify the following. a

2( x + 1) + 5( x − 1)

b

3( y + 2 ) + 2( y + 1)

c

5( a + 2 ) + 3( a + 4 )

d

8( c − 3) + 5( c + 3)

e

6( s + 6 ) + 2( 2 s − 1)

f

5( h + 7) + 2( 2 h − 7)

g

4( 3 x − 1) − 2( 2( x − 2 )

h

9( z + 5) − 7( z − 2 )

i

5( 2 c − 4 ) − 3( 3( c + 7)

j

5(5 g − 1) − 4( 4( g − 2 )

k

7( 2 u − 3) − ( u − 3)

l

−( 4 d − 1) − 3( d − 3)

Taylah was required to remove the grouping symbols and simplify. This was her solution. 6( 3 x − 2 ) − 22(( x + 3) = 1188 x − 12 12 − 2 x + 6 = 16x 16 x − 6 Where is the error in Taylah’s working?

9

Remove the grouping symbols and simplify if possible. a

x( x − 5) + x( x + 2 )

b

b(b + 3) + b(b + 1)

c

y( y − 3) + y( y + 8)

d

g ( 2 g + 3) − g ( g + 3)

e

v ( v − 7) + v (6 v + 4 )

f

b(5b − 1) − b( 2 + 4b )

g

2 u( u − 2 ) + u( u + 9)

h

4 n( n − 6 ) − n( n + 1)

i

3d ( d + 7) + d ( 2 d + 5)

j

e ( e + 2 ) − 7e ( e − 9)

k

6 k ( k − 3) + k ( k + 3)

l

t (5 − 3t ) + 7t ( 2 − t )

10

Expand and simplify the algebraic expression 2 ab( ab − 3) − ab( ab − 1). )

11

Simplify the following.

12

a

x 2 ( 2 x + 3) − 2( x + 1)

b

a 2 ( a + 2 ) − 4( 4( a + 3)

c

y 2 (5 y + 2 ) − 3( y + 7)

d

e

f

g

z ( 3 z − 1) + z 2 ( z − 5) x ( x 2 + 7) − x ( x 2 + 2 )

b(b + 7) − b 2 ( 3b + 2 ) e (7 − e ) − e 2 ( 2 e + 6 )

i

v( 2 − v 2 ) − v(1 − v 2 )

j

a( 2 a 2 − 1) + a( a 2 + 4 ) a 2 ( a + b ) − bb(( a + b )

k

x 2 ( x 2 + y ) − x( x( x + 3 y )

l

y 2 ( y + 4 z ) − y ( z 2 + 1)

h

Expand and simplify ( n + 4 r )n )n2 − (7n + r )n2.

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Chapter 2 — Algebraic manipulation

47

2.4 Factorising algebraic expressions

2.4

Factorising is the reverse process to expanding. For example, expanding the expression 5(2y − 3) produces 10y − 15, whereas factorising the expression 10y − 15 produces 5(2y − 3). The first step in factorising an expression is to find the largest factor of both terms or the highest common factor (HCF). In this case the HCF of 10y and 15 is 5. The HCF is written outside the grouping symbol and terms inside are found by dividing the HCF into each term. Factorising algebraic expressions 1 2 3 4

Find the largest factor of each term or the HCF. Write the HCF outside the grouping symbol. Divide the HCF into each term to find the terms inside the grouping symbols. Check the factorisation by expanding the expression.

Example 11

Factorising algebraic expressions

Factorise 3p − 6. Solution 1 2 3 4

Find the largest factor of each term (HCF is 3). Write the HCF or 3 outside the grouping symbol. Divide the HCF or 3 into each term to find the terms inside the grouping symbols. Check by expanding the expression.

Example 12

3p − 6 = 3 × p − 3 × 2 = 3 × ( p − 2) = 3( p − 2 )

Factorising algebraic expressions

Factorise 2x2 + 6x. Solution 1 2 3 4

Find the largest factor of each term (HCF is 2x). Write the HCF or 2x outside the grouping symbol. Divide the HCF or 2x into each term to find the terms inside the grouping symbols. Check by expanding the expression.

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2 x 2 + 6 x = 2 x × x + 22x × 3 = 2 x × ( x + 3) = 2 x( x + 3)

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48

Preliminary Mathematics General

Exercise 2D 1

2

Find the HCF to complete the factorisations. a 3m + 9 = ( m + 3) b 4 x − 16 16 =

( x − 4)

c

10 y + 20 =

( y + 2)

d

5 x − 30 30 =

( x − 6)

e

14 a + 56 =

( 2 a + 8)

f

6 w − 22 22 =

( 3w − 11)

g

8e + 12 12 =

h

9n − 12 12 =

( 3n − 4 )

Complete the following factorisations. 2y +8 = 2 a +

b

−

d

c

3

( 2 e + 3)

( 7x − 7 = 7(

e

14 v + 70 = 7

g

6d − 9 = 3

(

(

) )

+ −

)

)

( 8 h − 24 24 = 8 ( 5a + 20 20 = 5

f

15s − 45 = 5

h

10 + 5 x = 5

(

+ −

(

) )

− +

)

)

Jakob was required to factorise an algebraic expression. This was his solution. 2 x − 6 = 2( x − 6 ) Where is the error in Jakob’s working?

4

5

Factorise each of the following. a 5a + 220 b 3 x + 118 d 7 z + 221 e 32 + 4 d g 4 n + 110 h 16 c + 36 j 24 − 16 x k 28 g − 8

c f i l

8 p − 556 27 − 9t 15 f + 20 9w − 221

Anh’s working for a factorisation question is shown below. 6 y + 18 18 = 6( 6( y + 3) How can she check that the factorisation is correct?

6

Factorise each of the following by taking out a negative factor. −3q − 15 a −2 a − 12 b c −5m − 20 −16 − 4 y d −9 s + 90 e f −32 − 8h

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Chapter 2 — Algebraic manipulation

Development 7

8

9

Complete the following factorisations. a

x 2 + 4 x = x(

c

−2 e − 3e 3 = − e(

e

10r 2 − 14 r = 2 r (

g

q4 + 5q2 = q2 (

i

6 n3 + 8n = 2 n(

b

6 y − y 2 = y(

d

− 6 k − 7k 4 = − k(

f

9 x 2 + 3x = 3x(

)

h

− w 2 − 2w 3 = − w 2 (

)

j

6t 5 + 9t = 3t (

) ) )

) ) ) ) )

Fully factorise the following expressions. a

y2 − 5 y

b

10m 2 + 5m

c

4 x 2 − 1166 x

d

3v + v 2

e

8g + 4g3

f

4 d 2 + 1122 d 4

g

2 ab + 44a

h

5 xy 2 + 110 xy

i

6b 2 c − 1155b 2

j

deff − 2 e 2 f de

k

5 x 2 y + 33xy xy 2

l

r 3t 2 − r 2t 2

Stacey was required to completely factorise an algebraic expression. This was her solution. 12 x + 8 x 2 = 4( 4( 3 x + 2 x 2 ) Where is the error in Stacey’s working?

10

11

Factorise each of the following by taking out the highest common factor. 4 x + 12 12 y + 8 10 x − 5 y + 115 z a b c 2 a + 4b − 6 c d

9 y 2 + 6 y − 12

e

8 − 4r 4r + 6r 2

f

3 + 6h 6 h2 + 9 h

g

10v − 15v 2 + 25

h

8m − 4 mn + 12

i

9c + 15 15cd + 1188d

Factorise by using the HCF for each of the following expressions. a

2ab + ac + ag

b

4xx + xxyy + xxzz

c

7 g − gh + 1144 g 2

d

d 2 − 3d + de

e

b 2 + bc bc − 5b

f

k − 4 k 2 h + 2 kkh

g

xyz + 5 xy 2 − 2 x 2 y

h

2 abc + 55aa 2b 2 c 2 − 3ab

i

mn − 2 m 2 n + mn2

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Preliminary Mathematics General

2.5 Substitution Substitution involves replacing the pronumeral in an algebraic expression with one or more numbers. The resulting numerical expression is evaluated and expressed to the specified level of accuracy. 2.5 Substitution of values 1 2 3 4

Write the algebraic expression. Replace the variables in the expression with the numbers given in the question. Evaluate using the calculator. Write the answer to the specified level of accuracy and correct units if necessary.

Example 13

Substituting values

Evaluate 3a − 4b + c given a = 2, b = 5 and c = −10. Solution 1 2 3

Write the algebraic expression. Substitute the values for a, b and c into the algebraic expression. Evaluate.

Example 14

3a − 4b + c = 3 × 2 − 4 × 5 + −10 = 6 − 20 − 10 = −24

Substituting values

Evaluate the following, given a = 2. a

( 2 a + 5)

b

3a3

Solution 1 2 3 1 2 3

Write the algebraic expression. Substitute the value for a into the algebraic expression. Evaluate. Write the algebraic expression. Substitute the value for a into the algebraic expression. Evaluate.

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a

( 2 a + 5) = 2 × 2 + 5 = 9 =3

b

3a3 = 3 × 23 =3×8 = 24

Cambridge University Press

Chapter 2 — Algebraic manipulation

Exercise 2E 1

Evaluate these expressions given a = 3, b = 4 and c = 8. a

5aa + b

b

a + 4b

c

d

a2 + b2 abc 2

e

4bb × (−2 a) a

f

h

b2 ÷ c

i

a = 6 and b = -4 2 a = and b = 1 3

c

g 2

Find the value of 3a + 2b if: a a = 5 and b = 5 b d

3

4

e

Calculate the value of e2 - 3 if: a e=1 b e=3 d e=2 e e = -1 1 e= g h e = 3.1 2 Find ( w − 7) if: a w = 100 d w = 25 g

5

a = -7 and b = -2

w=9

f

c f i

c

e

w = 144 w = 49

h

w=1

i

b

Determine the value of 2 m − n 4 a m = 4 and n = 4 b c m = 20 and n = 8 d e m = 0.5 and n = -11 f

f

a−b+c 2ab 2ab c a = 0 and b = 0 1 a = 2 and b = 2 e = 10 e = -2 1 e= 5 w = 256 w = 400 1 w= 4

if: m = 10 and n = 2 m = 1 and n = -6 m = 2 and n = 0.25

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52

Preliminary Mathematics General

Development 6

Evaluate these expressions given x = 7, y = -5 and z = 21. a

x2 + z2 + y

b

c

4x + z −1

d

e

3 xy 2 z

f

y3 − 4 x z + 4 y2 6 y2 5 zx

7

The cross-sectional area of a solid is an annulus. It is evaluated using π ( R 2 − r 2 ) where R is the radius of the outer circle and r the radius of the inner circle. Find the area of an annulus if R is 8 cm and r is 4 cm. Answer correct to one decimal place.

8

Determine the value of

9

Evaluate

3

2 3

p 2 q given that p = 4 and q = 6.

2 y + 3 if y = 12.

10

Find the value of 2π decimal places.

11

Find the value of

12

1 Find the value of if f = 10 and c = 2. Give your answer correct to three decimal 2 π fc places.

13

3Rr Find the value of when R = 8.2 and r = 4.9. Give your answer correct to two R+r decimal places.

14

yA What is the value of ( y + 12 ) when y = 9 and A = 15. Give your answer correct to the nearest whole number.

l g

when l = 2.6 and g = 9.8. Give your answer correct to two

u 2 + 2 aas if u = 6, a = 7 and s = 2.

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Chapter 2 — Algebraic manipulation

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2.6 Linear equations An equation is a mathematical statement that says that two things are equal. It has an equal sign. For example, these are all equations: x+4=7

2 p = 118

6 y − 1 = 23

2.6 Linear equations have all their variables raised to the power of 1. The above three equations are linear equations. An equation such as x2 = 9 is not a linear equation as the variable is raised to the power of 2.

Solving an equation The process of finding the unknown value for the variable is called solving the equation. When solving an equation look to perform the opposite operation: • + is opposite to − • × is opposite to ÷ • x2 is opposite to x Make sure the equation remains balanced like a set of scales. The same operation needs to be done on both sides of the equal sign to keep the balance.

Solving an equation 1 2 3 4

Look to perform the opposite operation (+ is opposite to −, × is opposite to ÷). Add or subtract the same number to both sides of the equation OR Multiply or divide both sides of the equation by the same number. To solve two- or three-step equations, repeat the above steps as required. It is often easier to firstly add or subtract the same number to both sides of the equation.

When a solution has been reached it can be checked. The solution of the equation must satisfy the equation. Always check your solution by substituting your answer into the original equation. The left-hand side of the equation must equal the right-hand side.

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Preliminary Mathematics General

Example 15

Solving an equation

Solve the equation x + 4 = 7. Solution 1 2 3

x+4=7

Write the equation. The opposite operation to adding 4 is subtracting 4. Subtract 4 from both sides of the equation. Check that the solution is correct by substituting x = 3 into the original equation.

Example 16

−4

−4

x+ 4 = 7 x=3

Solving an equation

Solve the equation 2p = 18. Solution 1 2 3

Write the equation. The opposite operation to multiplying by 2 is dividing by 2. Divide both sides of the equation by 2. Check that the solution is correct by substituting p = 9 into the original equation.

Example 17

2 p = 118 2 p 18 = 2 2 p=9

Solving two-step equations

Solve the equation 4a + 5 = -1. Express your answer as a simplest fraction. Solution 1 2 3

Write the equation. The opposite operation to adding 5 is subtracting 5. Subtract 5 from both sides of the equation. The opposite operation to multiplying by 4 is dividing by 4. Divide both sides of the equation by 4.

( ) by writing it as a −6 4

4

Simplify the improper fraction

5

mixed number in simplified form −1 12 . Check that the solution is correct by substituting

( )

a = −1 12 into the original equation.

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4 a + 5 = −1 −5

−5

4a + 5 = − 1 4 a = −6 4 a −6 = 4 4 2 a = −1 4 1 = −1 2

Cambridge University Press

55

Chapter 2 — Algebraic manipulation

Example 18

Solving three-step equations

Solve the equation 2 y − 4 = 36 − 3 y. y Solution 1

Write the equation.

2

The opposite operation to subtracting 4 is adding 4. Add 4 to both sides of the equation.

3

4 5

2 y − 4 = 36 − 3 y +4

2 y = 40 − 3 y

The opposite operation to subtracting -3y is adding +3y. Add +3y to both sides of the equation.

+3 y

+3 y

2 y = 40 − 3 y 5 y = 40

5 y 40 = 5 5

The opposite operation to multiplying by 5 is dividing by 5. Divide both sides of the equation by 5. Check that the solution is correct by substituting y = 8 into the original equation.

Example 19

+4

2 y − 4 = 36 − 3 y

y=8

Solving a practical problem involving a linear equation

Seven is added to three times a number and the result is 22. a Write and equation using x to represent the number. b Solve the equation for x. Solution 1

Use mathematical symbols to replace the words in the statement. 7 for seven + for added 3 for three × for times = for result

2

The opposite operation to adding 7 is subtracting 7. Subtract 7 from both sides of the equation.

3 4

The opposite operation to multiplying by 3 is dividing by 3. Divide both sides of the equation by 3. Check that the solution is correct by substituting x = 5 into the original equation.

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a

7 + 3x 3 x = 22

b

7 + 3 x = 22 −7

−7

7 + 3 x = 22 3 x = 115 3 x 15 = 3 3 x=5

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56

Preliminary Mathematics General

Solving an equation using a graphics calculator Graphics calculators have an equation mode that may be used to solve any type of linear equation. The red letters on the keypad are used as the variable. After the equation is entered, set the required variable to zero and choose the ‘solver’ key. The calculator will show the answer and the values of the left-hand side and right-hand side of the equation.

Example 20

Solving a linear equation using a graphics calculator

Solve the equation 80 - 10y = 100. Solution 1 2 3

4 5 6

Select the EQUA menu. Select SOLVER (F3). Enter the equation. To place the variable (y) use the ALPHA key to access the letters in red above the keys. Highlight the required variable or Y = 0. Press SOLV and the calculator will show the answer. The values of the left (Lft) and right (Rgt) sides of the formula are shown and should be equal.

Example 21

Solving a linear equation using a graphics calculator

Solve the equation 6x + 5 = 7 + 5x. Solution 1 2 3

4 5

Select the EQUA menu. Select SOLVER (F3). Enter the equation. To place the variable (x) use the ALPHA key to access the letters in red above the keys. Highlight the required variable or X = 0. Press SOLV and the calculator will show the answer. The values of the left (Lft) and right (Rgt) sides of the formula are shown and should be equal.

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Chapter 2 — Algebraic manipulation

Exercise 2F 1

2

3

4

Solve the following linear equations. a y + 8 = 17 b x + 13 = 28 c + 7 = −4 d e m+9= 4 g h 8 + h = 111 9+r = 4 j 5=x+2 k 12 = m + 7 Solve the following linear equations. a a-7=3 b k-5=5 d s - 5 = -4 e z - 12 = -7 g 11 - v = 4 h 7-x=3 j 9=h-5 k 11 = f - 4 Solve the following linear equations. a 4x = 12 b 5w = 45 d 2t = -12 e 6h = -30 g 2w = 13 h 3c = -23 j 17 = 8k k -75 = 7d Solve the following linear equations. y d a b =4 =8 2 7 f a d e =5 =5 −3 −7 g j

d = −9 12 y 6= −2

h k

s = −3 −11 m 10 = 2

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c f i l

c f i l

c f i l

c f i l

a + 7 = 12 4 + d = −5 10 + q = −1 -3 = g + 5 d - 9 = 14 k - 7 = -9 6-j=7 -4 = c - (-1) 7v = 28 -4a = 40 7e = -8 -14 = -3e w =4 6 g =2 9 x = −4 −5 w 9= −9

Cambridge University Press

57

58

Preliminary Mathematics General

5

Solve the following linear equations. All solutions are integers. a 2x + 4 = 8 b 5y + 3 = 33 c 4q + 6 = 14 d 6d - 1 = 59 e 7v - 3 = 25 f 4m - 2 = 10 g 10g - 5 = -25 h 9x - 4 = -13 i 11p - 6 = -50 j 45 = 3v - 3 k 22 = 7n + 1 l 69 = 5b - 11

6

Solve the following linear equations. All solutions are integers. a 3 + 2y = 11 b 2 + 3c = 5 c 5 + 3a = 23 d 31 = 7y + 10 e 5 = 7 + 2m f 33 = 18 - 3d g 28 - 4q = 16 h 90 - 10r = 100 i 48 - 16e = -16 j 14 = 8 + 2h k 36 = 8 + 7d l 78 = 18 - 10w

7

Solve the following linear equations. Express your answer as a simplest fraction. a 4m + 1 = 26 b 3c + 16 = 27 c 2y + 13 = 16 d 28 = 16 + 5b e 37 = 6 + 4x f 11 = 28 + 8d g 2z - 3 = 4 h 5h - 7 = 14 i 6e - 2 = 13 j 21 = 2q + 10 k 32 = 3y - 5 l 17 = 9x + 9

8

Rajiv was required to solve the following equation for homework. This was his solution. 7x + 6 = 8 7 x = 114 x=2 Where is the error in Rajiv’s working?

9

Two is added to a number and the result is 7. a Write an equation using x to represent the number. b Solve the equation for x.

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Chapter 2 — Algebraic manipulation

59

Development 10

Solve the following linear equations. a 3x + 2 = 2x + 7 b 3y + 5 = 2y + 7 d 9b - 4 = 2 + 7b e 6d + 1 = 16 + 3d g 8 + a = 3a - 10 h 11 + 2w = 6w + 3

c f i

5v + 7 = 4v - 8 8z - 5 = 9 + z 4 + 3e = 9 - 2e

11

Matthew spent $93 on six show bags at the Royal Easter show, each costing the same price, $p. a Using p as the cost of one show bag, write an equation showing the cost of the six show bags. b Use the equation to find the cost of one show bag.

12

If 12 is added to a certain number, the result is three times the number. Find the number.

13

Solve the equation 2( 4 y − 1) − 55(( y − 2 ) = 117.

14

Isabella was required to solve the following equation for homework. This was her solution. Where is the error in Isabella’s working? 4 x − 3 = 11 + 6 x 2 x = 114 x=7

15

Ten is added to three times a number and the result is 19. a Write an equation using x to represent the number. b Solve the equation for x.

16

A number is increased by 4 and then this amount is doubled. The result is 20. a Write an equation for this information. b Find the number.

17

Four is added to twice a number and the result is 36. a Write an equation using n to represent the number. b Solve the equation for n.

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2.7 Equations with fractions Equations with fractions are solved in exactly the same way as other equations. Look to perform the opposite operation (+ is opposite to -, × is opposite to ÷) to both sides of the equation. Check your solution by substituting your answer into the original equation. 2.7 Example 22

Solve the equation

Solving an equation with a fraction

e + 4 = 110. 3

Solution 1 2 3 4

Write the equation. The opposite operation to adding 4 is subtracting 4. Subtract 4 from both sides of the equation. The opposite operation to dividing by 3 is multiplying by 3. Multiply both sides by 3. Check that the solution is correct by substituting e = 18 into the original equation.

Example 23

Solve the equation

e −4 −4 + 4 = 110 3 e =6 3 e 3× = 6 × 3 3 e = 18

Solving an equation with a fraction

3x − 5 = 6. 2

Solution 1 2 3 4 5

Write the equation. The opposite operation to dividing by 2 is multiplying by 2. Multiply both sides of the equation by 2. The opposite operation to subtracting 5 is adding 5. Add 5 to both sides of the equation. The opposite operation to multiplying by 3 is dividing by 3. Divide both sides of the equation by 3. Check that the solution is correct by substituting x = 5 23 into the original equation.

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3x − 5 =6 2 +5

+5

3 x − 5 = 12 3 x = 117 3 x 17 = 3 3 17 x= 3 2 =5 3

Cambridge University Press

Chapter 2 — Algebraic manipulation

Exercise 2G 1

Solve these equations. 1 x + 2 = 11 a 2 1 3a = 5 d 6 g

2

3

Solve these equations. k a = 10 10 3d d =6 4 k g 8= 3

Solve these equations. e a + 4 = 110 3 d g

4

x 1 =3 2 5

s −5 2 4y 5 = 1+ 3 4=

Solve these equations. 1 a x+3= 4 1 d (e + 2) = 1 3 1 g (6 + 2 c ) = 6 2

b e h

b e h

b

e h

b e h

1 2 y − 3 = 10 3 3 2 6s = 2 5 r 1 =1 5 4

x = 10 2 y 5=2+ 3 5r 7=2+ 9 6+

y+5=

m−4

f

5v = 2

i

d = −7 4 7a = 14 2 5q 20 = 3

2 3

1 ( h − 4) = 2 5 1 (11 − 5a ) = −1 4

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2 1 =1 5 3

c

c f i

c

f i

c f i

9 10 h 2 =4 2 7

v =4 12 2z =4 5 2n 16 = 3

d + 11 = 13 2 b 6 = 1− 9 2v 2= −8 7

a−2=

3 7

1 ( x + 3) = 1 2 1 (1 − 7u ) = −5 3

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62

Preliminary Mathematics General

Development 5

Solve these equations. a

b−5 =9 8

b

t−4 = −5 3

c

3x − 5 =6 2

d

1 + 2x 2 =7 3

e

4 + 2w = −6 5

f

2 + 3m =5 4

g

a+4 1 = 6 2

h

c−5 1 = 8 4

i

2e + 1 4 = 2 5

6

There are thirty-six times as many cars in Australia as trucks. Let C stand for the number of cars and T for the number of trucks. a Write an equation with C as the subject of the equation that correctly describes the relationship between the number of cars and trucks. b A local community has 120 trucks. How many cars are in the community?

7

Solve these equations. x x + =5 a 2 3

8

b

y y + = 12 6 2

c

n n + =3 5 10

d

c c − =7 2 4

e

s s − =2 5 6

f

m m − =4 7 9

g

3x x − =7 2 3

h

5a a − = 11 3 5

i

y 2y − =6 2 5

j

r r = −1 4 3

k

3w w = +8 10 2

l

3x 2 x = +2 2 3

Solve these equations. a

2y − 3=

y+3 2

b

x +1 − ( x − 1) = 1 2

c

2y − 3=

d

x + 4 x + 10 = 2 3

e

d + 6 2d + 4 = 3 4

f

3x − 5 2 x + 1 = 2 3

g

y +1 y +1 + =9 4 3

h

a + 4 2a − 3 − = −1 3 2

i

3x − 5 2 x + 1 − =2 2 3

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y+3 2

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Chapter 2 — Algebraic manipulation

63

2.8 Using formulas

2.8

A formula is a mathematical relationship between two or more variables. For example: • S = TD is a formula for relating the speed, distance and time. S, D and T are the variables. • P = 4L is a formula for finding the perimeter of a square, where P is the perimeter and L is the side length of the square. P and L are the variables. By substituting all the known variables into a formula, we are able to find the value of an unknown variable. Using a formula 1 2 3 4

Write the formula. Replace the variables in the formula with the numbers given in the question. Evaluate using the calculator. Write the answer to the specified level of accuracy and correct units if necessary.

Example 24

Using a formula

The cost of hiring a windsurfer is given by the formula C = 4t + 7 where C is the cost in dollars and t is the time in hours. Kayla wants to sail for 3 hours. How much will it cost her?

Solution 1 2 3 4

Write the formula. Substitute the value for t into the formula. Evaluate. Write your answer in words.

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C = 4t + 7 = 4 × 3+ 7 = 19 It will cost Kayla $19 to hire the windsurfer for 3 hours.

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Preliminary Mathematics General

Example 25

Using a formula

The length of a pendulum with a period of oscillation T is 2  T  l = 980    2π  Find l, correct to two decimal places, if T is 5.

Solution 1

Write the formula.

2

Substitute the 5 for T into the formula.

3

Evaluate.

4

Write answer in words correct to two decimal places.

Example 26 a b

 T  l = 980    2π 

 5  = 980 ×   2 × π  = 620.5922498 = 620.59

Using a formula

3

Perimeter, P, is the distance around the outside of a shape. Add all the sides to determine the perimeter. Write the formula. Substitute 27 for P into the formula.

4

Evaluate.

x x+3

x+4 2x − 5

Solution

2

2

The length of the pendulum is 620.59.

Write an expression for the perimeter of the trapezium. If the perimeter is 27 cm, calculate the value of x.

1

2

a

P = x + ( x + 4 ) + ( 2 x − 5) + ( x + 3) = 5x + 2

b

P = 5x + 2 27 = 5 x + 2 25 = 5 x x = 5 cm

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Chapter 2 — Algebraic manipulation

Using a formula and a graphics calculator Graphics calculators have an equation mode that may be used to enter a formula. The coloured letters on the keypad are used as the variable. After the formula is entered, set the unknown variable to zero and enter the values for the known variables. It is not necessary for the known variable to be the subject of the formula. Select the ‘solver’ key to obtain the answer. The calculator will show the answer and the values of the left-hand side and right-hand side of the equation.

Example 27

Using a formula and a graphics calculator

The circumference, C, of a circle with radius, r, is given by the formula C = 2π r. Find the circumference of a circle with a radius of 5 cm using a graphics calculator. Solution 1 2 3

4 5

6 7

Select the EQUA menu. Select SOLVER (F3). Enter the formula. To place the variables (C and r) use the ALPHA key to access the letters in red above the keys. Enter the known variable. The radius is 5 so R = 5. Highlight the required variable or C = 0.

Press SOLV and the calculator will show the answer. The values of the left (Lft) and right (Rgt) sides of the formula are shown and should be equal.

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Preliminary Mathematics General

Exercise 2H 1

If A = lb find the value of A when: a l = 16, b = 2 b l = 28, b = 7 c l = 30, b = 10

2

If A = s2 find the value of A when: a s=9 b s=6 c s = 32

3

If A = a

1 bbh find the value of A when: 2 b = 10, h = 4

b

b = 15, h = 2

c

b = 2, h = 3

4

If P = 2l + 2b find the value of P when: a l = 8, b = 10 b l = 2, b = 11 c l = 3, b = 9

5

Find the value of T (correct to one decimal place) in the formula T =

6

a

M = 1.7, v = 4 and r = 3.8

b

M = 2.1, v = 1 and r = 2.2

Use the formula a = 9 + a b c

7

b c 8

18 − b to find the value of a when: 2

b=8 b=1 b = -2

If p = a

Mv 2 if: r

12 x find the value of p when: x+4

x=7 x=5 x = -3

The cost of hiring a hall is given by the rule C = 30t + 1000 where C is the total cost in dollars and t is the number of hours for which the hall is hired. Find the cost of hiring the hall for: a 2 hours b 5.5 hours

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Chapter 2 — Algebraic manipulation

9

10

The distance, d km, travelled by a truck in t hours at an average speed of s km/h is given by the formula d = st. Find the distance travelled by a truck travelling at a speed of 70 km/h for 5 hours.

Given that H = a

E find the value of H when: T

E = 2.6 × 10−11 and T = 100

b

E = 7.8 × 10−6 and T = 20

11

The formula used to convert temperature from degrees Fahrenheit to degrees Celsius is C = 59 ( F − 32 ) . Use this formula to convert the following temperatures to degrees Celsius. Answer correct to the nearest whole number. a 40°F b 110°F

12

The formula v = u + at relates the velocity, acceleration and time. a Make u the subject of the formula. b Make a the subject of the formula. c Make t the subject of the formula.

13

The circumference, C, of a circle with radius, r, is given by the formula C = 2π r. a Make r the subject of the formula. b Find the radii of circles with the following circumferences. (Answer correct to two decimal places.) i 3 cm ii 6.9 mm

14

The body mass index is B = a b

m where m is the mass in kg and h is the height in m. h2 Make m the subject of the formula. Find m to the nearest whole number when: i B = 22.78 and h = 1.79 m ii B = 31.8 and h = 1.86 m

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68

Preliminary Mathematics General

Development 15

Find the value of v (correct to one decimal place) in the formula v = u 2 + 2 as if: a u = 4, a = 7 and s = 12 b u = -2, a = 10 and s = 5

16

Use the formula R = a

17

19

20

3V to find the value of R (correct to two decimal places) when: 4π b V = 44 c V = 100

Find the value of s (correct to one decimal place) in the formula s = ut ut + 12 aat 2 if: u = 2, a = 5 and t = 15 PRT The formula for calculating simple interest is I = where P is the principal, R is the 100 interest rate per annum and T is the time in years. Calculate the interest earned, correct to the nearest cent, on the following investments. a $10 000 at an interest rate of 9% p.a. for 3 years b $88 000 at an interest rate of 11.2% p.a. for 2 years 3 c $24 000 at an interest rate of 7 % p.a. for 2 years 4 a

18

V = 12

3

u = -5, a = 4 and t = 6

b

1 The volume of a cone is evaluated using V = π r 2 h where h is height and r is radius. a Write the formula with h as the subject. 3 b Calculate the height of a cone, correct to two decimal places, if the volume of the cone is 18 cm3 and the radius is 2 cm. 4 The volume of a sphere is given by the formula V = π r 3 where r is the radius. 3 a Write the formula with r as the subject. b What is the radius in metres of a spherical balloon with a volume of 2 m3? Answer correct to one decimal place.

14.8

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Chapter 2 — Algebraic manipulation

Adding and subtracting like terms

1

Multiplication and division of

1

algebraic terms

Study guide 2

Find the like terms. 2 Only add or subtract like terms. 3 Add or subtract the coefficients of the like terms. Write in expanded form and cancel any factors in a fraction. 2 Multiply and divide the coefficients and pronumerals. 3 Write the pronumerals in alphabetical order and express in index notation.

Expanding algebraic expressions

1

Multiply the term outside the grouping symbol by the a First term inside the grouping symbol. b Second term inside the grouping symbol. 2 Simplify and collect like terms if required.

Factorising algebraic expressions

1

Linear equations

1

Equations with fractions

•

Substitution

1

Using formulas

1

Find the largest factor of each term or the HCF. 2 Write the HCF outside the grouping symbol. 3 Divide the HCF into each term to find the terms inside the grouping symbols. 4 Check the factorisation by expanding the expression. Perform the opposite operation (+ and -, × and ÷). 2 Add/subtract the same number to both sides of the equation. 3 Multiply/divide both sides of the equation by the same number. Use the same steps as linear equations.

Write the algebraic expression. 2 Substitute the values and evaluate. Write the formula. 2 Substitute the values and evaluate.

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Review

Chapter summary – Algebraic manipulation

69

Review

70

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

2

Simplify 7xy - 5xy - 4xy. A -16xy B -2xy Simplify A

3

4

D

7 x2 16

D

96a2

6c × 3c. 5

9c 5

B

9c 2 5

C

18c 5

D

18c 2 5

-10d - 15e

B

-10d + 15e

C

-10d - 3e

D

-10d + 3e

4(4w2 + 6w)

B

4w(4w + 6)

C

8w(2w + 3)

C

b =1

D

16w(w + 8)

D

b=3

D

25

Solve the equation 11b - 9 = 26. b=

11 35

Find the value of A

9

B

7x 16

16xy

Factorise 16w2 + 24w by taking out the highest common factor.

A

8

7x 8

D

Expand -5(2d + 3e).

A

7

C

7 x2 8

5x 2 x + . 8 8

Simplify

A

6

6xy

Multiply and simplify the expression 4a × (8a) × 3. A -96a2 B -32a2 + 3 C -4a + 3

A

5

C

7

B

b=

11 15

4 11

a 2 + b 2 if a = 4 and b = 3. 14

B

C

5

What is the base of a triangle using the formula A = A

1.75

2 11

B

7

C

28

1 bbh if A is 14 and h is 4? 2 D 112

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Chapter 2 — Algebraic manipulation

Review

Sample HSC – Short-answer questions 1

Simplify these expressions. a 11st + 5 - 4st - 4

b

d + 4d2 – 2d –3d2

2

Simplify -3s + 5r - 3s + 3r by collecting the like terms.

3

Multiply and simplify: 7x × 5x a 2

4

5

6

7

b

2 w 3w 3 × 12 y 8 y 2

Add or subtract the following algebraic fractions. c c 5y 2 y a b + − 6 6 3 3

c

m 2m + 8 3

Remove grouping symbols and simplify if required. a

7(x - 1)

b

5(2 + 2r)

c

7(5x - 1)

d

3(4a - b)

e

7(d + 7e)

f

-2(8v - 2s)

g

-(9b - 2h)

h

4(w - 4) + 2(w + 2)

i

y(y + 2z)

Fully factorise these expressions. a

7b + 35

b

2v - 14

c

-3v + 15

d

6y + 9

e

4x - 14y

f

12x + 21y

g

21 - 15x

h

24b - 16c + 4

i

12s - 15v + 9

Solve the following linear equations. a

d-4=7

b

4h = 20

c

5+r=8

d

2t + 3 = 11

e

f

g

r = 12 6

h

5 = 2x - 5 2 1 v+3 =1 3 6

7n + 5 = 33 1 4m = 2 5

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71

i

Cambridge University Press

Review

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Preliminary Mathematics General

8

9

Solve the following linear equations. a 6x + 4 = 5x - 7 b 3 - 4r = 5r + 12 d 8(b - 1) = 2(b + 2) e y - 12 = 2(y - 7)

f

7(n - 2) = n + 4 6(2v - 7) = -(v + 1)

Solve these equations. a

10

c

u+3 =4 5

b

z z + = 12 4 8

c

1 ( x + 4) = 2 5

The distance d in km that a person can see the horizon from h in metres above the sea level, is given by the formula d = 5× a b

h . 2

Find d when h is 18 metres. Find h when d is 10 kilometres.

2V to find the value of R (correct to two decimal places) when: 3π

11

Use the formula R = a V=9 b V = 24 c V = 200

12

Einstein’s equation E = mc2 states that the energy E in joules equals the mass of m kg multiplied by the square of the speed of light c (3 × 108 m/s). Find the amount of energy produced by a: a mass of 500 kg b mass of 200 kg

13

The cost of hiring a hall is given by the formula C = 20t + 2000 where C is the total cost in dollars and t is the number of hours for which the hall is hired. a Make t the subject of the equation. b Find t when C is $2060.

3

Challenge questions 2

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C H A P T E R

3

Units of measurement and applications Syllabus topic — MM1 Units of measurement and applications Determine and convert appropriate units of measurement Convert units of area and volume Calculate the percentage error in a measurement Use numbers in scientific notation Express numbers to a certain number of significant figures Calculate and convert rates Find ratios of two quantities and use the unitary method Calculate repeated percentage changes

3.1 Units of measurement

3.1

Measurement is used to determine the size of a quantity. It usually involves using a measuring instrument. For example, to measure length, instruments that can be used include the rule, tape measure, caliper, micrometer, odometer and GPS. There are a number of systems of measurement that define their units of measurement. We use the SI metric system. 73 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Preliminary Mathematics General

SI units The SI is an international system of units of measurement based on multiples of ten. It is a version of the metric system which allows easy multiplication when converting between units. Units shown in red (below) are non-SI units approved for everyday or specialised use alongside SI units. Quantity

Name of unit

Symbol

Value

Length

Metre Millimetre Centimetre Kilometre Nautical mile

m mm cm km nm

Base unit 1000 mm = 1 m 100 cm = 1 m 1 km = 1000 m 1 nm = 1852 m

Area

Square metre Square centimetre Hectare

m2 cm2 ha

Base unit 10 000 cm2 = 1 m2 1 ha = 10 000 m2

Volume

Cubic metre Cubic centimetre Litre Millilitre Kilolitre

m3 cm3 L mL kL

Base unit 1 000 000 cm3 = 1 m3 1L = 1000 cm3 1000 mL = 1 L 1 kL = 1000 L

Mass

Kilogram Gram Tonne

kg g t

Base unit 1000 g = 1 kg 1 t = 1000 kg

Time

Second Minute Hour Day

s min h d

Base unit 1 min = 60 s 1 h = 60 min 1 d = 24 h

Converting between units A prefix is a simple way to convert between units. It indicates a multiple of 10. Some common prefixes are mega (1 000 000), kilo (1000), centi

( ) and milli ( ). 1 1000

1 100

Length, mass and volume × 1000 × 1000 × 100 × 10

mega kilo unit centi milli

Time

÷ 1000

× 24

÷ 1000

× 60

÷ 100

× 60

÷ 10

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days hours minutes seconds

÷ 24 ÷ 60 ÷ 60

Cambridge University Press

Chapter 3 — Units of measurement and applications

Example 1

75

Converting units of length

Complete the following. a 35 cm = mm

b

4500 m =

km

Solution 1

To change cm to mm multiply by 10.

a

2

To change m to km divide by 1000.

b

Example 2

35 cm = 35 × 10 mm = 350 mm 4500 = 4500 ÷ 1000 km = 4.5 km

Converting units of time

Complete the following. 3 h and 15 min = a

min

b

10 080 min =

d

Solution 1

To change hours to minutes, multiply by 60.

a

2

To change minutes to hours, divide by 60.

b

3

To change hours to days, divide by 24.

3 h 15 min = 3 × 60 + 15 min = 195 min 10 080 min = 10 080 ÷ 60 h = 168 h = 168 ÷ 24 d =7d

Converting area and volume units To convert area units, change the side length units and compare the values for area. =

1m

100 cm 100 cm

1m

1 m2 = 100 × 100 = 10 000 cm2 1 m2 = 10 000 cm2 1 or 1 cm 2 = m2 10 000

To convert volume units, change the side length units and compare the values for volume. 1m 100 cm

=

1m 1m

100 cm

100 cm

1 m3 = 100 × 100 × 100 = 1 000 000 cm3 1 m3 = 1000 000 cm3 1 or 1 cm 3 = m3 1 000 000

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76

Preliminary Mathematics General

Exercise 3A 1

2

3

4

Complete the following. a

5 cm m=

d

890 m =

g

9400 m =

j

49 000 cm =

mm m ccm kkm km

b

78 m =

e

57 cm =

h

600 mm =

k

22 000 m =

b

45 kg =

ccm mm cm kkm

c

2 km m=

m

f

6 km m=

ccm

i

8100 cm =

l

51 mm =

c

76 t =

f

0.52 52 t =

i

45 000 000 g =

l

60 000 g =

m cm

Complete the following. a

3t=

d

8100 kg =

g

e

4t=

g

6800 g =

kkg

h

9 300 000 g =

j

300 kg =

t

k

2300 g =

b

12 kL =

L

c

9 kL L=

mL m

f

300 kL =

i

210 000 mL =

l

8 000 000 mL =

kg

g g t

kkg

kkg kg t

kkg

Complete the following. a

2 L=

d

7800 kL =

L

e

50 L =

g

6100 L =

kkL

h

400 mL =

j

80 mL =

L

k

79 000 mL =

min

b

2 min =

s

c

20 d =

h

s

e

4.5 d =

h

f

10 h =

min

h

48 000 s =

min

i

96 h =

d

k

390 min =

h

l

780 s =

mL

L kL

mL m mL kL kL

Complete the following. a

2.5 h =

d

40 min =

g

720 min =

j

1080 h =

h d

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min

Cambridge University Press

Chapter 3 — Units of measurement and applications

5

What unit of length is most appropriate to measure each of the following? a Length of a pen b Height of a building c Thickness of a credit card d Distance from Sydney to Newcastle e Height of a person f Length of a football field

6

What unit of mass is most appropriate to measure each of the following? a Weight of an elephant b Mass of a mug c Bag of onions d Weight of a baby e Mass of a truck f Mass of a teaspoon of sugar

7

What unit of time is most appropriate to measure each of the following? a Lesson at school b Reheating a meal in a microwave c Age of a person d School holidays e Accessing the internet f Movie

8

There are 20 litres of a chemical stored in a container. a What amount of chemical remains if 750 mL is removed from the container? Answer in litres. b How many containers are required to make a kilolitre of the chemical?

9

Christopher bought 3 kg of sultanas. What mass of sultanas remains if he ate 800 grams? Answer in kilograms.

10

The length of the Murray River is 2575 km. The length of the Hawkesbury River is 80 000 m. What is the difference in their lengths? Answer in metres.

11

There are three tonnes of grain in a truck. What is the mass if another 68 kg of grain is added to the truck? Answer in kilograms.

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Preliminary Mathematics General

Development 12

A cyclist travels to and from work over a 1200-metre long bridge. Calculate the distance travelled in a week if the cyclist works for 5 days. Answer in kilometres.

13

Madison travels 32 km to work each day. Her car uses 1 litre of petrol to travel 8 km. a How many litres of petrol will she use to get to work? b How many litres of petrol will she use for 5 days of work, including return travel?

14

Arrange 500 m, 0.005 km, 5000 cm and 5 000 000 mm in: a Ascending order (smallest to largest) b Descending order (largest to smallest)

15

Complete the following. a

1 km 2 =

m2

b

1 m2 =

c

1 cm 2 =

mm 2

d

1000 cm 2 =

m2

e

2000 mm 2 =

f

5000 m 2 =

kkm m2

g

3.9 m 2 =

cm 2

h

310 km 2 =

m2

i

4.7 m 2 =

mm 2

j

74300 m 2 =

kkm m2

k

6500 mm 2 =

l

4000 cm 2 =

m2

cm 2

cm 2

mm 2

16

The area of a field is 80 000 square metres. Convert the area units to the following. a Square kilometres b Hectares

17

Jackson swims 30 lengths of a 50-metre pool. a How many kilometres does he cover? b If his goal is 4 kilometres, how many more lengths must he swim?

18

Eliza worked from 10.30 a.m. until 4.00 p.m. on Friday, from 7.30 a.m. until 2.00 p.m. on Saturday, and from 12 noon until 5.00 p.m. on Sunday. a How many hours did Eliza work during the week? b Express the time worked on Friday as a percentage of the total time worked during the week. Answer correct to the nearest whole number.

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3.2 Measurement errors There are varying degrees of instrument error and measurement uncertainty when measuring. Every time a measurement is repeated, with a sensitive instrument, a slightly different result will be obtained. The possible sources of errors include mistakes in reading the scale, parallax error and calibration error. The accuracy of a measurement is improved by making multiple measurements of the same quantity with the same instrument.

Accuracy in measurements The smallest unit on the measuring instrument is called the limit of reading. For example, a 30 cm rule with a scale for millimetres has a limit of reading of 1 mm. The accuracy of a measurement is restricted to ± 1 of the limit of reading. For example, 2 if the measurement on the ruler is 10 mm then the range of errors is 10 ± 0.5 mm. Here the upper limit is 10 + 0.5 mm or 10.5 mm and the lower limit is 10 – 0.5 mm or 9.5 mm.

1 cm

2

3

4

Every measurement is an approximation and has an error. The absolute error is the difference between the actual value and the measured value indicated by the instrument. The maximum value for an absolute error is 1 of the limit of reading. 2

Limit of reading

Absolute error

Smallest unit on measuring instrument

Measured value – Actual value Maximum value is 1 × limit of reading 2

Relative error gives an indication of how good a measurement is relative to the size of the quantity being measured. The relative error of a measurement is calculated by dividing the limit of reading by the actual measurement. For example, the relative error for the above measurement is

( ) = 0.005. The relative error is often expressed as a percentage and 0.5 10

called the ‘percentage error’. For example, the percentage error for the above measurement is 010.5 × 100 = 5% .

( )

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Preliminary Mathematics General

Relative error

Percentage error

 Absolute error  ±  Measurement 

 Absolute error  ± × 100%  Measurement 

Example 3 a b c d e

Finding the measurement errors

What is the length indicated by the arrow on the above ruler? 0 10 20 30 40 50 60 70 80 What is the limit of reading? What is the upper and lower limit for each measurement? Find the relative error. Answer correct to three decimal places. Find the percentage error. Answer correct to one decimal place. Solution 1 2

3 4

5

6 7

8 9 10

11

The arrow is pointing to 38 mm. Limit of reading is the smallest unit on the ruler (millimetre). Calculate half the limit of reading. Lower limit is the measured value minus 1 the limit of 2 reading. Upper limit is the measured value plus 1 the limit of 2 reading. Write the formula for relative error. Substitute the values for absolute error and the measurement. Evaluate correct to three decimal places. Write the formula for percentage error. Substitute the values for absolute error and the measurement. Evaluate.

a

Length is 38 mm.

b

Limit of reading is 1 mm.

c

1 2

× limit of reading =

1 2

×1

= 0.5 mm Lower limit = 38 − 0.5 = 37.5 mm Upper limit = 38 + 0.5 = 38.5 mm

d

 Absolute error  Relative error = ±   Measurement  0.5 38 = ± 0.013

=±

e

 Absolute error  Percentage error = ±  × 100%  Measurement 

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0.5 × 100% 38 = ± 1.3%

=±

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Exercise 3B 1

Four measurements of length are shown on the ruler below. B

A

0

10

30

40

50

60

D

70

80

90

100

What length is indicated by each letter? Answer to the nearest millimetre. What is the limit of reading? What the largest possible absolute error? What is the upper and lower limit for each measurement? Calculate the relative error, correct to three decimal places, for each measurement. Calculate the percentage error, correct to one decimal place, for each measurement.

a b c d e f

2

20

C

Two measurements of mass are shown on the scales below.

0

0 4.5 8 0

8

4kg

10lb

9lb

8

1lb

8

2lb

A

8 8

1kg

8

4kg

8

8

6lb 8 5lb

3kg

8

4lb

2kg

b c d e f

8 0

8

1lb

8

2lb

B

1.5

8

3.5

1kg

8

8lb

3lb 8

7lb

8

6lb 8 5lb

3kg

2.5

a

10lb

8

3lb

7lb

0.5

9lb

8

8lb 3.5

4.5

0.5

8

4lb

1.5

2kg 2.5

What mass is indicated by each letter? Use the outer scale. What is the limit of reading? What the largest possible absolute error? What is the upper and lower limit for each measurement? Calculate the relative error, correct to three decimal places, for each measurement. Calculate the percentage error, correct to one decimal place, for each measurement.

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Development 3

A dishwasher has a mass of exactly 49.6 kg. Abbey measured the mass of the dishwasher as 50 kg to the nearest kilogram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error to the nearest whole number.

4

An iPod has a mass of exactly 251 g. Jake measured the mass of the iPod as 235 g to the nearest gram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error correct to two decimal places.

5

An LCD screen has a mass of exactly 2.71 kg. Saliha measured the mass of the screen as 3 kg to the nearest kilogram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error correct to three decimal places.

6

A measurement was taken of a skid mark at the scene of a car accident. The actual length of the skid mark was 25.15 metres, however it was measured as 25 metres. a What is the absolute error? b Find the relative error. Answer correct to three decimal places. c Find the percentage error. Answer correct to one decimal place.

7 

The length of a building at school is exactly 56 m. Cooper measured the length of the building to be 56.3 m and Filip measured the building at 55.8 m. a What is the absolute error for Cooper’s measurement? b What is the absolute error for Filip’s measurement? c Compare the relative error for both measurements. Answer correct to four decimal places. d Compare the percentage error for both measurements. Answer correct to three decimal places.

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3.3 Scientific notation and significant figures Scientific notation 3.3

Scientific notation is used to write very large or very small numbers more conveniently. It consists of a number between 1 and 10 multiplied by a power of ten. For example, the number 4 100 000 is expressed in scientific notation as 4.1 × 106. The power of ten indicates the number of tens multiplied together. For example: 4.1 × 106 = 4.1 × (10 × 10 × 10 × 10 × 10 × 10) = 4 100 000 When writing numbers in scientific notation, it is useful to remember that large numbers have a positive power of ten and small numbers have a negative the power of ten. Writing numbers in scientific notation 1 2 3 4

Find the first two non-zero digits. Place a decimal point between these two digits. This is the number between 1 and 10. Count the digits between the new and old decimal point. This is the power of ten. Power of ten is positive for larger numbers and negative for small numbers.

Example 4

Expressing a number in scientific notation

The land surface of the earth is approximately 153 400 000 square kilometres. Express this area more conveniently by using scientific notation.

Solution 1 2 3 4 5

The first two non-zero digits are 1 and 5. Place the decimal point between these numbers. Count the digits from the old decimal point (end of the number) to position of the new decimal point. Large number indicates the power of 10 is positive. Write in scientific notation.

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1.534 1.53 400 000 − eight digits Power of 10 is +8 or 8 153 400 000 = 1.534 × 108

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Significant figures Significant figures are used to specify the accuracy of a number. It is often used to round a number. Significant figures are the digits that carry meaning and contribute to the accuracy of the number. This includes all the digits except the zeros at the start of a number and zeros at the finish of a number without a decimal point. These zeros are regarded as placeholders and only indicate the size of the number. Consider the following examples. • 51.340 has four significant figures: 5, 1, 3 and 4. • 0.00871 has three significant figures: 8, 7 and 1. • 56091 has five significant figures: 5, 6, 0, 9 and 1. The significant figures in a number not containing a decimal point can sometimes be unclear. For example, the number 8000 may be correct to 1 or 2 or 3 or 4 significant figures. To prevent this problem, the last significant figure of a number is underlined. For example, the number 8000 has two significant figures. If the digit is not underlined the context of the problem is a guide to the accuracy of the number. Writing numbers to significant figures 1 2 3

Write the number in scientific notation. Count the digits in the number to determine its accuracy (ignore zeros at the end). Round the number to the required significant figures.

Example 5

Writing numbers to significant figures

Write these numbers correct to the significant figures indicated. a 153 400 000 (3 significant figures) b 0.000 657 (2 significant figures) Solution 1 2 3 4

5 6 7 8

Write in scientific notation. Count the digits in the number. Round the number to 3 significant figures. Write answer in scientific notation correct to 3 significant figures.

a

153 400 000 = 1.534 × 108 1.534 has 4 digits 1.53 rounded to 3 sig. fig. 153 400 000 = 1.53 × 108

Write in scientific notation. Count the digits in the number. Round the number to 2 significant figures. Write answer in scientific notation correct to 2 significant figures.

b

0.000 657 = 6.57 × 10−4 6.57 has 3 digits 6.6 rounded to 2 sig. fig. 0.000 657 = 6.6 × 10−4

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Exercise 3C 1

2

Write these numbers in scientific notation. a 7600 c 590 000 e 35 000 g 77 100 000 i 95 400 000 000 Write these numbers in scientific notation. a 0.000 56 c 0.000 000 812 e 0.000 058 g 0.26 i 0.000 000 000 167

b d f h

b d f h

1 700 000 000 6 800 000 310 000 000 523 000 000 000

0.000 068 7 0.0043 0.000 003 12 0.092

3

A microsecond is one millionth of a second. Write 5 microseconds in scientific notation.

4

Sharks existed 410 million years ago. a Write this number in scientific notation. b Express this number correct to one significant figure.

5

Write each of the following as a basic numeral. a 1.12 × 105 c 5.2 × 103 e 2.4 × 102 g 3.9 × 106 i 6.4 × 104

6

Write each of the following as a basic numeral. a 3.5 × 10−4 c 1.63 × 10−7 e 4.9 × 10−2 g 4.12 × 10−8 i 3.0 × 10−9

b d f h

b d f h

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5.34 × 108 8.678 × 107 7.8 × 109 2.8 × 101

7.9 × 10−6 5.81 × 10−3 9.8 × 10−1 6.33 × 10−5

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7

Convert a measurement of 5.81 × 10−3 grams into kilograms. Express your answer in scientific notation.

8

Evaluate the following and express your answer in scientific notation. a (2.5 × 103) × (5.9 × 106) b (4.7 × 105) × (6.3 × 102) c (7.1 × 10−5) × (4.2 × 10−2) d (3.0 × 10−4) × (6.2 × 10−5)

9

Evaluate the following and express your answer in scientific notation. 9.1 × 105 7.2 × 107 4.8 × 10−4 a b c 2.8 × 10−2 4.8 × 10−3 3.2 × 10−5

10

Write these numbers correct to significant figures indicated. a 1 561 231 (2 sig. fig.) b 3 677 720 (4 sig. fig.) c 789 001 (5 sig. fig.) d 3 300 000 (1 sig. fig.) e 777 777 (3 sig. fig.) f 3 194 729 (5 sig. fig.) g 821 076 (4 sig. fig.) h 7091 (1 sig. fig.) i 49 172 (2 sig. fig.)

11

Write these numbers correct to significant figures indicated. a 0.0035 (1 sig. fig.) b 0.191 785 (4 sig. fig.) d 0.111 222 33 (6 sig. fig.) e 0.000 0271 (1 sig. fig.) g 0.008 12 (2 sig. fig.) h 0.092 71 (3 sig. fig.)

c f i

0.001 592 (3 sig. fig.) 0.019 832 6 (5 sig. fig.) 0.000 419 (2 sig. fig.)

12

A bacterium has a radius of 0.000 015 765 m. Express this length correct to two significant figures.

13

Convert a measurement of 2654 kilograms into centigrams. Express your answer correct to two significant figures.

14

Convert a measurement of 4 239 810 milligrams into grams. Express your answer correct to four significant figures.

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Development 15

If y = 12 x 2 , find the value of y when: a

16

17

b

x = 9.8 × 10−3

θ 2 πr where θ is the angle at the centre and r is the The arc length of a circle is l = 360° radius of the circle. Use this formula to calculate the arc length of a circle when θ = 30° and r = 7.4 × 108. Answer in scientific notation correct to one significant figure.

Given that V = a

13.3

x = 2.4 × 103

r

find the value of r in scientific notation when: h V = 5 × 104 and h = 9 × 106 b V = 6 × 10−7 and h = 4 × 102

18

Use the formula E = md2 to find d correct to three significant figures given that: a m = 0.08 and E = 5.5 × 109 b m = 2.7 × 103 and E = 1.6 × 104

19

Find x given x3 = 2.7 × 1012. Answer correct to four significant figures.

20

Light travels at 300 000 kilometres per second. Convert this measure to metres per second and express this speed in scientific notation.

21

Use the formula E = 3p − q to evaluate E given that p = 7.5 × 105 and q = 2.5 × 104. Answer in scientific notation correct to one significant figure.

22

The volume of a cylinder is V = πr2h where r is the radius of the cylinder and h is the height of the cylinder. Use this formula to calculate the volume of the cylinder if r = 5.6 × 104 and h = 2.8 × 103. Answer in scientific notation correct to three significant figures.

23

The Earth is 1.496 × 108 km from the Sun. Calculate the distance travelled by the Earth in a year using the formula c = 2πr. Answer in scientific notation correct to two significant figures.

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3.4 Calculations with ratios

3.4

A ratio is used to compare amounts of the same units in a definite order. For example, the ratio 3:4 represents 3 parts to 4 parts or 3 or 0.75 or 75%. 4

A ratio is a fraction and can be simplified in the same way as a fraction. For example, the ratio 15:20 can be simplified to 3:4 by dividing each number by 5. Equivalent ratios are obtained by multiplying or dividing each amount in the ratio by the same number. ÷3

÷3

15 : 12 = 5 : 4

×3

×3

5 : 4 = 15 : 12

15:12 and 5:4 are equivalent ratios. When simplifying a ratio with fractions, multiply each of the amounts by the lowest common denominator. For example, to simplify 1 : 3 multiply both sides by 8. This results in the 8 4 equivalent ratio of 1:6. Ratio A ratio is used to compare amounts of the same units in a definite order. Equivalent ratios are obtained by multiplying or dividing by the same number.

Dividing a quantity in a given ratio Ratio problems may be solved by dividing a quantity in a given ratio. This method divides each amount in the ratio by the total number of parts. Dividing a quantity in a given ratio 1 2 3 4

Calculate the total number of parts by adding each amount in the ratio. Divide the quantity by the total number of parts to determine the value of one part. Multiply each amount of the ratio by the result in step 2. Check by adding the answers for each part. The result should be the original quantity.

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Example 6

89

Dividing a quantity in a given ratio

Mikhail and Ilya were given $450 by their grandparents to share in the ratio 4:5. How much did each person receive? Solution 1 2 3 4 5

Calculate the total number of parts by adding each Total parts = 4 + 5 = 9 amount in the ratio (4 parts to 5 parts). 9 parts = $450 Divide the quantity ($450) by the total number of $450 1 part = = $5 50 parts (9 parts) to determine the value of one part. 9 Multiply each amount of the ratio by the result in 4 parts = 4 × $50 50 = $200 step 2 or $50. 5 parts = 5 × $50 50 = $250 Check by adding the answers for each part. The ($200 + $250 = $450) result should be the original quantity or $450. Mikhail receives $200 and Write the answer in words. Ilya receives $250.

The unitary method The unitary method involves finding one unit of an amount by division. This result is then multiplied to solve the problem. Using the unitary method 1 2

Find one unit of an amount by dividing by the amount. Multiply the result in step 1 by a number to solve the problem.

Example 7

Using the unitary method

A car travels 360 km on 30 L of petrol. How far does it travel on 7 L? Solution 1 2 3 4 5 6

Write a statement using information from the question. Find 1 L of petrol by dividing 360 km by the amount or 30. Multiply the 360 by a 7 to solve the problem. 30 Evaluate. Write answer to an appropriate degree of accuracy. Write the answer in words.

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30 L = 360 kkm 360 1 L= km 30 360 7 L= × 7 km 30 = 84 km The car travels 84 km.

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Exercise 3D 1

Express each ratio in simplest form. a 15:3 b 10:40 d 14:30 e 8:12

c f

g

9:18:9

h

5:10:20

i

j

1 1 : 2 5

k

2 3 : 3 7

l

24:16 49:14 1 1: 3 3 :1 4

2

A delivery driver delivers 1 parcel on average every 20 minutes. How many hours does it take to drop 18 parcels?

3

Divide 240 into the following ratios. a 2:1 b 3:2 c 1:5 d 7:5

4

A bag of 500 grams of chocolates is divided into the ratio 7:3. What is the mass of the smaller amount?

5

At a concert there were 7 girls for every 5 boys. How many girls were in the audience of 8616?

6

Molly, Patrick and Andrew invest in a business in the ratio 6:5:1. The total amount invested is $240 000. How much was invested by the following people? a Molly b Patrick c Andrew

7

In a boiled fruit cake recipe the ratio of mixed fruit to flour to sugar is 5:3:2. A 250 g packet of mixed fruit is used to make the cake. How much sugar and flour is required?

8

9

A 5 kg bag of potatoes costs $12.80. Find the cost of: a 1 kg b 10 kg c 14 kg d 6 kg The cost of 3 pens is $42.60. Find the cost of: a 1 pen b 4 pens c 6 pens d 10 pens

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Development 10

A punch is made from pineapple juice, lemonade and passionfruit in the ratio 3:5:2. a How much lemonade is needed if one litre of pineapple juice is used? b How much pineapple juice is required to make 10 litres of punch?

11

Angus, Ruby and Lily share an inheritance of $500 000 in the ratio of 7:5:4. How much will be received by the following people? a Angus b Ruby c Lily

12

Samantha and Mathilde own a restaurant. Samantha gets 3 of the profits and Mathilde 5 receives the remainder. a What is the ratio of profits? b Last week the profit was $2250. How much does Mathilde receive? c This week the profit is $2900. How much does Samantha receive?

13

A jam is made by adding 5 parts fruit to 4 parts of sugar. How much fruit should be added to 2 12 kilograms of sugar in making the jam?

14

A local council promises to spend $4 for every $3 raised in public subscriptions for a community hall. The cost of the hall is estimated at $1.75 million. How much does the community need to raise?

15

The ratio of $5 to $10 notes in Stephanie’s purse is 3:5. There are 24 notes altogether. What is the total value of Stephanie’s $5 notes?

16

Nathan makes a blend of mixed lollies using 5 kg jelly babies, 4 kg licorice and 1 kg skittles. What is the cost of the blend per kilogram to the nearest cent? Mixed lollies Jelly babies $5.95 per kg

17

Licorice

$6.95 per kg

Skittles

$11.90 per kg

The three sides of a triangle are in the ratio of 2:3:4. The longest side of the triangle is 12.96 mm. What is the perimeter of the triangle?

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3.5 Rates and concentrations Rates 3.5

A rate is a comparison of amounts with different units. For example, we may compare the distance travelled with the time taken. In a rate the units are different and must be specified. The order of a rate is important. A rate is written as the first amount per one of the second amount. For example, $2.99/kg represents $2.99 per one kilogram or 80 km/h represents 80 kilometres per one hour. Converting a rate 1 2 3 4

Write the rate as a fraction. First quantity is the numerator and 1 is the denominator. Convert the first amount to the required unit. Convert the second amount to the required unit. Simplify the fraction.

Example 8

Converting a rate

Convert each rate to the units shown. a 55 200 m/h to m/min b $6.50/kg to c/g Solution 1 2 3 4 5 6 7 8 9 10

Write the rate as a fraction. The numerator is 55 200 m and the denominator is 1 h. No conversion required for the numerator. Convert the 1 hour to minutes by multiplying by 60. Simplify the fraction. Write the rate as a fraction. The numerator is $6.50 and the denominator is 1 kg. Convert the $6.50 to cents by multiplying by 100. Convert the 1 kg to g by multiplying by 1000. Simplify the fraction.

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55 200 m 1h 55 200 m = 1 × 60 min = 920 m/min

a

55 200 =

b

6.50 =

$6.50 1 kg 6.50 × 100 c = 1 × 1000 g = 0.665 c/g

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Concentrations A concentration is a measure of how much of a given substance is mixed with another substance. Concentrations are a rate that has particular applications in nursing and agriculture. It often involves mixing chemicals. Concentrations may be expressed as: • weight per weight such as 10 g/100 g • weight per volume such as 5 g/10 mL • volume per volume such as 20 mL/10 L. Finding a percentage concentration 1 2

Write the two quantities as a fraction. Multiply the fraction by 100 to convert it to a percentage.

Example 9

Converting a concentration

A medicine is given as a concentration of 2.5 mL per 10 kg. What is the dosage rate for this medicine in mL/kg? Solution 1 2 3 4 5 6

Write the rate as a fraction. The numerator is 2.5 mL and the denominator is 10 kg. Divide the numerator by the denominator. Evaluate. Write answer to an appropriate degree of accuracy. Write the answer in words.

Example 10

2.5 mL 10 kg 2.5 mL = × 10 kg = 0.25 2 5 mL L//kg

2.5 mL/10 kg =

The dosage rate is 0.25 mL/kg.

Expressing as a percentage concentration

Express 6.2 g of sugar per 50 g as a percentage concentration. Solution

3

Write as a fraction. The first amount is divided by the second amount. Multiply the fraction by 100 to convert it to a percentage. Evaluate.

4

Write the answer in words.

1 2

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6.2 g 50 g 6.2 = × 100% 50 = 12.4%

6.2 g/50 g =

Percentage composition is 12.4%.

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Exercise 3E 1

Use the rate provided to answer the following questions. a Cost of apples is $2.50/kg. What is the cost of 5 kg? b Tax charge is $28/m². What is the tax for 7 m2? c Cost savings are $35/day. How much is saved in 5 days? d Cost of a chemical is $65/100 mL. What is the cost of 300 mL? e Cost of mushrooms is $5.80/kg. What is the cost of 12 kg? f Distance travelled is 1.2 km/min. What is the distance travelled in 30 minutes? g Concentration of a chemical is 3 mL/L. How many mL of the chemical is needed for 4 L? h Concentration of a drug is 2 mL/g. How many mL is needed for 10 g?

2

Express each rate in simplest form using the rates shown. a 300 km on 60 L [km per L] b 15 m in 10 s [m per s] c $640 for 5 m [$ per m] d 56 L in 0.5 min [L per min] e 78 mg for 13 g [mg per g] f 196 g for 14 L [g per L]

3

Convert each rate to the units shown. a 39 240 m/min [m/s] c 88 cm/h [mm/h] e 0.4 km/s [m/s] g 6.09 g/mL [mg/mL] i 12 600 mg/g [mg/kg]

b d f h

2 m/s [cm/s] 55 200 m/h [m/min] 57.5 m/s [km/s] 4800 L/kL [mL/kL]

4

Mia earns $37.50 per hour working in a cafe. a How much does Mia earn for working a 9-hour day? b How many hours does Mia work to earn $1200? c What is Mia’s annual income if she works 40 hours a week? Assume she works 52 weeks in the year.

5

Patrick mixes 35 mL of a pesticide per 20 L as a percentage concentration. a Express this concentration in litres per litre. b What is the percentage concentration?

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Development 6

A tap is dripping water at a rate of 70 drops per minute. Each drop is 0.2 mL. a How many millilitres of water drip from the tap in one minute? b How many litres of water drip from the tap in a day?

7

Natural gas is charged at a rate of 1.4570 cents per MJ. a Find the charge for 12 560 MJ of natural gas. Answer to the nearest dollar. b The charge for natural gas was $160.27. How many megajoules were used?

8

Olivia’s council rate is $2915 p.a. for land valued at $265 000. Lucy has a council rate of $3186 on land worth $295 000 from another council. a What is Olivia’s council charge as a rate of $/$1000 valuation? b What is Lucy’s council charge as a rate of $/$1000 valuation?

9

Mira’s car uses 9 litres of petrol to travel 100 kilometres. Petrol costs $1.50 per litre. a What is the cost of travelling 100 kilometres? b How far can she drive using $50 worth of petrol? Answer to the nearest kilometre.

10

A motor bike is moving at a steady speed. When the speed is 90 km/h the bike consumes 5 litres of petrol for every 100 kilometres travelled. a The petrol tank holds 30 litres. How many kilometres can the bike travel on a full tank of petrol when its speed is 90 km/h? b When the speed is 110 km/h the bike consumes 30% more petrol per kilometre travelled. Calculate the number of litres per 100 kilometres consumed when the bike travels at 110 km/h.

11

A plane travelled non-stop from Los Angeles to Sydney, a distance of 12 027 kilometres in 13 hours and 30 minutes. The plane started with 180 kilolitres of fuel, and on landing had enough fuel to fly another 45 minutes. a What was the plane’s average speed in kilometres per hour? Answer to the nearest whole number. b How much fuel was used? Answer to the nearest kilolitre.

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3.6 Percentage change Percentage change involves increasing or decreasing a quantity as a percentage of the original amount of the quantity. 3.6

Percentage increase 1 2

Add the % increase to 100%. Multiply the above percentage by the amount.

Example 11

Percentage decrease 1 2

Subtract the % decrease from 100%. Multiply the above percentage by the amount.

Calculating the percentage change

The retail price of a toaster is $36 and is to be increased by 5%. What is the new price? Solution 1 2 3 4 5

Add the 5% increase to 100%. Write the quantity (new price) to be found. Multiply the above percentage (105%) by the amount. Evaluate and write using correct units. Write the answer in words.

Example 12

100% + 5% = 105% New price = 105% of $36 = 1.05 × 36 = $37.80 New price is $37.80.

Calculating repeated percentage changes

Increase $75 by 20% and then decrease the result by 20%. Solution 1 2 3 4 5 6 7 8 9

Add the 20% increase to 100%. Write the quantity (new price) to be found. Multiply the above percentage (120%) by the amount. Evaluate and write using correct units. Subtract the 20% decrease from 100%. Write the quantity (new price) to be found. Multiply the above percentage (80%) by the amount. Evaluate and write using correct units. Write the answer in words.

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100% + 20% = 120% New price = 120% of $75 = 1.20 × 75 = $90 100% − 20% = 80% New price = 80% of $90 = 0.80 × 90 = $72 New price is $72.

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97

Exercise 3F 1

What is the amount of the increase in each of the following? a Increase of 10% on $48 b Increase of 30% on $120 c Increase of 15% on $66 d Increase of 25% on $88 e Increase of 40% on $1340 f Increase of 36% on $196 g Increase of 4.5% on $150 h Increase of 1 % on $24 2

2

What is the amount of the decrease in each of the following? a Decrease of 20% on $110 b Decrease of 60% on $260 c Decrease of 35% on $320 d Decrease of 75% on $1096 e Decrease of 6% on $50 f Decrease of 32% on $36 g Decrease of 12.5% on $640 h Decrease of 1 14 % on $56

3

David Jones clearance sale has a discount of 30% off the retail price of all clothing. Find the amount saved on the following items. a Men’s shirt with a retail price of $80 b Pair of jeans with a retail price of $66 c Ladies jacket with a retail price of $450 d Boy’s shorts with a retail price of $22 e Jumper with a retail price of $124 f Girl’s skirt with a retail price of $50

30%

origi

OFF

nal p

rice

4

A manager has decided to award a salary increase of 6% for all employees. Find the new salary awarded on the following amounts. a Salary of $46 240 b Salary of $94 860 c Salary of $124 280 d Salary of $64 980

5

Molly has a card that entitles her to a 2.5% discount at the store where she works. How much will she pay for the following items? a Vase marked at $190 b Cutlery marked at $240 c Painting marked at $560 d Pot marked at $70

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Development 6

A used car is priced at $18 600 and offered for sale at a discount of 15%. a What is the discounted price of the car? b The car dealer decides to reduce the price of this car by another 15%. What is the new price of the car?

7

Find the repeated percentage change on the following. a Increase $100 by 20% and then decrease the result by 20%. b Increase $280 by 10% and then increase the result by 5%. c Decrease $32 by 50% and then increase the result by 25%. d Decrease $1400 by 5% and then decrease the result by 5%. e Increase $960 by 15% and then decrease the result by 10%. f Decrease $72 by 12.5% and then increase the result by 33 13 % .

8

An electronic store offered a $30 discount on a piece of software marked at $120. What percentage discount has been offered?

9

The cost price of a sound system is $480. Retail stores have offered a range of successive discounts. Calculate the final price of the sound system at the following stores. a Store A: Increase of 10% and then a decrease of 5% b Store B: Increase of 40% and then a decrease of 50% c Store C: Increase of 25% and then a decrease of 15% d Store D: Increase of 30% and then a decrease of 60%

10

The price of a clock has been reduced from $200 to $180. a What percentage discount has been applied? b Two months later the price of the clock was increased by the same percentage discount. What is new price of the clock?

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Chapter 3 — Units of measurement and applications

Units of measurement

× 1000 × 1000 × 100 × 10

mega kilo unit centi milli

÷ 1000

× 24

÷ 1000

× 60

÷ 100

× 60

÷ 10

Study guide 3

days hours minutes seconds

÷ 24 ÷ 60 ÷ 60

10 000 cm2 = 1 m2 1 ha = 10 000 m2 1 000 000 cm3 = 1 m3 Writing numbers in scientific notation

Writing numbers in significant figures

1

Find the first two non-zero digits. 2 Place a decimal point between these two digits. 3 Power of ten is number of the digits between the new and the old decimal point. (Small number – negative value, Large number – positive value)

1

Write the number in scientific notation. 2 Count the digits in the number to determine its accuracy. 3 Round the number to the required significant figures.

Ratios

A ratio is used to compare amounts of the same units in a definite order. Equivalent ratios are obtained by multiplying or dividing by the same number.

Unitary method

1

Converting a rate

1

Percentage change

1

Find one unit of an amount by dividing by the amount. 2 Multiply the result in step 1 by the number. Write the rate as a fraction. First quantity is the numerator and 1 is the denominator. 2 Convert the first amount to the required unit. 3 Convert the second amount to the required unit. 4 Simplify the fraction. Add the % increase or subtract the % decrease from 100%. 2 Multiply the above percentage by the amount.

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Review

Chapter summary – Units of measurement and applications

99

Review

100

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

2

3

4

Convert 7.5 metres to millimetres. A 0.0075 mm B 75 mm

C

D

7500 mm

How many square millimetres are in a square centimetre? A 10 B 100 C 1000

D

10 000

Write 4 500 000 in scientific notation. A 4.5 × 10−6 B 4.5 × 10−5

D

4.5 × 106

D

0.066

C

750 mm

4.5 × 105

Express 0.0655 correct to two significant figures. A 0.06 B 0.07 C 0.065

5

The ratio of adults to children in a park is 5:9. How many adults are in the park if there are 630 children? A 70 B 126 C 280 D 350

6

A 360 gram lolly bag is divided in the ratio 7:5. What is the mass of the smaller amount? A 150 g B 168 g C 192 g D 210 g

7

A hose fills a 10 L bucket in 20 seconds. What is the rate of flow in litres per hour? A 0.0001 B 30 C 1800 D 7200

8

Which of the following is the slowest speed? A 60 km/h B 100 m/s C 10 000 m/min

D

6000 m/h

9

The concentration of a drug is 3 mL/g. How many mL are required for 30 g? A 0.1 mL B 10 mL C 27 mL D 90 mL

10

What is the new price when $80 is increased by 20% then decreased by 20%? A $51.20 B $76.80 C $80.00 D $115.20

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101

1

There are six tonnes of iron ore in a train. What is the mass (in tonnes) if another 246 kg of iron ore is added to the train?

2

Complete the following. a

3

b c

5

m2

b

4000 cm 2 =

mm 2

c

3 km 2 =

m2

A field has a perimeter of exactly 400 m. Lily measured the field to be 401.2 m using a long tape marked in 0.1 m intervals.

a

4

500 cm 2 =

Calculate the limit of reading. What is the absolute error for Lily’s measurement? What is the percentage error for Lily’s measurement? Answer correct to three decimal places.

Write each of the following as a basic numeral. a 4.8 × 106 b 6.25 × 10−4

c

1.9 × 102

Write these numbers in scientific notation. a 50 800 b 0.0036

c

381 000 000

6

Evaluate the following and express your answer in scientific notation. 4.6 × 10 4 a (7.2 × 105) × (2.1 × 104) b 2.3 × 10 −2

7

Convert a measurement of 3580 tonnes into milligrams. Express your answer in scientific notation correct to two significant figures.

8

Find the value of 45 × 154 and express your answer in scientific notation correct to two significant figures.

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Sample HSC – Short-answer questions

Review

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Preliminary Mathematics General

9

Simplify the following ratios. a 500:100 d 10:15:30 g

10

e h

20:30 12:9 3 1 : 4 2

c f

28:7 56:88

A 5 kg bag of rice costs $9.20. What is the cost of the following amounts?

a d 11

4.8:1.6

b

10 kg 7 kg

b e

40 kg 500 g

Convert each rate to the units shown. a $15/kg to $/g c 120 cm/h to mm/min e 14 L/g to mL/kg

c f

b d f

3 kg 250 g

14 400 m/h to m/min 4800 kg/g to kg/mg $3600/g to c/mg

12

A car travels 960 km on 75 litres of petrol. How far does it travel on 50 litres?

13

Daniel and Ethan own a business and share the profits in the ratio 3:4. a The profit last week was $3437. How much does Daniel receive? b The profit this week is $2464. How much does Ethan receive?

14

Jill has a shareholder card that entitles her to a 5% discount at a supermarket. How much will she pay for the following items? Answer to the nearest cent. a Breakfast cereal at $7.60 b Milk at $4.90 c Coffee at $14.20 d Cheese at $8.40

15

An electrician is buying a light fitting for $144 at a hardware store. He receives a clearance discount of 15% then a trade discount of 10%. How much does the electrician pay for the light fitting? Challenge questions 3

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C H A P T E R

4

Statistics and society, data collection and sampling Syllabus topic — DS1 Statistics and society, data collection and sampling Recognise the process of statistical inquiry Appreciate the role of statistical methods in quality control Classify data as quantitative or categorical Distinguish between random, stratified and systematic samples Design an effective questionnaire

4.1 Statistical inquiry Statistical inquiry is a process of gathering statistics that involves six steps: posing questions, collecting data, organising data, displaying data, analysing data and writing a report. The information gained from a statistical inquiry is a vital part of our society. Many people believe that information is more important than the natural resources as a source of social and economic power.

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Collecting data Collecting data involves deciding what to collect, locating it and collecting it. Data comes from either primary or secondary sources. • Primary sources – interviewing people, conducting questionnaires or observing a system in operation • Secondary sources – data collected or created by someone else such as information gathered from newspapers, books and the internet. It is important that procedures are in place to ensure the collection of data is accurate, up to date, relevant and secure. If the data collected comes from unreliable sources or is inaccurate, the information gained from it will be incorrect.

Organising data Organising data is the process that arranges, represents and formats data. It is carried out after the data is collected. The organisation of the data depends on the purpose of the statistical inquiry. For example, to store and search a large amount of data, the data needs to be categorised. Frequency tables are used organise ungrouped and grouped data. Organising gives structure to the data.

Summarising and displaying data Displaying data is the presentation of the data and information. Information must be well organised, readable, attractively presented and easy to understand. Information is often displayed using graphs such as dot plots, sector graphs, histograms, line graphs, stem-and-leaf plots and box-and-whisker plots. Data is summarised using statistics such as the mean, median, mode and standard deviation.

Analysing data Analysing data is the process that interprets Statistical inquiry data and transforms it into information. 1 Pose questions. It involves examining the data and giving 2 Collect the data. meaning to it. When data has been ordered 3 Organise the data. and given meaning by people, it is called 4 Summarise and display the data. information. The particular type of analysis 5 Analyse the data and draw conclusions. depends on the format of the data and the 6 Write a report. information that is required. Graphs are used to analyse the data. They make it easy to interpret data by making instant comparisons and revealing trends. Graphs help people to make quick and accurate decisions. ISBN: 9781107627291 © The Powers Family Trust 2013 Photocopying is restricted under law and this material must not be transferred to another party

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Example 1

105

Case study of a statistical inquiry

James is an accountant working for a large retail company. He has been asked to complete a statistical inquiry to reduce the company’s phone costs. James identified seven departments that often use the phone. James performed the following steps: 1 Collecting data — James accessed the Department Number of calls latest phone bill. Automotive 3450 2 Organising data — James categorised the Gardening 2804 phone data into the seven departments. 3 Summarising and displaying data — Hardware 4320 James presented the data in the table that Jewellery 4506 is shown opposite. Kitchen 2567 4 Analysing data — Phone costs will have to be paid by each department. Ladies wear 3633 James knows there will be a rise in the Men’s wear 3760 cost of calls. He wants to make some projections of the increase to the phone budget and calculate the average amount that each department can spend on calls. James produced the spreadsheet shown below.

5

The spreadsheet has formulas in cells C7:C17. The formula entered in cell C7 is =$B$4*B7. To analyse and draw conclusions James was able to modify the cost of the call in cell B4 and observe the changes to the cost. Writing a report — James wrote a report that included the effects of a 10% increase in the call costs and the effect on each of the seven departments.

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Exercise 4A 1

Copy and complete the following sentences. a Statistical is a process of gathering statistics that involves posing questions, collecting, organising, displaying, analysing and writing a report. b Data comes from either or secondary sources. c data is the process that arranges, represents and formats data. d Information must be well organised, , attractively presented and easy to understand. e Data that has been ordered and given some meaning by people is called . f help people to make quick and accurate decisions.

2

True or false? a Interviewing people is a secondary source of information. b Data collected from unreliable sources results in incorrect information. c Information is often displayed using graphs. d Data is summarised using statistics such as dot plots, sector graphs, histograms, line graphs, stem-and-leaf plots and box-and-whisker plots. e Analysing data is the process that interprets data and transforms it into information.

3

List the six steps involved in a statistical inquiry.

4

Why is information gained from a statistical inquiry a vital part of our society?

5

Explain the difference between primary and secondary sources.

6

What is the purpose of frequency tables?

7

List some of the types of graphs used to display data.

8

Why are graphs used to analyse data?

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Development 9

Privacy is the ability of an individual to control their personal data. Organisations are collecting a huge amount of data about our personal lives and attitudes on various issues. Every time you fill out a form, use a transaction card, or ‘surf the Net’, data is collected. It is possible for people to access this data and combine it. This combined information would provide a very accurate picture of you. a Why would your personal data be of great value to retailers and advertising people? b How would you feel if a person accessed your application to build a house and sold it to a bricklayer? c Are you concerned about receiving email from an unknown organisation? How would you feel if this organisation paid to receive your email address? d Why must you consider issues of privacy and ethics when collecting statistical data?

10

List some of the information that has been collected on you. How can you check whether this information is accurate? What are your privacy rights? Explore the Office of the NSW Privacy Commissioner (www.privacy.nsw.gov.au) to answer these questions.

11

The Data-matching Act permits certain agencies to check records held by different government departments, such as the tax office and the departments responsible for social security, employment and education. It aims to catch people who are cheating the welfare system. a Do you think this is an invasion of privacy? Give a reason. b Is the Data-matching Act benefiting our society? Give a reason.

12

Why is it important to consider how the questions are asked when conducting a questionnaire for a statistical inquiry?

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4.2 Classification of data There are many types of data that can be collected. Data is divided into two broad classifications called quantitative and categorical. 4.2

Quantitative data Quantitative data is numerical data. For example, if we asked each student in the class their height we would expect to get a variety of answers. However, each answer is a number. Quantitative data is further classified as discrete or continuous. •

•

Discrete data – data that can only take exact numerical values. For example, the number of sisters will give rise to the numbers such as 0, 1 or 2. Counting a quantity often results in discrete data. Continuous data – data that can take any numerical value (sometimes within specified interval). For example, a student’s height will give rise to numbers such as 171.2 cm and 173.5 cm. Measuring a quantity often results in continuous data.

Categorical data Categorical data is data that can be divided into categories. It uses labels not numbers. Categorical data is further classified as nominal or ordinal. • Nominal data – uses a name or label that does not indicate order. For example, a student’s gender could be classified as an ‘F’ for females and an ‘M’ for males. • Ordinal data – uses a name or label that does indicate order. For example, the quality of work could be classified as an ‘A’ for excellent, ‘B’ for good or ‘C’ for satisfactory. It shows a sequence A, B and C. Categorical data has no quantity or amount associated with each category.

Classification of data 1

Quantitative data – is numerical data. a Discrete data – data that can only take exact numerical values b Continuous data – data that can take any numerical value. 2 Categorical data – classified by the name of the category it belongs to. a Nominal data – name does not indicate order b Ordinal data – name does indicate order.

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Example 2

109

Classifying data as a categorical or quantitative

Classify the data from these situations as quantitative or categorical. a The heart rate of a group of personal trainers b The most watched television show in Australia c The number of people living in Smith Ave d The reasons for people travelling to work by train. Solution 1

2 3

4

The heart rate, such as 70 beats per minute, can be measured and results in a number. A television show, such as the news, does not result in a number. The number of people living in Smith Ave, such as 27, can be counted and results in a number. The reason for travelling to work by train, such as it is cheaper, does not result in a number.

Example 3

a

The heart rate is quantitative data.

b

A television show is categorical data. The number of people living in Smith Ave is quantitative data.

c

d

The reasons for travelling to work is categorical data.

Classifying data as a discrete or continuous

Classify the following quantitative data as discrete or continuous. a The number of pets in your family b The perimeter of the school. Solution 1

The number of pets can be counted and is exact.

a

2

The perimeter of the school is a measurement of distance and assumes a value.

b

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T he number of pets is discrete data. T he perimeter of the school is continuous data.

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Exercise 4B 1

Classify the data from these situations as quantitative or categorical. a The favourite colour of Jenny’s friends b The number of people travelling in a car c Each student in the year is weighed in kilograms d People rating their doctor on personal service (high, medium or low) e The number of students in each class f The IQ of a group of students g Responses to a survey question (agree or disagree) h A person’s lucky number i A female’s favourite mobile phone j Distance from Sydney to Wollongong k The cost of bread at the supermarket l The community’s preferred leader m The number of computers in the school.

2

Classify the following quantitative data as discrete or continuous. a The price paid for a can of soft drink

b c d e f g h i j k

The number of people at a concert The time between trains The number of pages in the newspaper The amount of water used in the past month The number of people in your immediate family The numbers drawn in this week’s lotto The length of the cricket pitch The distance measured for the long jump at the world championships The score achieved from a quiz consisting of 10 questions The height of the tallest person in the world.

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3 

111

State whether the following is categorical, discrete or continuous data. a The heights of members of a football team b The distance to drive to the train station

c d e f g h i j k l m n o p

The different types of ice creams The quality of food in a restaurant The eye colour of a group of people The number of pets in a household The time to swim 50 metres The number of goals scored in the first match of the season Today’s most fashionable style of dress The number of computers in the building Replies given to a questionnaire (Yes or No) The perimeter of Joel’s block of land The width of the Anzac Bridge The number of people killed on the roads due to speed The stopping distance for a car travelling at 60 km/h The most popular type of car sold in the past twelve months.

4

The hospital measures the weight and length of every new baby. a Classify the data as quantitative or categorical. b Is this data discrete or continuous?

5

A coffee shop is conducting a survey on the drinking habits of its customers. One of the questions was: ‘How many cups of coffee do you drink each day?’ a State whether the data is quantitative or categorical. b Is this data discrete or continuous?

6

The government collected data on their latest policy proposal. The people surveyed answered ten questions and were given three choices for each question: Agree, Disagree or Not sure. What type of data has the government collected?

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Development 7

In a survey customers were asked to rate the service they received by selecting one of the following: outstanding, excellent, good, satisfactory or needs improvement

Describe the type of data that would result from this question. 8

A teacher marks a class’s assessment task and awards a mark out of 100 for each student. Describe the type of data that that has been collected by the teacher.

9

A marketing poll was conducted that asked about a person’s employment status: unemployed, receiving education, part-time job or full-time job

Describe the type of data that would result from this poll. 10

Emma is planning to build a new restaurant. She conducted a survey of the community. One of the questions asked was ‘How far in kilometres would you be prepared to travel to get to a good restaurant?’ The options given were: 5 km, 10 km, 20 km, 50 km

Describe the type of data that would result from this question. 11

The local community recorded the amount of rainfall each day for the past 3 months. Some of the data is shown below. Rainfall 23rd April

10 mm

24th April

0 mm

25th April

25 mm

Describe the data collected by the local community.

12

The police department collected data on fatal crashes. One of the questions it asked was: ‘What was the age of the driver involved in a fatal crash?’ a Describe the type of data collected by the police department. b The question was modified to give the interviewee six choices: 17–24 yrs, 25–34 yrs, 35–44 yrs, 45–54 yrs, 55–64 yrs, 65+ yrs

Describe the type of data that would result from these choices.

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4.3 Sample types

4.3

Collecting data from every person in a population is called a census. However, a census may be costly in terms of money and time. For this reason, samples are taken from a population and estimates made about the population are based on the sample. A sample is only part of a population. For example, if all the students in your school are regarded as the population then a sample of this population is the students in your class. A sample must be large enough to give a good representation of the population, but small enough to be manageable. There are many different types of sampling including a random sample, stratified sample and systematic sample.

Random sample A random sample occurs when members of the population have an equal chance of being selected. For example, six students are selected at random from the entire school population. Lotto is another good example of random sampling. A sample of 6 numbers is chosen from 40 numbers. Random samples are simple and easy to use for small populations. However, for large populations, it is possible to miss out on a particular group.

Stratified sample A stratified sample occurs when categories or strata of a population are chosen and then members from each category are randomly selected. For example, one student is selected from each year 7, 8, 9, 10, 11 and 12. Each year group is a category in a stratified sample. Some other common types of categories are age, sex, religion or marital status. A stratified sample is useful when the categories are simple and easy to determine. However, care needs to be taken when selecting categories to avoid any bias in the data.

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Systematic sample Systematic sample occurs when the population is divided into a structured sample size. For example, the students in the school population are put in alphabetical order and the 100th student, 200th student, 300th student, … are selected. A systematic sample is often used by a manufacturer to ensure the machines are working correctly. Here the manufacturer might test a machine every 30 minutes or check the 50th item on a production line. Systematic sampling results in a gap between each selection.

Random sample

Stratified sample

Systematic sample

Members of the population have an equal chance of being selected.

Categories of a population are chosen. Members then are randomly selected from each category.

Population is divided into a structured sample size. Members are then selected in a certain order from this structure.

Example 4

Distinguishing sample types

A retirement village has 63 residents, 42 women and 21 men. Decide whether each sample of resident would be random, stratified or systematic. a Every seventh resident b Six of the women and three of the men c Nine names picked from a hat containing the names of the residents. Solution 1 2 3

The population has been divided into a structured sample size – 7th, 14th, 21st, … 63rd The population has been divided into categories – women and the men. Sample is taken at random.

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a

S ystematic sample

b

S tratified sample

c

R andom sample.

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115

Exercise 4C 1

State whether a census or a sample is the most appropriate way to collect this data. a Information on the shopping experience of people in the city b John collecting the height of his best friends c The travelling habits of the Jones family to work d Australians watching the grand final e Number of people eating toast for breakfast f Length of time every AAA battery lasts g Number of people entering a gym between 5 p.m. and 6 p.m. h Holly collecting the amount of time spent on the internet by her class i The world’s reaction to climate change j Shop manager’s reaction to a drop in sales.

2

State whether the sample is random, stratified or systematic. a Police officer breathalysing every tenth person b Each person is given a raffle ticket and the tickets are drawn out of a hat c Twenty people aged under 30 and twenty people aged over 30 travelling on a bus d A business has 240 married and 120 unmarried employees. A sample was chosen to include 10 of the married and 5 of the unmarried employees. e Students were sorted into alphabetical order and each third student selected. f Individuals were randomly selected using their tax file number. g Every 12th jogger selected from an alphabetical list h Randomly selecting 10 cards from a normal deck of cards i Ten girls and ten boys randomly selected from a concert j Ten people who arrive at a shopping centre each day completed the survey.

3

Michael uses a random sample to survey 10% of the local community. In the local community there are 810 males and 920 females. How many people does Michael need to survey?

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4

Amelia plans to conduct a random sample to survey the netball players in the local association. There are 2850 players in the local association and she plans to survey 171 players. What percentage of the population is her sample?

5

Paige uses a stratified sample to survey 10% of her school population. At the school there are 80 teachers and 1160 students. a How many people should complete the survey? b How many teachers should complete the survey? c How many students should complete the survey?

6

Tyler uses a stratified sample to survey 25% of his swimming club. He uses their sex as a category and selects a random group of female and male swimmers. There are 88 female swimmers and 112 male swimmers. a How many swimmers are in the entire population? b How many female swimmers are in the sample? c How many male swimmers are in the sample?

7

Osman uses a stratified sample to survey 7.5% of his chat room friends. He uses marital status as a category and selects a random group of married and unmarried friends. There are 200 married and 240 unmarried friends. a How many friends are in the entire population? b How many married friends are in the sample? c How many unmarried friends are in the sample?

8

Taylia uses a stratified sample to survey 20% of the senior students from her school. There are 205 year 11 students and 180 year 12 students. How many students should Taylia choose from year 12?

9

Ming uses a stratified sample to survey 12 1 % of the junior students from his school. 2 There are 88 year 7, 120 year 8, 104 year 9 and 128 year 10 students. a How many students are in the entire population? b How many students should Ming choose in following years? i Year 7 ii Year 8 iii Year 9 iv Year 10

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117

Development 10

A survey was conducted in a school on whether Australia should remain a constitutional monarchy or become a republic. The results are shown below. Male

Female

Total

Republic

51

  79

130

Monarchy

23

  47

  70

Total

74

126

200

a b c

How many males surveyed did not prefer Australia to change to a republic? What percentage of people is in favour of changing to a republic? This survey is not a good random sample of all Australians. Why?

11

A sample of 30 students is taken from a primary school with an enrolment of 420 students from kindergarten to year 6. The sampling is designed so that the same proportion of each year of the sample matches the population. There are 4 students from year 1 in the sample. How many year 1 students are there in the school population?

12

A store has 400 employees of which 208 are females and 192 are males. The store intends to survey 25 of their employees. A stratified survey is to be conducted. a How many females should be surveyed? b How many males should be surveyed?

13

Identify any possible issues with each of the following survey questions. a Do you like the government’s new policy? Yes/No b Alan is a lazy boss who should be forced to pay his diligent workers more money. Agree or disagree?

14

Kayla surveyed a group of 15 people at the Tamworth country music festival on their music preferences. She used this data to make judgements for the entire population of NSW. a Do you think her judgements will be accurate? Give a reason. b What would be a more appropriate method of sampling music preferences?

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4.4 Designing a questionnaire A questionnaire or survey is a series of questions to gather specific information. The interviewee may be given time to complete the questionnaire or it could be carried out faceto-face or online. Questionnaire may contain open ended and closed questions. Questionnaires allow data to be collected from a large number of people quickly and with little expense. There are a number of principles for effective questionnaire design such as: • simple language – questions are easily read and understood by the interviewee • unambiguous questions – questions can only be interpreted one way • respect for privacy – privacy is the ability of an individual to control their personal data • freedom from bias – biased data is unfairly skewed. For example, ‘Have you stopped cheating in exams? Yes/No’ assumes the interviewee is a cheat.

Designing a questionnaire 1 2 3 4

Use simple language. Make the questions unambiguous. Respect the privacy of the interviewee. Ensure the questionnaire is free from bias.

Data must be carefully interpreted to ensure that the resulting information is valid. For example, can the results of a questionnaire be generalised to a larger group of people? The reliability of the data is also an issue. If a similar research were conducted at another time and place would the results be the same? It is important that the information gained from the questionnaire be accurate. After the data is collected and checked for accuracy it must be carefully analysed. Entering the data into a table is a good way to examine the data. It enables totals to be easily calculated and comparisons between different results to be made.

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Exercise 4D 1

Examine the questionnaire below. Questionnaire – Carter’s Place Please take a minute to complete the survey and put it on the counter.

How often have you eaten here?  Never before  1–4 times  5–9 times  More than 10 times

How do you rate the staff?  Excellent  Good  Average  Disappointing

How do you rate the food?  Excellent  Good  Average  Disappointing

How do you rate our service?  Excellent  Good  Average  Disappointing

Are the meals reasonably priced?  Yes  No

What features would bring you back to Carter’s Place?  Menu  Wine list  Service  Price  Open late  Atmosphere

Do you like our music?  Excellent  Good  Average  Disappointing Thank you for completing this survey. a b c d e f g

What is the structure of the questionnaire? What type of data would be collected from the questionnaire? How many are closed questions? How many are open ended questions? Design an additional question focused on the decorations. Design an additional question focused on the location. Design an additional question focused on improving the menu.

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Development 2

A pizza shop is having problems meeting its costs. It needs to decide whether it should increase or decrease its prices. a Estimate or visit a local pizza shop to obtain a list of current price structures and types of pizzas.

Create a questionnaire that could be used to estimate the effects of different prices on revenue and the types of pizzas offered. c Survey students on their pizza preferences and use this information to make a recommendation to the owner of the pizza shop. 3 A fitness centre needs to attract more members. It would like to encourage people to live a healthier lifestyle. a Estimate or visit a local fitness centre to obtain a list of the services that are offered and the current price structures. b Create a questionnaire that could be used to estimate the effects of different services and price structures. c Survey students on their fitness needs and use this information to make a recommendation to the manager of the fitness centre. b

4

Harley is unhappy with her mobile phone plan. She is wondering whether other people have similar issues. a Investigate a range of mobile phone planes that are currently available. Gather data on the rates and charges being offered. b Use the above data to create a survey on people’s opinion of their mobile phone plane and their current rates and charges. c Take a sample of students in your school and conduct the above survey.

The following question is to be included in a survey about the local community: ‘Why do you like your local member of parliament?’ Is this a good question? Justify your answer.

5 

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Chapter 4 — Statistics and society, data collection and sampling

Statistical inquiry

1 2 3 4 5 6

Classification of data

Pose questions. Collect the data. Organise the data. Summarise and display the data. Analyse the data. Write a report.

Quantitative (Numbers)

Discrete Continuous

Sample types

Study guide 4

Data that can only take exact numerical values such as 0 or 1. Data that can take any numerical value such as 71.25.

Categorical (Category)

Classified by the name of the categories.

Random sample

Members of the population have an equal chance of being selected.

Stratified sample

Categories of a population are chosen such as male/ female. The members are randomly selected from each category.

Systematic sample

Population is divided into a structured sample size. The members are orderly selected in a certain order from this structure, such as each third person in alphabetical order.

Designing a

1

questionnaire

2 3 4

Nominal Ordinal

Name does not indicate order. Name does indicate order.

Use simple language. Make questions unambiguous. Respect the privacy of the interviewee. Ensure the questionnaire is free from bias.

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Review

Chapter summary – Statistics and society, data collection and sampling

121

Review

122

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

A step in a statistical inquiry that arranges, represents and formats data. A Analysing B Collecting C Displaying D Organising

2

A step in a statistical inquiry that interprets and transforms data into information. A Analysing B Collecting C Displaying D Organising

3

Source of data collected or created by someone else. A Primary B Observing C Secondary D Survey

4

Type of data that can only take particular numerical values. A Constant B Continuous C Discrete D Number

5

‘The number of rooms in your house’. What is the classification for this data? A Categorical B Continuous C Discrete D Text

6

‘Do you agree? Yes/No’. What is the classification for this data? A Categorical B Continuous C Discrete D Text

7

Which of the following is an example of collecting data from a school using a census? A Class opinion B All the students’ opinion C School captain’s opinion D Year 11 opinion

8

Which of the following is an example of a random sample? A First ten shoppers in the store B Ten shoppers who used EFT C Ten shoppers drawn from a box D Ten shoppers who spent the most

9

Sorting each person in alphabetical order and selecting every fifth person. How would you describe this type of sample? A Quantitative B Random C Stratified D Systematic

10

Sample designed to include four boys and four girls from a surf club. How would you describe this type of sample? A Quantitative B Random C Stratified D Systematic

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1

True or false? a Data comes from either primary, secondary or tertiary sources. b Analysing data is the process that arranges, represents and formats data. c Information that has been ordered and given some meaning by people is called data. d Data collected from reliable sources results in correct information.

2

Classify the data from these situations as quantitative or categorical. a Most popular student in the class

b c d e f g h i 3

Ava’s favourite beach in Australia The call cost on a mobile phone Blake’s school high jump record The amount of annual leave The hair colour of the students in your class The number of websites accessed in the past 24 hours The digital download time for a 4 MB file The average age of the people living in NSW.

Classify the following quantitative data as discrete or continuous. a The number of televisions in a house b The height of the tree in the local park c The quantity of petrol used on a trip from Bega to Kiama d The number of party pies in a 500 gram pack e The score (0, 1 or 2) on the latest computer game f The shirt size of the boys in year 11 g The amount of water Tran drank during the day h The number of students in your class.

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Sample HSC – Short-answer questions

Review

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Preliminary Mathematics General

Chapter summary – Earning 4 A marketing poll was conductedMoney on a person’s sleeping patterns: Good, Satisfactory, Unsatisfactory

Describe the type of data that would result from this poll.

5

State whether the sample is random, stratified or systematic. a Manufacturer selects every 40th product on the assembly line b An equal number of city and country people are randomly chosen for a survey c Random number generator is used to select 5 people in the class d Selecting every 100th supporter at a football match e Drawing the names of people from a hat f Dividing the participants into people with tertiary qualifications and those with no qualifications. Randomly selecting 20 people from each group g Choosing 5 raffle tickets, each with a person’s name from a container h Sorting the names in alphabetical order and selecting every second name i Grouping people according to the car they own and selecting an equal number of people from each group.

6

Sharif uses a stratified sample to survey 10% of employees. He uses their age as a category. There are 330 employees over 40 years of age and 670 employees under 40. a How many employees should complete the survey? b How many employees over 40 years of age should complete the survey? c How many employees under 40 years of age should complete the survey?

7

A school has 600 students of which 315 are boys and 285 are girls. The school intends to survey 40 of their students. A stratified survey is to be conducted. a How many boys should be surveyed? b How many girls should be surveyed?

8

A cinema session had 84 patrons: 48 women and 36 men. Decide whether a sample of patrons would be random, stratified or systematic. a Four of the women and three of the men b Every sixth patron c Twelve names picked from a hat containing the names of the patrons.

9

The school canteen would like to know the preferred brand of juice to stock. Describe a suitable sample to conduct a survey.

10

The following question is to be included in a survey at a swimming centre: ‘Why do you like the swimming centre?’ Is this a good question? Justify your answer. Challenge questions 4

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C H A P T E R

5

Interpreting linear relationships Syllabus topic — AM2 Interpreting linear relationships Graphing linear functions from everyday situations Calculating the gradient and vertical intercept Using and interpreting graphs of the form y = mx + b Solving simultaneous linear equations from a graph Using linear functions to model and interpret practical situations

5.1 Graphing linear functions A linear function makes a straight line when graphed on a number plane. There are many everyday situations that result in a linear function such as the distance travelled as a function of the time (d = 50t). The graph of d = 50t is shown below. 5.1

Distance travelled

d

Distance (km)

200 150 100 50

1

2 3 Time (h)

4

t

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Independent and dependent variables The linear function d = 50t has two variables d (distance) and t (time). Time (t) is the independent variable, as any number can be substituted for this variable such as t = 1. Conversely, the distance (d) is the dependent variable is it depends on the number substituted for the independent variable. That is, when time is 1 (t = 1) then the distance is d = 50 × 1 or 50. Graphing a linear function 1 2 3

Construct a table of values with the independent variable as the first row and the dependent variable as the second row. Draw a number plane with the independent variable on the horizontal axis and the dependent variable as the vertical axis. Plot the points. Join the points to make a straight line.

Graphing a linear function from a table of values

Example 1

The table below shows the cost of postage (c) as a function of the weight of the parcel (w). Weight (w) Cost (c) a b

1

2

3

4

5

1.2

2.4

3.6

4.8

6.0

Draw a graph of cost (c), against the weight of the parcel (w). Use the graph to determine the cost of a parcel if the weight is 2.5 kg. Solution

2 3 4

5

Draw a number plane with the weight of parcel (w) as the horizontal axis and the cost of postage (c) as the vertical axis. Plot the points (1, 1.2), (2, 2.4), (3, 3.6), (4, 4.8) and (5, 6.0). Join the points to make a straight line. Find 2.5 kg on the horizontal axis and draw a vertical line. Where this line intersects the graph, draw a horizontal line to the vertical axis. Write the answer in words.

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a

c Cost of postage Cost of postage in $

1

b

6 5 4 3 2 1 w 1 2 3 4 5 Weight of parcel in kg

2.5 kg would cost about $3.

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Example 2

Graphing a linear function

Draw the graph of y = 2x - 1. Solution 1 2

3 4 5

Draw a table of values for x and y. Let x = -2, -1, 0, 1 and 2. Find y using the linear function y = 2x - 1.

x

-2

-1

0

1

2

y

-5

-3

-1

1

3

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points (-2, -5), (-1, -3), (0, -1), (1, 1) and (2, 3). Join the points to make a straight line.

y 3 2 1 −3 −2 −1 0 −1

1

2

3

x

−2 −3 −4 −5

Example 3

Graphing a linear function using a graphics calculator

Use a graphics calculator to draw the graph of y = 2x - 1. Solution

5

Select the Graph menu. Enter the formula y = 2x - 1 by typing 2X–1 at Y1. The graph of Y1 = 2X–1 is the same as y = 2x - 1. Edit the axes to an appropriate scale. Select SHIFT F3 for the V-Window. Enter the Xmin = –3, Xmax = 3 , Ymin = –5, Ymax = –3. Press EXE to exit V-Window.

6

Select F6 to draw the graph.

1 2 3 4

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Exercise 5A 1

Chocolates are sold for $12 per kg. The table below shows weight against cost. Weight (w)

1

2

3

4

5

Cost (c)

12

24

36

48

60

Which is the dependent variable? Draw a graph of weight against cost.

a c

2

Time (t)

1

2

3

4

5

6

Cost (c)

0.20

0.40

0.60

0.80

1.00

1.20

Which is the dependent variable? Draw a graph of time against cost.

c

b d

Mass (m)

3

6

9

12

15

Time (t)

2.2

3.7

5.2

6.7

8.2

Draw a graph of mass against time.

b

Use the graph to find t if m is 10.

Complete the following table of values for each linear function. y=x+1

a

x

0

b

1

2

3

4

a

0

2

4

6

8

-1

0

1

2

y b = 3a + 4

c

y = 2x x

y

-2

-1

d

0

1

2

b 5

Which is the independent variable? Use the graph to find t if c is 0.90.

Soraya conducted a science experiment and presented the results in a table.

a

4

d

Mobile phone call costs are charged at a rate of 20 cents per minute.

a

3

Which is the independent variable? Use the graph to find c if w is 1.5.

b

q = -p + 1 p

-2

q

Use the table of values from the above question to graph these linear functions. a y=x+1 b y = 2x c b = 3a + 4 d q = -p + 1

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Development 6

The cost (c) of apples is $2.50 per kilogram and is determined by the formula c = 2.5w where (w) is the weight of the apples. a Construct a table of values for the weight against cost. Use 0, 1, 2, 3 and 4 for w. b Draw the graph of the weight (w) against the cost (c). c How many kilograms of apples can be purchased for $15?

7

The age of a computer (t) in years to its current value (v) in $100 is v = -5t + 30.

a b c d e

Construct a table of values for the age against current value. (t = 0, 1, 2, 3, 4) Draw the graph of the age (t) against current value (v). What is the initial cost of the computer? What will be the current value of the computer after two years? When will the computer be half its initial cost?

8 

The cost of hiring a taxi is $3 flag fall and $2 per kilometre travelled. a Construct a table of values using 0, 10, 20, 30 and 40 as values for kilometres travelled (d) and calculating cost of the taxi (C). b Draw the graph of the kilometres travelled (d) against cost of taxi (C).

9

Emily was born on Jack’s tenth birthday. a Construct a table of values using 0, 1, 2, 3 and 4 as values for Emily’s age (E) and calculating Jack’s age (J). b Draw the graph of Emily’s age (E) against Jack’s age (J).

10

One Australian dollar (AUD) was converted to 1.20 New Zealand dollars (NZD). a Construct a table of values using 0, 10, 20, 30 and 40 as values for AUD and calculate the NZD using the above conversion. b Draw the graph of the AUD against NZD.

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5.2 Gradient and intercept Gradient 5.2

The gradient of a line is the slope or steepness of the line. It is calculated by dividing the vertical rise by the horizontal run. The larger the gradient, the steeper the slope. The letter m is often used to indicate gradient. Vertical rise Horizontal run

Gradient (or m) =

Vertical r rtical rise Horizontal run

Positive gradients are lines that go up to the right or are increasing. Conversely, negative gradients are lines that go down to the right or are decreasing.

+

−

Positive gradient

Example 4

Negative gradient

Finding the gradient of a line

Find the gradient of a line through the points (1, 1) and (3, 4). Solution 1 2 3 4

5 6 7 8

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points (1, 1) and (3, 4). Draw a line between the two points. Construct a right-angled triangle by drawing a vertical and a horizontal line. The line is positive as it slopes towards the right. Determine the vertical rise (4 – 1 = 3). Determine the horizontal run (3 – 1 = 2). Substitute 3 for the vertical rise and 2 for the horizontal run into the formula.

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y

(3, 4)

4 3

Vertical rise

2 1 0

(1, 1) Horizontal run 1

2

3

4

x

Vertical r rtical rise Horizontal run 3 =+ 2

Gradient or m =

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Intercept The intercept of a line is where the line cuts the axes. The intercept on the vertical axis is called the y-intercept and is denoted by the letter b. The intercept on the horizontal axis is called the x-intercept and is denoted by the letter a. Gradient

Intercept

Gradient of a line is the slope of the line. The intercept of a line is where the line cuts the axes. Vertical intercept is often denoted by b. Vertical r rtical rise Gradient (or m) = Horizontal run

Example 5

Finding the gradient and vertical intercept

Find the gradient and vertical intercept for the line y = -2x + 1. Solution 1 2

3 4 5 6

7 8 9 10 11 12

Draw a table of values for x and y. Let x = –1, 0 and 1. Find y using the linear function y = -2x + 1. Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points (–1, 3), (0, 1) and (1, –1). Draw a line between these points. Construct a right-angled triangle by drawing a vertical and a horizontal line. The line is negative as it slopes towards the left. Determine the vertical rise (3 – 1 = 2). Determine the horizontal run (0 – –1 = 1). Substitute 2 for the vertical rise and 1 for the horizontal run into the formula. Evaluate. The line cuts the vertical axis at 1.

x

-1

0

1

y

3

1

-1

y 3 Vertical rise

2

1 Horizontal run 0 −2 −1

Vertical intercept

1

2

−1

Vertical rise Horizontal run 2 =− 1 = −2

Gradient or m =

Intercept on the vertical axis is 1.

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x

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Exercise 5B 1

Find the gradient of the following lines. y

a

y

b

4

2

3

1

2

−2 −1 0 −1

1 0

1

2

c

3

x

4

d

y

8

16

4

12 1

x

2

x

2

−2

y

−2 −1 0 −4

1

8 4

−8

0

1

2

3

x

4

2

What is the gradient of the line that joins these points? a (0, 1) and (2, 5) b (1, 3) and (2, –2) c (2, –1) and (4, –2)

3

What is the intercept on the vertical axis for the following lines? y

a

y

b

4

4 3

2

2 1 −3 −2 −1 0 −1

1

2

3

x

−4

−2

0

2

4

x

−2 −4

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133

Development 4

Plot the following points on a number plane and join them to form a straight line. Determine the gradient and y-intercept for each line. a

c

5

x

0

1

2

3

4

y

2

4

6

8

10

x

0

2

4

6

8

y

-1

1

3

5

7

b

d

x

-2

-1

0

1

2

y

3

1

-1

-3

-5

x

0

3

6

9

12

y

0

1

2

3

4

Draw a graph of these linear functions and find the gradient and y-intercept. a d

y=x+3 2 y=− x−3 3

b

y = -x + 1

c

e

y + 3 = 4x

f

1 x +1 2 2x - y = 0 y=

6

The distance (d) a train travels in kilometres is calculated using the formula d = 150t where (t) is the time taken in hours. a Construct a table of values using 0, 1, 2, 3 and 4 as values for t. Calculate the distance (d). b Draw the graph of the distance (d) against the time (t). c What is the gradient of the graph? d What is the intercept on the vertical axis?

7

Meat is sold for $16 per kilogram. a Construct a table of values using 0, 1, 2, 3 and 4 as values for the number of kilograms (n). Calculate the cost (c) of the meat. b Draw the graph of the cost (c) against the number of kilograms (n). c What is the gradient of the graph? d What is the intercept on the vertical axis?

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5.3 Gradient–intercept formula

5.3

When the equation of a straight line is written in the form y = mx + b it is called the gradient–intercept formula. The gradient is m, the coefficient of x, and the y-intercept is b, the constant term. The independent variable in the formula is x and the dependent variable in the formula is y. The gradient–intercept formula is useful in modelling relationships in many practical situations. However, the variables are often changed to reflect the situation. For example, the formula c = 25n + 100 has c as the cost of the event ($) and n as the number of guests. These letters are the dependent and independent variables.

Gradient–intercept formula Linear equation – y = mx + b. m – Slope or gradient of the line (vertical rise over the horizontal run). b – y-intercept. Where the line cuts the y-axis or vertical axis.

Example 6

Finding the gradient and y-intercept from its equation

Write down the gradient and y-intercept from each of the following equations. a y = −2 + 5x b y=8−x c y = 6x d y − 3x = 4 Solution 1 2 3 1 2 3

Write the equation in gradient–intercept form. Gradient is the coefficient of x. y-intercept is the constant term.

a

y = −2 + 5x y = 5x − 2 Gradient is 5, y-intercept is –2

Write the equation in gradient–intercept form. Gradient is the coefficient of x. y-intercept is the constant term.

b

y=8−x y = −1x + 8 Gradient is –1, y-intercept is 8

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1 2 3 1 2 3

Write the equation in gradient– intercept form. Gradient is the coefficient of x. y-intercept is the constant term.

c

y = 6x y = 6x + 0 Gradient is 6, y-intercept is 0

Write the equation in gradient– intercept form. Gradient is the coefficient of x. y-intercept is the constant term.

d

y − 3x = 4 y = 3x + 4 Gradient is 3, y-intercept is 4

135

Sketching graphs of linear functions Sketching a straight-line graph requires at least two points. When an equation is written in gradient–intercept form, one point on the graph is immediately available: the y-intercept. A second point can be quickly calculated using the gradient or by substituting a suitable value of x into the equation.

Example 7

Sketching a straight-line graph from its equation

Draw the graph of y = 3x + 1. Solution 1 2 3 4 5 6

7 8

Write the equation in gradient–intercept form. Gradient is the coefficient of x or 3. y-intercept is the constant term or 1. Mark the y-intercept on the y-axis at (0, 1). Gradient of 3 (or 3 ) indicates a vertical rise 1 of 3 and a horizontal run of 1. Start at the y-intercept (0, 1) and draw a horizontal line, 1 unit in length. Then draw a vertical line, 3 units in length. The resulting point (1, 4) is a point on the required line. Join the points (0, 1) and (1, 4) to make the straight line.

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y = 3x + 1 Gradient is 3, y-intercept is 1 y (1, 4)

4 3

Rise = 3

2 (0, 1) −2

−1

1 0 −1

Run = 1 1

2

x

Cambridge University Press

136

Preliminary Mathematics General

Exercise 5C 1

Write down the gradient and y-intercept from each of the following equations. a y = 4x + 2 b y = 3x - 7 c y = 5x + 0.4 d y = 1.5x - 2 1 e y= x+3 f y = 5 - 3x 2 g y=x h y = 2 + 5x

2

Write down the equation of a line that has: a gradient = 3 and y-intercept = 2 c gradient = –4 and y-intercept = –1

3

d

Find the equation of the following line graphs. y

a

y

b

3

2

2

1

1 −2 −1 0 −1

1

2

3

−2 −1 0 −1

x

x

2

y

d

12

1 −2 −1 0 −1

1

−2

y

c

1

2

3

x

−2 −3

4

gradient = –2 and y-intercept = 10 gradient = 0.5 and y-intercept = 1

b

8 4 1

2

3

4

x

A straight line has the equation y = 2x + 3. a What are the gradient and the y-intercept? b Sketch the straight line on a number plane using the gradient and y-intercept.

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Chapter 5 — Interpreting linear relationships

137

Development 5

It is known that y varies directly with x. When x = 4, y = 12. a Write a linear equation in the form y = mx to describe this situation. b Draw the graph of y against x.

6

Kalina’s pay (p) is directly proportional to the number of hours (h) she works. For an 8-hour day she receives $168. a Write a linear equation in the form p = mh to describe this situation. b Draw the graph of p against h.

7

A bike is travelling at constant speed. It travels 350 km in 7 hours. a Write a linear equation in the form d = mt to describe this situation. b Draw the graph of d against t.

8

Sketch the graphs of the following equations on the same number plane. a y = 2x b y = 2x + 1 c y = 2x + 2 d y = -x e y = -x - 1 f y = -x - 2 g What do you notice about these graphs?

9

Sketch the graphs of the following equations on the same number plane. 1 a y=x+1 b y = 3x + 1 c y = x +1 2 d y = -x - 2 e y = -2x - 2 f y = -3x - 2 g What do you notice about these graphs?

10

Sketch the graphs of the following equations using the gradient–intercept formula. 2 1 a y= x+2 b y = 0.25x - 3 c y=2− x 3 3 d y = - 0.5x - 3 e y+x=5 f 4x + y = 8 g

2x + y + 6 = 0

h

x + 4y = 0

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i

3x - y = -3

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138

Preliminary Mathematics General

5.4 Simultaneous equations

5.4

Two straight lines will always intersect unless they are parallel. The point at which two straight lines intersect can be found by sketching the two graphs on the one set of axes and reading off the coordinates at the point of intersection. When the point of intersection is found it is said to be solving the equations simultaneously. Solving two linear equations simultaneously from a graph 1 2 3

Draw a number plane. Graph both linear equations on the number plane. Read the point of intersection of the two straight lines.

Example 8

Finding the solution of simultaneous linear equations

By drawing their graphs find the simultaneous solution of y = 2x + 3 and y = -x. Solution 1

2 3 4 5 6

7

Use the gradient–intercept form to determine the gradient and y-intercept for each line. Gradient is the coefficient of x. y-intercept is the constant term. Draw a number plane. Sketch y = 2x + 3 using the y-intercept of 3 and gradient of 2. Sketch y = -x using the y-intercept of 0 and gradient of –1. Find the point of intersection of the two lines (–1, 1).

y = 2x + 3 Gradient is 2, y-intercept is 3 y = -x Gradient is –1, y-intercept is 0 y 3 y = 2x + 3 2 1 −3 −2 −1 0 −1

Simultaneous solution is the point of intersection.

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2

3

−2 −3

8

1

y = −x

Simultaneous solution is x = -1 and y = 1

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x

139

Chapter 5 — Interpreting linear relationships

Exercise 5D 1

What is the point of intersection for each of these pairs of straight lines? y

a

y

b

4

3

3 y=x+3

2

2

1

1

−1 0 −1

−3 −2 −1 0 −1

1

−2

2

x

3

d

y = 0.5x + 1

4

x

5

3 2

1

−2

3

y y = −2x + 1

2

−3 −2 −1 0 −1

2

−3

y 3

1

−2 y = x − 2

y = −2x

c

y = −x + 2

1

2

3

y = −1x − 2

1

x −3

−2

−1 0 −1

−3

1

2

x

3

−2 y = 4x + 1 −3

2

Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? x

0

1

2

3

4

y

0

2

4

6

8

y 8 7 6 5 4

x

0

1

2

3

4

3

y

6

5

4

3

2

2 1 1

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2

3

4

x

Cambridge University Press

Preliminary Mathematics General

3

4

Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? x

-2

-1

0

1

2

y

-6

-5

-4

-3

-2

x

-2

-1

0

1

2

y

6

3

0

-3

-6

The graph opposite shows the cost of producing boxes of sweets and the income received from their sale. a Use the graph to determine the number of boxes which need to be sold to break even. b How much profit or loss is made when 4 boxes are sold? c How much profit or loss is made when 1 box is sold? d What are the initial costs?

y 6 5 4 3 2 1 −3 −2 −1 0 −1 −2 −3 −4 −5 −6

1 2 3

x

y Income

50 Dollars ($)

140

40 Costs

30 20 10 1

2

3 4 Boxes

5

x

y

The graph opposite shows the cost of producing a pack of batteries and the income received from their sale. a Use the graph to determine the number of packs that need to be sold to break even. b How much profit or loss is made when 5 packs are sold? c How much profit or loss is made when 20 packs are sold? d What are the initial costs?

400

Income

350 Dollars ($)

5

300 250

Costs

200 150 100 50 x 5 10 15 20 Packs

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Chapter 5 — Interpreting linear relationships

141

Development 6

Draw the graphs of the following pairs of equations and find their simultaneous solution. a y = x + 1 and y = -2x b y = 3x - 3 and y = x + 1 c y = 5x + 1 and y = 3x - 7 d y = x and y = 4x + 3

7

Zaina buys and sells books. Income received by selling n books is calculated using the formula I = 16n. Costs associated in selling n books are calculated using the formula C = 8n + 24. a Draw the graph of I = 16n and C = 8n + 24 on same number plane. b What are the initial costs? c Use the graph to determine the number of books needed to be sold to break even. d How much profit or loss is made when 6 books are sold?

8 

Amy and Nghi work for the same company and their wages are a and b respectively. a Amy earns $100 more than Nghi. Write an equation to describe this information. b The total of Amy’s and Nghi’s wages is $1500. Write an equation to describe this information. c Draw a graph of the above two equations on the same number plane. Use a as the horizontal axis and b as the vertical axis. d Use the intersection of the two graphs to find Amy’s and Nghi’s wage.

9

A factory produces items whose costs are $200 plus $40 for every item. The factory receives $45 for every item sold. a Write an equation to describe the relationship between the: i costs (C) and number of items (n) ii income (I) and number of items (n) b Draw a graph to represent the costs and income for producing the item. c How many items need to be sold to break even?

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Preliminary Mathematics General

5.5 Linear functions as models Linear modelling occurs when a practical situation is described mathematically using a linear function. For example, the gradient–intercept form of a straight-line graph can sometimes be used to model catering costs. A catering company charges a base amount of $100 plus a rate of $25 per guest. Using this information, we can write down a linear equation to model the cost of the event. Let c be the cost of the event ($) and n be the number of guests, we can write c = 25n + 100. Note: The number of guests (n) must be greater than zero and a whole number. Catering cost c The graph of this linear model has been drawn 1500 opposite. There are two important features of 1000 this linear model: 500 1 Gradient is the rate per guest or $25. n 2 The c-intercept is the base amount or $100. 5 10 15 20 25 30 35 40 45 50

Example 9

Using graphs to make conversions

The graph opposite is used to convert Australian dollars to euros. Use the graph to convert: a 50 Australian dollars to euros b 15 euros to Australian dollars

EUR

Australian dollars to euros

30 25 20 15 10 5 10 20 30 40 50

Solution 1 2

Read from the graph (when AUD = 50, EUR = 30). Read from the graph (when EUR = 15, AUD = 25).

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a

30 EUR

b

25 AUD

Cambridge University Press

AUD

Chapter 5 — Interpreting linear relationships

Example 10

143

Interpreting linear models

Water is pumped into a partially full tank. The graph gives the volume of water V (in litres) after t minutes. a How much water is in the tank at the start? b How much water is in the tank after 10 minutes? c The tank holds 1600 L. How long does it take to fill? d Find the equation of the straight line in terms of V and t. e Use the equation to calculate the volume of water in the tank after 7 minutes. f How many litres are pumped into the tank each minute?

V

Volume of water

1600 1400 1200 1000 800 600 400 200 2 4 6 8 10 12

Solution 1 2 3 4 5 6 7 8 9 10 11 12

Read from the graph (when t = 0, V = 300). Read from the graph (when t = 10, V = 1400). Read from the graph (when V = 1600, t = 12). Find the gradient by choosing two suitable points such as (0, 400) and (12, 1600). Calculate the gradient (m) between these points using the gradient formula. Determine the vertical intercept (400). Substitute the gradient and y-intercept into the gradient–intercept form y = mx + b. Use the appropriate variables (V for y, t for x). Substitute t = 7 into the equation. Evaluate. Check the answer using the graph.

a

e

V = 100t + 400 = 100 × 7 + 400 = 1100 L

The rate at which water is pumped into the tank is the gradient of the graph. (m =100)

f

100 L/min

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c

400 L 1400 L 12 minutes

d

m=

b

Rise b = 400 Run 1600 − 400 = 12 − 0 = 100

y = mx + b V = 100t + 400

Cambridge University Press

t

144

Preliminary Mathematics General

Exercise 5E 1

Water is pumped into a partially full tank. The graph gives the volume of water V (in litres) after t minutes. a How much water is in the tank at the start? v Volume of water b How much water is in the tank after 5 minutes? 3000 c How much water is in the tank after 8 minutes? 2500 d The tank holds 2500 L. How long does it take to fill? 2000 e Use the graph to calculate the volume of water in the 1500 tank after 7 minutes. 1000 500 2

2

The conversion graph opposite is used to convert Australian dollars to Chinese yuan. Use the graph to convert: a 80 Australian dollars to yuan b 50 Australian dollars to yuan c 100 yuan to Australian dollars d 350 yuan to Australian dollars e What is the gradient of the conversion graph?

CNY

4

6

8 10

t

Australian dollars to Chinese yuan

500 400 300 200 100 20 40 60 80 100

A post office charges according to the weight of a parcel. Use the step graph to determine the postal charges for the following parcels. a 50 g b 900 g c 200 g d 800 g

Postal charges 4 Cost ($)

3

AUD

3 2 1 200 400 600 800 1000 Weight (g)

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145

Development A new piece of equipment is purchased by a business for $120 000. Its value is depreciated each month using the graph opposite. a What was the value of the equipment after Value of equipment v 32 months? b What was the value of the equipment after one 120 year? 100 c When does the line predict the equipment will 80 have no value? 60 d Find the equation of the straight line in terms 40 of v and t. 20 e Use the equation to predict the value of the equipment after 2 months. 8 16 24 32 40 48 f By how much does the equipment depreciate Months in value each month? $ 1000

4

The amount of money transacted through ATMs has increased with the number of ATMs available. The graph below shows this increase. a What was the amount of money transacted through ATMs when there were 500 000 machines? b How many ATM machines resulted in an amount of 75 billion? c Find the equation of the line in terms of amount of money transacted, A, and the number of ATMs, N. d Use the equation to predict the amount of money transacted when there were A 350 000 machines. 125 e Use the equation to predict how much money will be transacted through ATM 100 machines when there are 1 000 000 machines. 75 Amount (billions of $)

5

Amount of transactions through ATMs

50 25

100 200 300 400 500 Number of machines (thousands) © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

t

Cambridge University Press

N

146

Preliminary Mathematics General

6

A phone company charges a monthly service fee, plus the cost of calls. The graph below gives the total monthly charge, C dollars, for making n calls. This includes the service fee. a How much is the monthly service fee? Total monthly b How much does the company charge if you make charge C 20 calls a month? 70 c How many calls are made if the total monthly charge 60 is $30? 50 d Find the equation of the line in terms of total monthly 40 charge (C) and the number of calls (n). 30 20 10 20 40 60 80 100

14.7

n

7 

A company charges the following parking fees: $10 per hour for up to 3 hours, $15 up to 6 hours and $20 for over 6 hours. a Draw a step graph to illustrate the parking fees, with the Time (h) on the horizontal axis and Cost ($) on the vertical axis. b What is the cost to park for 4 hours? Use the step graph. c Liam arrived in the parking area at 10.30 a.m. and left at 1.00 p.m. How much did he pay for parking? d Ruby arrived in the parking area at 5.15 p.m. and left at 11.15 p.m. How much did she pay for parking?

8

Tomas converted 100 Australian dollars to 40 British pounds. a Draw a conversion graph with Australian dollars on the horizontal axis and British pounds on the vertical axis. b How many British pounds is 40 Australian dollars? Use the conversion graph. c How many Australian dollars is 10 British pounds? Use the conversion graph. d Find the gradient and vertical intercept for the conversion graph. e Write an equation that relates Australian dollars (AUD) to British pounds (GBP).

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Chapter 5 — Interpreting linear relationships

Graphing linear functions

3

Construct a table of values with the independent variable as the first row and the dependent variable as the second row. Draw a number plane with the independent variable on the horizontal axis and the dependent variable as the vertical axis. Plot the points. Join the points to make a straight line.

•

Gradient of a line is the slope of the line.

•

Gradient (or m) =

•

The intercept of a line is where the line cuts the axes.

•

Linear equation in the form y = mx + b. m – Slope or gradient of the line. b – y-intercept. Sketching a straight line requires at least two points. When an equation is written in gradient–intercept form, one point on the graph is immediately available: the y-intercept. A second point can be quickly calculated using the gradient.

1

2

Gradient and intercept

Gradient–intercept formula

•

Simultaneous equations

1 2 3

Linear functions as models

Study guide 5

Vertical rise Horizontal run

Draw a number plane. Graph both linear equations on the number plane. Read the point of intersection of the two straight lines.

Linear modelling occurs when a practical situation is described mathematically using a linear function.

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Review

Chapter summary – Interpreting linear relationships

147

Review

148

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

An equation that compares the age of a fax machine (t) in years to its current value (v) is v = −40t + 150. What is the value of the fax machine after two years? A 70 B 108 C 150 D 230

2

What is the gradient of the line drawn opposite? 2 3 A B 3 4 4 3 C D 3 2

3

Using the graph opposite, what is the y-intercept of this line? A –2 B –1 C 1 D 2

y 4 2 −2

−1

0

1

−2

2

−4

4

A straight line has the equation of y = −3x + 1. What is the y-intercept? A –3 B –1 C 1 D 3

5

A car is travelling at a constant speed. It travels 60 km in 3 hours. This situation is described by the linear equation d = mt. What is the value of m? A 0.05 B 3 C 20 D 60

6

What is the point of intersection of the lines y = x + 1 and y = −x + 1? A (0, 0) B (0, 1) C (1, 0) D (1, 1)

7

What is the equation of the line drawn opposite? A c=n B c = n + 30 C c = 30n D c = 8n + 240

8

Using the graph opposite, what is the charge for 12 months? A 24 B 36 C 240 D 360

Monthly charge

c 240 180 120 60 2

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x

4

6

8

Cambridge University Press

n

Chapter 5 — Interpreting linear relationships

149

Review

Sample HSC – Short-answer questions 1

An internet access plan charges an excess fee of $8 per GB. Data (d)

1

2

3

4

5

6

Cost (c)

8

16

24

32

40

48

a b c d

Which is the dependent variable? Which is the independent variable? Draw a graph of data against cost. Use the graph to find d if c is 20.

2

One Australian dollar (AUD) was converted to 0.90 Japanese yen (JPY). a Construct a table of values using 0, 10, 20, 30 and 40 as values for AUD and calculate the JPY using the above conversion. b Draw the graph of the AUD against JPY.

3

What is the gradient of the line that joins these points? a (1, 5) and (3, 7) b (–2, 1) and (0, 4) c (–3, –1) and (2, –11)

4

Draw a graph of these linear functions and find the gradient and y-intercept. a y = x + 1 b y = -2x + 5 c y = 3x - 2

5

Find the equations of the following line graphs. a y

b

y

2

4

1

2

−2 −1 0 −1

1

2

x

−2

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−4 −2 0 −2

2

4

x

−4

Cambridge University Press

Preliminary Mathematics General

Chapter summary Earning Money 6 The graph opposite– shows the cost of growing a rose and the income received from the sale

80 Dollars ($)

of the roses. a Use the graph to determine the number of roses which need to be sold to break even. b How much profit or loss is made when 1 rose is sold? c How much profit or loss is made when 4 roses are sold? d What are the initial costs?

Income

60 Costs

40 20 1

7

A motor vehicle is purchased by a business for $30 000. Its value is then depreciated each month using the graph opposite. a What was the value of the motor vehicle after 24 months? b What was the value of the motor vehicle after one year? c Find the equation of the straight line in terms of v and t. d Use the equation to predict the value of the motor vehicle after 6 months. e When does the line predict that the motor vehicle will have no value? f By how much does the motor vehicle depreciate in value each month?

Dollars ($ 1000)

Review

150

8

v

2 3 Roses

4

Motor vehicle

30 24 18 12 6 t 12 24 36 48 Time (months)

The table below shows the speed v (in km/s) of a rocket at time t seconds. Time (t)

1

2

3

4

5

Speed (v)

1.5

3

4.5

6

7.5

a b c

Draw a number plane with time (t) on the horizontal axis and speed (v) on the vertical axis. Plot the points from the table and draw a straight line through these points. Extend the straight line to predict the rocket’s speed when t = 6 seconds. Challenge questions 5

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C H A P T E R

61

Investing Earning money money Area of study Syllabus topic — — FM1 FM2 Earning Investing money money Calculate simple payments interest from a salary Calculate Draw and wages describe using simple hourly andrate, compound overtime interest rates graphs and allowances Calculate the annual future leave value, loading compound and bonuses interest and present value Calculate Use prepared earnings tablesbased to calculate on commission, final amount piecework and interest and royalties Determineaccounts Compare deductions from and different calculate financial net pay institutions A Compare accounts Calculate the appreciated from different value onfinancial investments institutions Calculate the dividend and dividend yield on shares

6.1 Simple interest

6.1

Interest is the amount paid for borrowing money or the amount earned for lending money. There are different ways of calculating interest. Simple interest (or flat interest) is a fixed percentage of the amount invested or borrowed and is calculated on the original amount. For example, if we invest $100 in a bank account that pays interest at the rate of 5% per annum (per year) we would receive $5 each year. That is, 5 Interest = $100 × = $5 100 This amount of interest would be paid each year. Simple interest is always calculated on the initial amount, or the principal. Simple interest I = Prn I – Interest (simple or flat) earned for the use of money P – Principal is the initial amount of money borrowed r – Rate of simple interest per period expressed as a decimal n – Number of time periods 151 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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152

Preliminary Mathematics General

Example 1

Finding simple interest

Calculate the amount of simple interest paid on an investment of $12 000 at 10% simple interest per annum for 3 years.

Solution 1 2 3 4

Write the simple interest formula. Substitute P = 12 000, r = 0.10 and n = 3 into the formula. Evaluate. Write the answer in words.

I=P Prn = 12 000 × 0.10 × 3 = $3600 Simple interest is $3600.

Amount owed or future value The interest is added to the principal to determine the amount owed on a loan or the future value of an investment. Amount owed or future value A=P+I A – Amount or final balance I – Interest (simple or flat) earned P – Principal is the initial quantity of money

Example 2

Calculating the amount owed

Find the amount owed on a loan of $50 000 at 7% per annum simple interest at the end of two years and six months. Solution 1 2 3 4 5 6 7

Write the simple interest formula. Substitute P = 50 000, r = 0.07 and n = 2.5 into the formula. Evaluate. Write the amount owed formula. Substitute P = 50 000 and I = 8750 into the formula. Evaluate. Write the answer in words.

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I = Prn = 50 000 × 0.07 × 2.5 = $8750 A=P+I = 50 000 + 8750 = $58 750 Amount owed is $58 750. Cambridge University Press

Chapter 6 — Investing money

Example 3

153

Finding simple interest using a graphics calculator

Joel plans to make an investment of $200 000 at 7 12 % p.a. simple interest for 2 years. Answer the following questions by using a graphics calculator. a How much simple interest will Joel earn? b What is the total value of his investment at the end of 2 years?

Solution 1 2

Select the TVM (Time, Value, Money) menu. Select Simple Interest (F1).

6

Enter the time period n = 2 × 365 = 730 (simple interest period is calculated in days). Enter the interest rate I% = 7.5. Enter the principal or present value PV = -200 000. In the TVM mode all money we pay out is negative and money we receive is positive. In this example $200 000 is deposited or paid out. To calculate the simple interest, select SI.

7

Write the answer in words.

8

To calculate the total amount owed, select SFV (Simple Final Value). Write the answer in words.

3 4 5

9

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a

Joel will earn $30 000. b

Total value is $230 000.

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154

Preliminary Mathematics General

Exercise 6A 1

2

Calculate the amount of simple interest for each of the following. a Principal = $15 000, Interest rate = 13% p.a., Time period = 3 years b

Principal = $2000, Interest rate = 6 12 % p.a., Time period = 7 years

c

Principal = $200 000, Interest rate = 9 14 % p.a., Time period = 2 years

d

Principal = $3600, Interest rate = 9% p.a., Time period = 3 12 years

e

Principal = $40 000, Interest rate = 7.25% p.a., Time period = 5 14 years

Calculate the amount owed for each of the following. a Principal = $500, Simple interest rate = 5% p.a., Time period = 4 years b Principal = $900, Simple interest rate = 3% p.a., Time period = 7 years c

Principal = $4000, Simple interest rate = 8 12 % p.a., Time period = 3 years

d

Principal = $6900, Simple interest rate = 10% p.a., Time period = 4 12 years

e

Principal = $10 000, Simple interest rate = 6.75% p.a., Time period = 2 14 years

3

The simple interest rate is given as 4.8% per annum. a What is the interest rate per quarter? b What is the interest rate per month? d What is the interest rate per six months? e What is the interest rate per nine months?

4

Calculate the amount of simple interest for each of the following. a Principal = $800, Interest rate = 12% p.a., Time period = 1 month. b Principal = $1600, Interest rate = 18% p.a., Time period = 6 months. c Principal = $60 000, Interest rate = 9.6% p.a., Time period = 3 months. d Principal = $20 000, Interest rate = 6% p.a., Time period = 9 months.

5

Andrew takes a loan of $30 000 for a period of 6 years, at a simple interest rate of 14% per annum. Find the amount owing at the end of 6 years.

6

A loan of $1800 is taken out at a simple interest rate of 15.5% per annum. How much interest is owing after 3 months?

7

A sum of $100 000 was invested in a fixed-term account for 4 years. Calculate: a the simple interest earned if the rate of interest is 5.5% per annum b the future value of the investment at the end of 4 years

8

Joshua invested $1200 at 8% per annum. What is the simple interest earned between 30 September and 1 January?

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Development 9

Sophie decides to buy a car for $28 000. She has saved $7000 for the deposit and takes out a loan over two years for the balance. The flat rate of interest charged is 12% per annum. What is the total amount of interest to be paid?

10

Domenico has borrowed $24 000 to buy furniture. He wishes to repay the loan over four years. Calculate the simple interest on the following rates of interest. a 8% per annum for the entire period b 9% per annum after a 6-month interest-free period c 10% per annum after a 12-month interest-free period

11

Create the spreadsheet below.

6A

a b

Cell D5 has a formula that calculates the simple interest. Enter this formula. The formula for cell E5 is ‘=A5 + D5’. Fill down the contents of E6 to E12 using this formula.

12

Isabelle buys a TV for $1400. She pays it off monthly over 2 years at an interest rate of 11.5% per annum flat. How much per month will she pay?

13

Riley wants to earn $4000 a year in interest. How much must he invest if the simple interest rate is 10% p.a.?

14

Samira invests $16 000 for 2 12 years. What is the minimum rate of simple interest needed for her to earn $3000? Gurrumul pays back $20 000 on a $15 000 loan at a flat interest rate of 10%. What is the term of the loan?

15

16

Harry borrowed $300 000 at a flat rate of interest of 8.5% per annum. This rate was fixed for 2 years on the principal. He pays back the interest only over this period. a How much interest is to be paid over the 2 years? b After paying the fixed rate of interest for the first year, Harry finds the bank will decrease the flat interest rate to 7.5% if he pays a charge of $1000. How much will he save by changing to the lower interest rate for the last year?

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6.2 Simple interest graphs When graphing simple interest make the horizontal axis the time period and the vertical axis the interest earned. Simple interest will increase by a constant amount each time period. This will result in a straight-line graph. Simple interest graphs 1 2 3

Construct a table of values for I and n using the simple interest formula. Draw a number plane with n the horizontal axis and I the vertical axis. Plot the points. Join the points to make a straight line.

Example 4

Constructing a simple interest graph

Draw a graph showing the amount of simple interest earned over a period of 4 years if $1000 is invested at 6% p.a. Use the graph to estimate the interest earned after 8 years. Solution 1 2 3 4 5

6

7 8 9

Write the simple interest formula. Substitute P = 1000, r = 0.06 and n into the formula. Draw a table of values for I and n. Let n = 0, 1, 2, 3 and 4. Find the interest (I) using I = 60n. Draw a number plane with n as the horizontal axis and I as the vertical axis. Plot the points (0, 0), (1, 60), (2, 120), (3, 180) and (4, 240). Extend the line to estimate the value of I when n = 8. Read the graph to estimate I. (I = 480 when n = 8). Write the answer in words.

I=P Prn = 1000 × 0.06 × n = 60n n

0

1

2

3

4

I

0

60

120

180

240

Simple interest on $1000 at 6% p.a.

I 500 400 300 200 100

1 2 3 4 5 6 7 8

n

Interest after 8 years is approximately $480.

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Chapter 6 — Investing money

Example 5

157

Simple interest graphs using a graphics calculator

Use a graphics calculator to draw two graphs showing the amount of simple interest earned for 5 years if $9000 is invested at: a 8% p.a. b 5% p.a.

Solution 1 2 3

Write the simple interest formula. Substitute P = 9000, r = 0.08 and n into the formula. Select the Graph menu.

4

Enter the formula I = 720n by typing 720X at Y1. The graph of Y1 = 720X is the same as I = 720n.

5

7

Edit the axes to an appropriate scale. Select SHIFT F3 for the V-Window. Enter the Xmin = 0, Xmax = 5, Ymin = 0, Ymax = 3500, scale = 500. Press EXE to exit V-Window.

8

Select F6 to draw the graph.

1

Write the simple interest formula. Substitute P = 9000, r = 0.05 and n into the formula.

6

2 3

4

I= P Prn = 9000 × 0.08 × n = 720 n a

b

I=P Prn = 9000 × 0.05 × n = 450n

Enter the formula I = 450n by typing 450X at Y2. The graph of Y2 = 720X is the same as I = 450n. Select F6 to draw the two graphs.

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Exercise 6B 1

Nicholas invested $1000 at 7% per annum simple interest for 4 years. a Simplify the simple interest formula (I = Prn) by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

0

1

2

3

4

I c d e

Use the graph to find the interest after 2 12 years.

f

Extend the graph to find the interest after 6 years. Find the time when the interest is $210.

g

2

Draw a number plane with n as the horizontal axis and I as the vertical axis. Plot the points from the table of values. Join the points to make a straight line.

Melissa invested $600 at 5% per annum simple interest for 5 years. a Simplify the simple interest formula (I = Prn) by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

0

1

2

3

4

5

I c d e f g

Draw a number plane with n as the horizontal axis and I as the vertical axis. Plot the points from the table of values. Join the points to make a straight line. Use the graph to find the interest after 3 12 years. Extend the graph to find the interest after 6 years. Find the time when the interest is $360.

3

Use a graphics calculator. a Draw the graph from question 1. b Use the TRACE feature to confirm the answers to parts e, f and g of question 2. c Draw the graph from question 2. d Use the TRACE feature to confirm the answers to parts e, f and g to question 2.

4

Draw a graph showing the amount of simple interest earned over a period of 4 years if $1000 is invested at 4% p.a. Use the graph to estimate the interest earned after 6 years.

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159

Development 5

Chloe is comparing three different interest rates for a possible investment. a Draw a graph to represent the interest earned over 5 years on: i $5000 invested at 5% per annum simple interest ii $5000 invested at 7% per annum simple interest iii $5000 invested at 9% per annum simple interest b

How much does each investment earn after 2 12 years?

c

How much does each investment earn after 5 years? Find the time for each investment to earn $1000 in interest.

d

6

Mick is comparing three different interest rates for a possible investment. a Draw a graph to represent the interest earned for 6 months on: i $100 000 invested at 6% p.a. simple interest ii $100 000 invested at 9% p.a. simple interest iii $100 000 invested at 12% p.a. simple interest b How much does each investment earn after 1 month? c How much does each investment earn after 6 months? d Find the time for each investment to earn $2000 in interest.

7

The table below gives details for a fixed-term deposit. Time period

14.5

Simple interest rate per annum

Less than 3 months

6.5%

3 to 6 months

7.0%

6 to 12 months

7.5%

12 to 24 months

8.1%

24 to 48 months

8.3%

Chris has $50 000 to invest in a fixed-term deposit. Draw a separate graph to represent the interest earned after 12 months given these investments. a Fixed-term deposit for 3 months b Fixed-term deposit for 6 months c Fixed-term deposit for 12 months

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Preliminary Mathematics General

6.3 Compound interest

6.3

Compound interest is calculated from the initial amount borrowed or principal plus any interest that has been earned. It calculates interest on the interest. For example, if $100 is invested at a compound interest rate of 10% per annum. First year – Interest = $100 × 0.10 × 1 = $10 Amount owed = $100 + $10 = $110 Second year – Interest = $110 × 0.10 × 1 = $11 Amount owed = $110 + $11 = $121 Third year – Interest = $121 × 0.10 × 1 = $12.10 Amount owed = $121 + $12.10 = $133.10 These calculations show the interest earned increased each year. In the first year it was $10, the second year $11 and the third year $12.10. Compound interest A = P(1 + r)n or FV = PV(1 + r)n A – Amount (final balance) or future value of the loan P – Principal is the initial quantity of money or present value of the loan r – Rate of interest per compounding time period expressed as a decimal n – Number of compounding time periods

Calculating compound interest The compound interest is calculated by subtracting the principal from the amount borrowed or invested. Alternatively, finance companies provide an investment calculator as an estimate to the value of an investment.

Interest earned or owed I = A - P A – Amount or final balance I – Interest (compound) earned P – Principal is the initial quantity of money

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Chapter 6 — Investing money

Example 6

161

Finding the compound interest

Paige invests $5000 over 5 years at a compound interest rate of 6.5% p.a. Calculate: a the amount of the investment after 5 years, correct to the nearest cent b the interest earned after 5 years, correct to the nearest cent. Solution

3

Write the compound interest formula. Substitute P = 5000, r = 0.065 and n = 5 into the formula. Evaluate.

A = P(1 + r )n = 5000(1 + 0.065)5 = 6850.433317 = $6850.4433

4

Write answer in words.

Amount is $6850.43.

5

Write the amount borrowed formula. Substitute P = 5000 and I = 6850.43 into the formula. Evaluate. Write in words.

I = A− P = 6850.43 − 5000 = $1850.43

1 2

6 7 8

Example 7

Interest earned is $1850.43.

Finding compound interest using a graphics calculator

James borrowed $50 000 for 4 years at 11% p.a. interest compounding monthly. What is the amount owed after the 4 years? Solution 1 2

3 4 5 6

7 8

Select the TVM (Time, Value, Money) menu. Select Compound Interest (F2).

Enter the time period n = 4 × 12 = 48 (number of compounding time periods). Enter the interest rate I% = 11. Enter compounding periods per year P/Y = 12. Enter the principal or present value PV = 50 000. In the TVM mode, all money we pay out is negative and money we receive is positive. To calculate the amount or future value, select FV. Write answer using appropriate accuracy and in words.

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James will owe $77 479.90.

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Exercise 6C 1

2

Calculate the amount owed, to the nearest cent, for each of the following. a Principal = $800, Compound interest rate = 4% p.a., Time period = 3 years b

Principal = $9000, Compound interest rate = 6 12 % p.a., Time period = 4 years

c

Principal = $12 000, Compound interest rate = 11% p.a., Time period = 2 12 years

d

Principal = $22 000, Compound interest rate = 5.5% p.a., Time period = 4 14 years

Calculate the amount of compound interest to the nearest cent for each of the following. a Principal = $25 000, Interest rate = 7% p.a., Time period = 5 years b

Principal = $300 000, Interest rate = 10 14 % p.a., Time period = 3 years

c

Principal = $6500, Interest rate = 13% p.a., Time period = 1 12 years

d

Principal = $80 000, Interest rate = 8.25% p.a., Time period = 3 14 years

3

Amy is investing $20 000 with AMP. What sum of money will she receive if she invested for 4 years at 8% p.a. compound interest? Answer to the nearest cent.

4

Use the formula A = P(1 + r)n to calculate the value of an investment of $10 000, over a period of 2 years with an interest rate of 0.8% compounding monthly. Answer to the nearest cent.

5

Ryan invested $20 000 for 5 years at 12% p.a. interest compounding monthly. What is the amount of interest earned in the first year? Answer to the nearest cent.

6

Find the amount of money in a bank account after 6 years if an initial amount of $4000 earns 8% p.a. compound interest, paid quarterly. Answer to the nearest cent.

7

Christopher invested $13 500 over 7 years at 6.2% p.a. interest compounding quarterly. Calculate the: a value of the investment after 7 years to the nearest cent. b compound interest earned to the nearest cent.

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Development 8

What sum of money would Bailey need to invest to accumulate a total of $50 000 at the end of 4 years at 6% p.a. compound interest? Answer to the nearest cent.

9

Calculate the amount that must be invested at 9.3% p.a. interest compounding annually to have $70 000 at the end of 3 years. Answer to the nearest cent.

10

What sum of money needs to be invested to accumulate to a total of $100 000 in 10 years at 7.25% p.a. compound interest? Answer to the nearest cent.

11

Create the spreadsheet below.

6C

a b

Cell D5 has a formula that calculates the compound interest. Enter this formula. The formula for cell E5 is ‘= D5 - A5’. Fill down the contents of E6 to E12 using this formula.

12

How much more interest is earned on a $40 000 investment if the interest at 6% p.a. is compounded annually over 6 years, compared with the simple interest at 6% p.a. earned over the same time?

13

Hassan has $50 000 to invest for two years. Which is the better investment? Why? Investment 1 Simple Interest rate 4% p.a.

Investment 2 Compound Interest rate 4% p.a.

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Preliminary Mathematics General

6.4 Compound interest graphs When graphing compound interest, make the horizontal axis the number of time periods and the vertical axis the amount (final balance). Compound interest will increase by a different amount each time period. This will result in an exponential curve. Compound interest graphs 1 2 3

Construct a table of values for A and n using the compound interest formula. Draw a number plane with n the horizontal axis and A the vertical axis. Plot the points. Join the points to make an exponential curve.

Example 8

Constructing a compound interest graph

Draw a graph showing the amount of the loan over a period of 8 years if $1000 is invested at a compound interest rate of 6% p.a. Use the graph to estimate the amount of the loan after 7 years. Solution 1 2 3 4 5

6

7 8

9

Write the formula. Substitute P = 1000, r = 0.06 and n into the formula. Draw a table of values for A and n. Let n = 0, 2, 4, 6 and 8. Find the amount earned. Draw a number plane with n as the horizontal axis and A as the vertical axis. Plot the points (0, 1000), (2, 1124), (4, 1262), (6, 1419) and (8, 1594). Draw an exponential curve (not a straight line) between the points. Read the graph to estimate I when n = 7 years (A = 1500 when n = 7). Write the answer in words.

A = P(1 + r )n = 1000 × (1.06 )n n

0

2

4

6

8

A

1000

1124

1262

1419

1594

Compound interest at 6% p.a.

A 1600 1500 1400 1300 1200 1100 1000 1

2

3

4

5

6

7

8

n

Loan amount after 7 years is about $1500.

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Chapter 6 — Investing money

Example 9

165

Compound interest graphs using a graphics calculator

Use a graphics calculator to draw two graphs showing the amount earned for 5 years if $9000 is invested at: a 8% p.a. compounding annually b 5% p.a. compounding annually.

Solution 1 2 3

Write the compound interest formula. Substitute P = 9000, r = 0.08 and n into the formula. Select the Graph menu.

4

Enter the formula A = 9000 × (1.08)n by typing 9000×1.08^X at Y1. The graph of Y1=9000×1.08^X is the same as A = 9000 × (1.08)n.

5

Edit the axes to an appropriate scale. Select SHIFT F3 for the V-Window. Enter the Xmin = 0, Xmax = 5, Ymin = 9000, Ymax = 13500, scale = 500.

6

7

Press EXE to exit V-Window.

8

Select F6 to draw the graph.

1

Write the compound interest formula. Substitute P = 9000, r = 0.05 and n into the formula. Enter the formula A = 9000 × (1.05)n by typing 9000×1.05^X at Y1. The graph of Y1=9000×1.05^X is the same as A = 9000 × (1.05)n. Select F6 to draw the two graphs.

2 3

4

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A = P(1 P(1 + r )n = 9000 × (1.08)n a

b

A = P(1 + r )n = 9000 × (1.05)n

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Preliminary Mathematics General

Exercise 6D 1

Ava invested $2000 at 4% per annum interest compounding annually for 4 years. a Simplify the compound interest formula A = P(1 + r)n by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. Answer to nearest dollar. n

0

1

2

3

4

A c d e

Use the graph to find the amount after 3 12 years.

f

Extend the graph to find the amount after 5 years. Find the time when the amount is approximately $2120.

g

2

Draw a number plane with n as the horizontal axis and A as the vertical axis. Plot the points from the table of values. Join the points to make a curve.

Dylan invested $800 at 7% p.a. compound interest, paid annually, for 5 years. a Simplify the compound interest formula A = P(1 + r)n by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

0

1

2

3

4

5

A c d

Draw a number plane with n as the horizontal axis and A as the vertical axis. Plot the points from the table of values. Join the points to make a curve.

e

Use the graph to find the amount after 2 12 years.

f

Extend the graph to find the amount after 7 years. Find the time when the amount is approximately $1025.

g

3

Use a graphics calculator. a Draw the graph in question 1. b Use the TRACE feature to confirm the answers to parts e, f and g in question 1. c Draw the graph in question 2. d Use the TRACE feature to confirm the answers to parts e, f and g in question 2.

4

Draw a graph showing the amount of the loan over a period of 3 years if $1000 is invested at a compound interest rate of 7% p.a. Use the graph to estimate the amount of the loan after 6 years.

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167

Development 5

Bailey is comparing three different interest rates for a possible investment. a Draw a graph to represent the future value over 10 years on: i $1000 invested at 4% per annum interest compounding biannually ii $1000 invested at 6% per annum interest compounding biannually iii $1000 invested at 8% per annum interest compounding biannually b

What is the approximate value of each investment after 3 12 years?

c

What is the approximate value of each investment after 5 years? Find the approximate time for each investment to earn $500 in interest.

d

14.6

6

Laura is comparing three different interest rates for a possible investment. a Draw a graph to represent the amount earned over 4 years on: i $100 000 invested at 6% p.a. interest compounding monthly ii $100 000 invested at 9% p.a. interest compounding monthly iii $100 000 invested at 12% p.a. interest compounding monthly b What is the approximate value of each investment after 6 months? c What is the approximate value of each investment after 4 years? d Find the time for each investment to earn $20 000 in interest.

7

The table below gives details for an investment product. The compound interest earned is paid quarterly. Investment

Rate of compound interest

A

4% p.a.

B

6% p.a.

C

8% p.a.

D

10% p.a.

Ethan is prepared to invest $50 000 in the above product. a Draw a graph to represent the future value of these investments after 3 years. b What is the approximate value of investment B after 2 years? c What is the approximate value of investment C after 18 months? d Find the time for investment D to earn $5000 in interest.

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Preliminary Mathematics General

6.5 Using prepared tables

6.5

Investment problems are made easier by using tables. The table below shows the future value or final balance when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. For example, the value of $1 after 4 periods at an interest rate of 6% per period is 1.2625. Period

3%

4%

5%

6%

1

1.0300

1.0400

1.0500

1.0600

2

1.0609

1.0816

1.1025

1.1236

3

1.0927

1.1249

1.1576

1.1910

4

1.1255

1.1699

1.2155

1.2625

Using a prepared table 1 2 3

Determine the time period and rate of interest. Find the intersection of the time period and rate of interest in the table. Multiply the number in the intersection with the money invested.

Example 10

Using prepared tables

Use the above table to calculate the future value of: a $20 000 invested for 3 years at 6% p.a. compounded annually b $130 000 invested for 2 years at 6% p.a. compounded six-monthly. Solution 1 2 3 4 5 1

2 3 4

Time period is 3, interest rate is 6%. Find the intersection value from the table. (1.1910) Multiply intersection value by the money invested. Evaluate. Answer the question in words. Period is 4 (n = 2 × 2), interest rate is  6  3%  r = 2 % . Find the intersection value from the table. (1.1255) Multiply intersection value by the money invested. Evaluate and answer the question in words.

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a

Intersection value is 1.1910 FV = 1.1910 × 20000 = $23820 Future value is $23 820.

b

Intersection value is 1.1255 FV = 1.1255 × 130000 = $146 315 Future value is $146 315.

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Chapter 6 — Investing money

169

Exercise 6E 1

The table below shows the future value when $1 is invested at the given simple interest rate for the given number of periods. Period

2%

4%

6%

8%

10%

12%

1

1.02

1.04

1.06

1.08

1.10

1.12

2

1.04

1.08

1.12

1.16

1.20

1.24

3

1.06

1.12

1.18

1.24

1.30

1.36

4

1.08

1.16

1.24

1.32

1.40

1.48

Use the above table to calculate the future value of: a $1000 invested for 2 years at a simple interest rate of 10% p.a. b $36 000 invested for 4 years at a simple interest rate of 2% p.a. c $48 000 invested for 3 years at a simple interest rate of 8% p.a. d $6000 invested for 1 year at a simple interest rate of 12% p.a. e $20 000 invested for 3 years at a simple interest rate of 4% p.a. 2

The table below shows the future value when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. Period

1%

4%

8%

12%

16%

20%

1

1.0100

1.0400

1.0800

1.1200

1.1600

1.2000

2

1.0201

1.0816

1.1664

1.2544

1.3456

1.4400

3

1.0303

1.1249

1.2597

1.4049

1.5609

1.7280

4

1.0406

1.1699

1.3605

1.5735

1.8106

2.0736

5

1.0510

1.2167

1.4693

1.7623

2.1003

2.4883

6

1.0615

1.2653

1.5869

1.9738

2.4364

2.9860

Use the above table to calculate the future value of the following (answer correct to nearest dollar). a $10 000 invested for 5 years at 1% p.a. compounded annually b $50 000 invested for 3 years at 20% p.a. compounded annually c $2000 invested for 2 years at 8% p.a. compounded annually d $500 000 invested for 1 year at 16% p.a. compounded annually e $78 000 invested for 6 years at 12% p.a. compounded annually

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3

The table below shows the future value when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. Period

5%

10%

15%

20%

25%

1

1.05

1.10

1.15

1.20

1.25

2

1.10

1.21

1.32

1.44

1.56

3

1.16

1.33

1.52

1.73

1.95

4

1.22

1.46

1.75

2.07

2.44

Use the table to calculate the future value of the following (answer correct to nearest dollar). a $1000 invested for 3 years compounding annually at an interest rate of 10% p.a. b $2000 invested for 2 years compounding annually at an interest rate of 25% p.a. c $500 invested for 4 years compounding annually at an interest rate of 5% p.a. d $400 invested for 3 years compounding annually at an interest rate of 20% p.a. e $5000 invested for 1 year compounding annually at an interest rate of 15% p.a. f $7000 invested for 2 years compounding annually at an interest rate of 20% p.a. 4

The table below shows the compounded value of $1. Period

1%

2%

3%

4%

5%

1

1.010

1.020

1.030

1.040

1.050

2

1.020

1.040

1.061

1.082

1.103

3

1.030

1.061

1.093

1.125

1.158

4

1.041

1.082

1.126

1.170

1.216

5

1.051

1.104

1.159

1.217

1.276

Use the table to find the amount received on the following investments. Answer correct to the nearest dollar. a Principal = $5000, Interest rate = 1%, Time period = 4 years b Principal = $15 000, Interest rate = 3%, Time period = 1 years c Principal = $65 000, Interest rate = 2%, Time period = 5 years d Principal = $200 000, Interest rate = 5%, Time period = 2 years e Principal = $60 000, Interest rate = 4%, Time period = 3 years f Principal = $100 000, Interest rate = 1%, Time period = 5 years g Principal = $90 000, Interest rate = 3%, Time period = 4 years

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171

Development 5

The table below shows the future value when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. Period

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1

1.010

1.015

1.020

1.025

1.030

1.035

1.040

2

1.020

1.030

1.040

1.051

1.061

1.071

1.082

3

1.030

1.046

1.061

1.077

1.093

1.109

1.125

4

1.041

1.061

1.082

1.104

1.126

1.148

1.170

5

1.051

1.077

1.104

1.131

1.159

1.188

1.217

6

1.062

1.093

1.126

1.160

1.194

1.229

1.265

7

1.072

1.110

1.149

1.189

1.230

1.272

1.316

8

1.083

1.126

1.172

1.218

1.267

1.317

1.369

Use the table to calculate the future value of the following (answer correct to nearest dollar). a $170 000 invested for 1 month at 30% p.a. compounded monthly b $89 000 invested for 7 months at 42% p.a. compounded monthly c $240 000 invested for 3 months at 6% p.a. compounded quarterly d $75 000 invested for 2 years at 4% p.a. compounded quarterly e $5800 invested for 3 years at 3% p.a. compounded six-monthly f $380 000 invested for 2 years at 2% p.a. compounded six-monthly

6 

The table below shows the future value when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. Period

5%

10%

15%

20%

25%

3

1.1576

1.3310

1.5209

1.7280

1.9531

4

1.2155

1.4641

1.7490

2.0736

2.4414

Use the table to find the amount of money which could be invested to give the following (answer correct to the nearest dollar). a $20 000 at the end of 4 years, at 10% p.a. interest compounded annually b $80 000 at the end of 3 years, at 20% p.a. interest compounded annually c $1 000 000 at the end of 4 years, at 15% p.a. interest compounded annually d $650 000 at the end of 3 years, at 5% p.a. interest compounded annually

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6.6 Financial institutions: costs Banks and financial institutions charge their customers for maintaining accounts. It is called an account servicing fee and is charged per month. Some accounts charge a transaction fee for using ATM, telephone, internet or branch services. However, some accounts have unlimited transactions or a set number of free transactions. Most banks will also charge a fee for using another institution’s ATM. In addition to the above charges institutions have a range of penalty fees including: • Periodic payment dishonour – direct debt but insufficient funds in the account • Overdrawn account fee – payment made but insufficient funds in the account • Cheque dishonour fee – insufficient funds to clear a cheque • Late payment fee – minimum payment not received on the due date • Over-limit fee – exceeding the credit card limit. Many financial institutions provide information to their customers on how to reduce bank fees. It involves managing your transactions, knowing your account balance, not using other banks’ ATM and using internet banking.

Example 11

Calculating costs associated with bank accounts

Evgenia’s bank charges an account servicing fee of $12 per month, ATM transaction fee of $0.40 and branch enquiry fee of $0.50. What are Evgenia’s banking costs for the past three months if she made 15 ATM withdrawals and 4 branch enquiries? Solution 1 2 3 4 5

Write the quantity (charge) to be found. Multiply the account servicing fee by 3. Multiply the number of ATM transactions by $0.40. Multiply the number branch enquiries by $0.50. Write the answer in words.

Charge = 3 × 12 + (15 × 0.40) × (4 × 0.50) = $44

Evgenia is charged $44 in bank charges.

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Exercise 6F 1

Maxim’s bank charges an account servicing fee of $15 per month, ATM transaction fee of $0.45 and telephone enquiry fee of $0.25. What is Maxim’s banking costs for the past two months if he made 13 ATM withdrawals and 6 telephone enquiries?

2

The graph below shows the fees collected from banks in the last 5 years. The deposit fee income and the credit card fee income are shown in the first two columns. The third column shows the total fee income. 4500 4000 3500

Millions ($)

3000 2500 2000 1500 1000 500 0

1

2

3 Years

4

5

Deposit fee income Credit card fee income Total fee income a b c d e f

What was the total fee income in year 2? What was the credit card fee income in year 5? What was the deposit fee income in year 4? How much has the deposit fee income increased from year 1 to year 5? How much has the credit card fee income increased from year 1 to year 5? Calculate the percentage increase in total fee income from year 1 to year 5.

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Development 3 

The table below shows the banking charges of an overseas bank. Activity

Charge

ATM withdrawal fee

$0.50

Branch withdrawal

$0.60

Cheque dishonour fee

$5.00

Telephone enquiry

$0.20

Find the cost of the following activities. a Ten ATM withdrawals and six cheque dishonour fees b Three branch withdrawals and five telephone enquiries c Seven telephone enquiries and two dishonoured cheques d Two ATM withdrawals, five telephone enquiries and six branch withdrawals e Eleven ATM withdrawals, ten telephone enquiries and one branch withdrawal f Thirty ATM withdrawals, twenty telephone enquiries and ten branch withdrawals

4 

The table below shows the banking charges for four banks. Periodic Overdrawn Bank payment account

a b c d e f g h i j

Cheque dishonour

Late payment

A

$35

$35

$35

$35

B

$35

$30

$45

$25

C

$45

$38

$30

$35

D

$40

$40

$35

$35

What is the cost of an overdrawn account with bank B? What is the cost of a cheque dishonour fee at bank D? Which bank has the lowest late payment fee? Which bank has the highest overdrawn account fee? Calculate the difference in the periodic payment fee between banks C and D. Calculate the difference in the cheque dishonour fee between banks B and C. What is the average overdrawn account fee for these banks? What is the average periodic payment for these banks? Which bank has the lowest overall bank fees? Which bank has the highest overall bank fees?

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6.7 Appreciation and inflation Appreciation 6.7

Appreciation is the increase in value of items such as art, gold or land. This increase in value is often expressed as the rate of appreciation. Calculating the appreciation is similar to calculating the compound interest. For example, a painting worth $100 000 that has an annual rate of appreciation of 10% will be worth $110 000 after one year (an increase of $10 000). In the second year its value will increase by $11 000. The amount of appreciation has increased.

Appreciation A = P(1 + r)n or FV = PV(1 + r)n A – Amount (final balance) or future value P – Principal is the initial quantity of money or present value r – Rate of appreciation per compounding time period expressed as a decimal n – Number of compounding time periods

Example 12

Finding the appreciated value

Joel bought a unit for $290 000. If the unit appreciates at 9% p.a., what is its value after 7 years? Answer to the nearest dollar.

Solution 1 2 3 4 5

Write the formula for appreciation. Substitute P = 290 000, r = 0.09 and n = 7 into the formula. Evaluate. Write answer to the correct degree of accuracy. Answer the question in words.

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A = P(1 + r )n = 290 000(1 + 0.09)7 = 530 131.345 = $530 131 Unit is valued at $530 131.

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Inflation Inflation is a rise in the price of goods and services or Consumer Price Index (CPI). It is measured by comparing the prices of a fixed basket of goods and services. If inflation rises then a person’s spending power decreases. The inflation rate is the annual percentage change in the CPI. In Australia, the Reserve Bank aims to keep the inflation rate in a 2% to 3% band. Inflation Inflation rate is the annual percentage change in the CPI. Use the formula A = P(1 + r)n to calculate the future value of an item following inflation.

Example 13 a

b

Finding the price of goods following inflation

What is the price of a $650 clothes dryer after one year following inflation? (Inflation rate is 2.6% p.a.) What is the price of a $400 clothes dryer after three years following inflation? (Inflation rate is 3.2% p.a.)

Solution 1 2 3 4 1 2 3 4

Write the formula for inflation. Substitute P = 650, r = 0.026 and n = 1 into the formula. Evaluate correct to 2 decimal places. Write the answer in words. Write the formula for inflation. Substitute P = 400, r = 0.032 and n = 3 into the formula. Evaluate correct to 2 decimal places. Write the answer in words.

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a

A = P(1 P(1 + r )n = 650(1 650(1 + 0.026)1 = $666.90

Clothes dryer will cost $666.90. b A = P(1 P(1 + r )n = 400(1 400(1 + 0.032)3 = $439.64 Clothes dryer will cost $439.64.

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Exercise 6G 1

A vintage car was bought for $70 000 and appreciated at the rate of 6% p.a. What will be the value of the car after 4 years? Answer correct to the nearest cent.

2

The price of a house has increased by 4.5% for each of the last two years. It was bought for $490 000 two years ago. What is the new current value?

3

William bought the following antiques. a Tall boy valued at $4450. Each year its value appreciated by 5%. Calculate the value of the tall boy after 3 years. Answer correct to the nearest cent. b Table valued at $6200. Each year its value appreciated by 4%. Calculate the value of the table after 5 years. Answer correct to the nearest cent. c Chair valued at $1250. Each year its value appreciated by 9%. Calculate the value of the chair after 4 years. Answer correct to the nearest cent.

4

The collection of dolls was valued at $1500 four years ago. If it appreciated at 12% p.a., find its current value. Answer correct to the nearest cent.

5

The price of a diamond ring has increased from $3400 to $5300 during the past five years due to inflation. What is the rise in the price of the ring?

6

The average inflation for the next five years is predicted to be 3%. Calculate the price of the following goods in five years time. Answer correct to the nearest cent. a 3 L of milk for $3.57 b Loaf of bread for $3.30 c 250 g honey for $4.50 d 800 g of eggs for $5.20

7

If the inflation rate is 5% p.a., what would you expect to pay to the nearest dollar in four years time for a house that costs: a $280 000? b $760 000? c $324 000? d $580 000? e $1 260 000? f $956 000?

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Development 8

The graph below shows the value of cricket memorabilia for the past 6 years. a What was the value of the memorabilia after 4 years? b What was the initial value? c How much did the memorabilia appreciate each year? d Find the equation of the straight-line graph in terms of V and n. Cricket memorabilia

V ($) 300 200 100 1

2

3

4

5

6

n

9

The following blocks of land have increased in value this year. What is the rate of appreciation? Answer correct to two decimal places. a $328 000 to $352 000 b $256 000 to $278 000

10

Hayley invested $700 on rugby memorabilia. It appreciated over 7 years at 8.2% p.a. interest compounding annually. a What is the value of the investment after 7 years? Answer correct to the nearest cent. b Calculate the amount of appreciation.

11

An investment is appreciating at a rate of 8% of its value each year. Kumar decides to invest $20 000. a What will be the investment’s value after 5 years? Answer correct to the nearest cent. b How much does the investment increase during the first 5 years? c When will the investment at least double in value? Answer to the nearest year.

12

The cost of a certain car has increased during the past two years from $45 200 to $49 833 following inflation. What was the annual inflation rate?

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6.8 Shares and dividends Shares A share or stock is a part ownership in a company. Shares are bought and sold on the stock market or stock exchange, such as the ASX (Australian Securities Exchange). The ASX provides current information about share prices, market data, tools and resources for investment. When a company is first listed on the ASX the initial price is called the face value. The price of a share will change according to the performance of the company. The current price of a share is called the market price. The amount paid when a share is bought is called the cost price or issued price. The amount paid when a share is sold is called the selling price. Shares are bought and sold using a broker. A broker receives a brokerage fee when shares are traded or a percentage value of the transaction. There are many different types of brokers, such as CommSec. Goods and Services Tax (GST) of 10% is charged for buying and selling shares.

Example 14

Calculating the cost of shares

Lucy bought 500 shares at a market value of $6.80 each. Brokerage costs incurred were $33 including GST. What is the total cost of purchasing the shares? Solution 1 2 3 4

Write the quantity (cost) to be calculated. Cost = (500 × 6.80 8 0) + $ 3 3 Multiply 500 by $6.80 and add the brokerage of $33. = $3433 Evaluate. Cost of the shares is $3433. Write answer using correct units and in words.

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Dividends The owners of a share are entitled to share in the company’s profits. Profits are given to shareholders as a dividend. Dividend is a payment given as an amount per share or as a percentage of the issued price. Dividends are issued twice per year: interim and final dividend. The dividend yield is the annual rate of return. It is calculated by dividing the annual dividend by the share’s market price.

Dividend History

Type

Cents per share

Frank %

Ex Dividend Date

Interim

62.00

100

8/5/2010

1/7/2010

Final

74.00

100

8/11/2009

21/12/2009

Interim

62.00

100

14/5/2009

2/7/2009

Final

69.00

100

9/11/2008

15/12/2008

Interim

56.00

100

15/5/2008

3/7/2008

Dividend Pay Date

Dividends Dividend is a payment given as an amount per share or a percentage of the issued price. Dividend yield =

Example 15

Annual dividend × 100% Dividend = Dividend yield × Market Price Market price

Calculating the dividend and dividend yield

The share price of a company is $28.42. a The predicted dividend yield is 3.5%. What would be the dividend? b The company decides to pay a dividend of $1.12. What is the dividend yield?

Last Trade 28.420

Today’s % Change Volume -0.280 -0.976

51645

Trades 63

Solution 1 2

3 4 5

6

Write the quantity to be found. Express the dividend yield as a decimal and multiply it by the share price. Evaluate correct to two decimal places. Write the formula to calculate the dividend yield. Substitute the annual dividend of 1.12 and the market price of 28.42 into the formula. Evaluate correct to two decimal places.

a

Dividend = 3.5% of $28.42 = 0.035 × $28.42 = $0.99 Dividend is $0.99

b

Annual dividend × 100% Market pricee 1.112 = × 100% 28.42 = 3.94 94%

Dividend yield =

Dividend yield is 3.94%

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Exercise 6H 1

A broker charges a fee of $33 to trade shares. What is the total cost for these shares? a 150 shares, market price of $19.70 b 340 shares, market price of $2.41 c 60 shares, market price of $92.35 d 2000 shares, market price of $1.68 e 208 shares, market price of $49.61 f 3900 shares, market price of $56.23

2

Molly buys 3000 shares with a market price of $6.20. She pays a brokerage fee of $22.50. How much does she pay altogether?

3

Calculate the dividend yield (to two decimal places) on the following shares. a Market price of $33.70 and a dividend of $0.84 per share b Market price of $22.08 and a dividend of $1.63 per share c Market price of $20.58 and a dividend of $1.00 per share d Market price of $37.72 and a dividend of $2.11 per share e Market price of $45.43 and a dividend of $2.86 per share f Market price of $4.00 and a dividend of 10 cents per share

4

A company with a share price of $8.40 declares a dividend of 56 cents. What is the dividend yield correct to two decimal places?

5

Calculate the dividend received on the following shares. (Answer correct to two decimal places.) a 500 shares with a market price of $4.80 and a dividend yield of 5.2% b 80 shares with a market price of $88.10 and a dividend yield of 4.4% c 2200 shares with a market price of $9.56 and a dividend yield of 6.1% d 890 shares with a market price of $22.30 and a dividend yield of 1.9% e 3400 shares with a market price of $56.30 and a dividend yield of 4.6% f 780 shares with a market price of $12.58 and a dividend yield of 8.1%

6

A company with a market price of $1.30 has a dividend yield of 9.2%. What is the dividend correct to two decimal places?

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Development 7

Tipeni bought 100 shares in a bank for $35.60 each. He sold them two years later for $48.90 each and paid a brokerage fee of $32.95. a What is the profit made on these shares? b What is the profit as a percentage of the cost of the shares?

8

Cooper bought 2500 shares for $5.60 each. a Cooper is charged a brokerage fee of 6.2 cents per share. What is the total cost of purchasing the shares? b Two months later a dividend of 36 cents per share was paid. What was the total dividend Cooper received? c Cooper sold the shares after receiving the dividend for $5.75 each. He was charged a brokerage fee of 6.2 cents per share. What was the profit on these shares?

9

The dividend yield on a company was 5%. How much is the dividend if you owned 500 shares with a market value of $4.80?

10

A company started trading on the ASX with a face value of $8.40. 8.5 8.0 7.5 7.0 6.5 May a b c

Jun

Jul

What was the share price at the beginning of June? How much has the share price decreased during the 3 months? Express the decrease in the share price as a percentage of the face value.

11

A company has an after tax profit of $128 million. There are 200 million shares in the company. What dividend per share will the company declare if all the profits are distributed to shareholders?

12

Lauren owns 400 $3 ordinary shares and 300 $2 preference shares. The current prices of the ordinary shares and preference shares are $4.20 and $3.60 respectively. The dividend on the ordinary shares is 55c and on the preference shares is 3%. Calculate Lauren’s total dividend.

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Chapter 6 — Investing money

Study guide 6

Simple interest

I = Prn A=P+I I – Interest (simple or flat) earned for the use of money P – Principal is the initial amount of money borrowed r – Rate of simple interest per period expressed as a decimal n – Number of time periods A – Amount or final balance

Simple interest graphs

1 2 3

Construct a table of values for I and n using I = Prn. Draw a number plane – n horizontal axis, I the vertical axis. Plot the points and join them to make a straight line.

Compound interest

A = P(1 + r)n or FV = PV(1 + r)n I=A-P A – Amount (final balance) or future value of the loan P – Principal is the initial quantity of money r – Rate of interest per compounding time period (decimal) n – Number of compounding time periods A – Amount or final balance I – Interest (compound) earned

Compound interest

1

graphs

2 3

Using prepared tables

1 2 3

Financial institutions

• •

Appreciation and inflation Shares and dividends

Construct a table of values for A and n using A = P(1 + r)n. Draw a number plane – n horizontal axis, A the vertical axis Plot and join the points and to make an exponential curve. Determine the time period and rate of interest. Find the intersection of the time period and rate of interest in the table. Multiply the number in step 2 with the money invested. Charges – account servicing charge, transaction fees Penalties – overdrawn fee, cheque dishonour fee, late payment fee

Use the formula A = P(1 + r)n for appreciation and inflation. Inflation rate is the annual percentage change in the CPI. Dividend is a payment given as an amount per share or a percentage of the issued price. Dividend yield =

Annual dividend × 1100% 00% Market price

Dividend = Divi Dividend Di vide dend nd yiel yield yi eldd × M Marke ar t price arke

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Review

Chapter summary – Investing money

183

Review

184

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

What is the simple interest on $500 at 8% p.a. for 4 years? A $40 B $160 C $660

D

$1600

2

What is the simple interest rate when $8000 increases to $8600 over 3 years? A 2.5% p.a. B 3% p.a. C 6% p.a. D 7.5% p.a.

3

Using the graph what is the interest after 3 12 years? Simple interest at 4% p.a.

I 240 200 160 120 80 40

n 1 A

$120

2

3

4

5

6 B

$140

C

$160

D

$240

4

What was the amount of the investment shown in the graph in question 3? A $40 B $100 C $240 D $1000

5

James borrows $3000 at 10% p.a. interest compounding annually. What is the amount owed after 2 years? (Answer to the nearest dollar.) A $3030 B $3060 C $3600 D $3630

6

What is the interest earned for 3 years on $6000 at 9% p.a. interest compounding monthly? (Answer to the nearest dollar.) A $1770 B $1852 C $7770 D $7852

7

A painting was bought for $460 000 and appreciated at the rate of 7% p.a. What will be the value of the painting after 4 years? (Answer to the nearest dollar.) A $473 016 B $492 200 C $588 800 D $602 966

8

Chloe bought 800 shares at a market value of $10.20 each. Brokerage costs incurred were $38 including GST. What is the total cost of purchasing the shares? A $48.20 B $8160 C $8198 D $30 400

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1

What is the simple interest on $1250 at a flat rate of 8% p.a. over 3 years?

2

Susan wants to earn $9000 a year in interest. How much must she invest if the simple interest rate is 14% p.a.? Answer to the nearest dollar.

3

Matthew takes out a simple interest loan of $14 000 over 3 years at 7% p.a. a How much interest does he pay each year? b How much interest does Matthew pay altogether for the loan?

4

Nicholas invested $1000 at 5% per annum simple interest for 6 years. a Simplify the simple interest formula (I = Prn) by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

0

1

2

3

4

5

I c d e

Draw a number plane with n as the horizontal axis and I as the vertical axis. Plot the points from the table of values. Join the points to make a straight line. What is the interest earned after 5 12 years?

5

Riley invested $120 000 with a superannuation fund. What sum of money will he receive, to the nearest dollar, if invested for 4 years at: a 3% p.a. compound interest? b 6% p.a. compound interest?

6

Calculate the amount that must be invested at 5.8% p.a. interest compounding annually to have $50 000 at the end of 4 years. Answer to the nearest dollar.

7

Farid invested $24 000 over 8 years at 9.8% p.a. interest compounding quarterly. What is the value of the investment after 8 years? Answer to the nearest dollar.

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Review

Sample HSC – Short-answer questions

Review

186

Preliminary Mathematics General

Chapter summary – Earning Money 8 The table below shows the future value when $1 is invested at the given interest rate for the given number of periods. The interest is compounded per period. Period

3%

6%

9%

12%

15%

2

1.061

1.124

1.188

1.254

1.323

4

1.126

1.262

1.412

1.574

1.749

6

1.194

1.419

1.677

1.974

2.313

8

1.267

1.594

1.993

2.476

3.059



Use the table to calculate the future value of the following, correct to nearest dollar. a $40 000 invested for 8 years at 15% p.a. compounded annually b $350 000 invested for 6 years at 9% p.a. compounded annually c $400 000 invested for 4 years at 12% p.a. compounded six-monthly d $64 000 invested for 2 years at 12% p.a. compounded quarterly

9

The table (right) shows the charges for using a bank account. a What is the charge for 7 ATM withdrawals and 5 cheque dishonour fees? b What is the charge for 10 branch withdrawals and 12 telephone enquiries?

Activity

Charge

ATM withdrawal fee

$0.45

Branch withdrawal

$0.50

Cheque dishonour fee

$7.00

Telephone enquiry

$0.30

10

An investment is appreciating at a rate of 4% of its value each year. Ruby decides to invest $480 000. a What will be the investment’s value after ten years? Answer to the nearest dollar. b How much does the investment increase during the first ten years?

11

The average inflation for the next five years is predicted to be 2.5%. Calculate the price of the following goods in three years time. Answer to the nearest cent. a 2 L of soft drink for $2.80 b Apple pie for $4.60

12

Mia bought 50 shares in a bank for $82.64 each. She sold them two years later for $68.90. The brokerage fee paid was $35.95 each time. What was the loss on these shares?

13

A company with a share price of $2.42 declares a dividend of 12 cents. What is the dividend yield, correct to two decimal places?

14

Ebony purchases 1000 shares at $16.90 per share. Her broker charges $20 plus 1.5% of the purchase price. Calculate the brokerage for this purchase. Challenge questions 6

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HSC Practice Paper 1 Section I Attempt Questions 1–15 (15 marks) Allow about 20 minutes for this section 1

Which of the following is the correct simplification of 11x4 - 7x4? (A) 4 (B) 4x (C) 4x4 (D) 4x8

2

What is the new running cost of a vehicle if $60 is increased by 20% then decreased by 20%? (A) $38.40 (B) $57.60 (C) $60.00 (D) $86.40

3

Elizabeth works for $19.20 per hour for eight hours each day on Thursday and Friday. On Saturday she works for six hours at time-and-a-half. How much does Elizabeth earn in total for Thursday, Friday and Saturday? (A) $268.80 (B) $307.20 (C) $480.00 (D) $691.20

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Preliminary Mathematics General

4

What is the equation of line m? (A)

y = 3x − 3

(B)

y=

(C)

y = −3 x − 3

(D)

y=−

y

2

x −3 3

x −3 3

m

3 1 −33 −22 −11 0 −1

1

2

3

x

−2 −3

5

Simplify 6 - 4(2x - 1). (A) 4x - 2 (B) 4x + 6 (C) 7 - 8x (D) 10 - 8x

6

Which of the following is the highest pay? (A) $1442.22 per week (B) $2884.68 per fortnight (C) $6247.50 per month (D) $75 000 per annum

7

Arrange the numbers 4.8 × 10-2, 4.0 × 10-1 and 5.6 × 10-2 in ascending order. (A) 4.8 × 10-2, 5.6 × 10-2, 4.0 × 10-1 (B) 4.0 × 10-1, 4.8 × 10-2, 5.6 × 10-2 (C) 5.6 × 10-2, 4.8 × 10-2, 4.0 × 10-1 (D) 4.0 × 10-1, 5.6 × 10-2, 4.8 × 10-2

8

A survey required mobile phone users to write down their age last birthday. Which of the following terms best describes this data? (A) Categorical (B) Continuous (C) Discrete (D) Stratified k Find the value of m, correct to one decimal place, given k = 24 and the formula m = . 5 (A) 1.0 (B) 2.2 (C) 2.4 (D) 4.8

9

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HSC Practice Paper 1

189

10

A four-litre tin of paint is made using a mixture of blue, white and green paint in the ratio 3 : 5 : 2. How much blue paint is needed per tin? (A) 300 mL (B) 900 mL (C) 1200 mL (D) 2000 mL

11

A country has 30% of the population between the ages of 20 and 30. How many people aged between 20 and 30 should be included in a sample of 250 people? (A) 30 (B) 45 (C) 60 (D) 75

12

A car is travelling at a constant speed. It travels 60 km in 3 hours. This situation is described by the linear equation d = mt. What is the value of m? (A) 0.05 (B) 3 (C) 20 (D) 60

13

What is the value of x if 2x - 7 = 21? (A) 3 (B) 7 (C) 14 (D) 28

14

Police checked the blood alcohol content of every fifth driver passing an intersection. What is this method of sampling? (A) Census (B) Random (C) Stratified (D) Systematic

15

What is the point of intersection of the lines y = x + 1 and y = -x + 1? (A) (0, 0) (B) (0, 1) (C) (1, 0) (D) (1, 1)

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Preliminary Mathematics General

Section II Attempt Questions 16–18 (45 marks) Allow about 70 minutes for this section All necessary working should be shown in every question. Question 16 (a)

(b)

Marks

(15 marks)

Nathan is an auto-electrician who is entitled to a 15% trade discount. In addition, he is given a 10% reduction on the discounted price if he pays cash. Nathan bought $480 worth of electrical equipment and was given both discounts. (i) How much does Nathan pay for the electrical equipment? (ii) How much money is saved using the discounts? (iii) Express the overall savings as a percentage of the retail price.

1 1 1

If x = 6 and y = -2, calculate the value of each of the following: (i) x-y (ii) xy (iii) x2 - 5y2

1 1 1

(c)

A truck has a load of six boxes of equal weight. The total weight of the boxes is 4.68 tonnes. What is the weight of each box in kilograms?

(d)

Chloe agreed to a car loan for $3500 at 6.5% p.a. compounding annually for 4 years. (i) How much does Chloe have to repay? (ii) How much interest did Chloe pay during the 4 years?

1 2

Simplify: (i) 12x - 2(x + 1) (ii) 48x6y3 × 6xy3

1 1

(e)

(f)

Alexis’s parents want to give her $10 000 for her wedding in five years’ time. They have found an account that will earn 8% p.a. simple interest. What is the amount of money they need to invest in this account to total $10 000 interest in 5 years’ time?

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2

2

Cambridge University Press

HSC Practice Paper 1

Question 17 (15 marks) (a)

191

Marks

y 3 2 1 −3 −2 −1 0 −1

1

2

3

x

−2 −3 (i) (ii) (iii)

(b)

Solve these equations. (i) 11 = x - 4 (ii)

(c)

(d)

(e)

(f)

1 1 1

What is the gradient of this straight line? What is the y-intercept of this straight line? What is the equation of this straight line?

1

8x − 2 = 4x 3

2

Riley received a gross fortnightly salary of $2858. His pay deductions were $745.20 for tax, $305.13 for superannuation and $21.20 for union fees. (i) What was his fortnightly net pay? (ii) What percentage of his gross income was deducted for tax? (Answer correct to 1 decimal place.) (iii) Riley is paid an annual leave loading of 17.5% of 4 weeks gross pay. Calculate his annual leave loading.

1 1 1

Sarah conducted a survey of students’ opinions about the school uniform. She selected the first five people who were not in school uniform for the survey. Why might the results of this survey be biased?

2

Light travels at a speed of 2.9979 × 108 metres per second. How many kilometres does light travel in one hour? Answer in scientific notation correct to three significant figures.

2

Grace earned $369.60 in a week when she worked 27 hours at the normal rate and 4 hours at time-and-a-half. What is Grace’s pay rate per hour?

2

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Preliminary Mathematics General

Question 18 (15 marks) (a)

Marks

The graph below shows the cost of making dresses and the income received from their sale.

Dollars ($)

100 80 Costs

60 40

Income

20 0 (i) (ii) (iii)

(b)

(c)

(d)

(e)

1

2

3 Dress

4

5

How much profit or loss is made when 1 dress is sold? How much profit or loss is made when 5 dresses are sold? Use the graph to determine the number of dresses that need to be sold to break even.

1 1 1

Oscar invests $100 000 for 8 months in a term deposit. This investment offers a flat rate of 6% per annum interest. What is the interest earned from this investment?

3

Volume of a sphere is given by the formula V = 4 π r 3 where r is the radius. 3 (i) Write the formula with r as the subject. (ii) What is the radius in metres of a spherical balloon with a volume of 2 m3? Answer correct to two decimal places.

2

Hannah bought 1000 shares for $6.40 each. She received a dividend of 5.5% on her purchase price. Hannah sold these shares for $7.20 each after 3 months. Calculate Hannah’s profit from the dividend and the sale of these shares. Logan records the details of each CD in his music collection. State the type of data that would be recorded for each of the following. (i) Title of the CD. (ii) Number of tracks on the CD. (iii) Playing time for each track. © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1

3

1 1 1 Cambridge University Press

C H A P T E R

7

Displaying and interpreting single data sets Syllabus topic — DS2 Displaying and interpreting single data sets Create frequency tables to organise ungrouped and grouped data Calculation and interpretation of range and interquartile range Create frequency and cumulative frequency graphs Draw radar charts to display data Create a box-and-whisker plot from a five-number summary Create sector graphs and divided bar graphs for categorical data

7.1 Frequency tables A frequency table is a listing of the outcomes and how often (frequency) each outcome occurs. The outcomes are often listed under a heading called ‘score’. The tally of the frequency and the final count are listed in separate columns. A frequency table is also called a frequency distribution. 7.1

Score Lowest score

Highest score

Tally

Frequency

17

|

1

18

|||| |

6

19

||||

5

20

|||| ||

7

21

|||

3

Lowest frequency

Highest frequency

Frequency table 1 2 3

Scores or outcomes are listed in the first column in ascending order. Tally column records the number of times the score occurred (groups of 5s). Frequency column is the total count of each outcome. 193

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194

Preliminary Mathematics General

Example 1

Reading information from a frequency table

The following frequency table shows the results of a test out of 20. a How many students scored 15 in the test? b What was the score that occurred the most times? c How many students completed the test?

Score

Solution 1 2 3

Read the frequency of 15. Find the largest number in the frequency column. Add the frequency column to find the total number of students.

Example 2

a b c

Tally

Frequency

14

|||

3

15

||||

5

16

|||| ||

7

17

|||| |||| |

11

18

|||

3

Five students scored 15. Score of 17 occurred the most. Total = 3 + 5 + 7 + 11 + 3 = 29

Constructing a frequency table

The temperatures for 39 days are shown below. Construct a frequency table. 19

20

18

23

27

25

26

27

28

27

25

24

24

19

25

22

21

28

26

26

22

20

25

20

22

24

24

22

21

24

25

26

25

27

21

23

23

22

25

Solution 1 2 3 4 5

Draw a table with three columns and label them score, tally and frequency. List the temperatures in the score column from the lowest (18) to the highest (28). Record a mark in the tally column for each temperature. Count the tally marks and write the total in the frequency column. Add the frequency column to find the total number of scores. This should match the total number of temperatures (39).

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Score

Tally

Frequency

18

|

1

19

||

2

20

|||

3

21

|||

3

22

||||

5

23

|||

3

24

||||

5

25

|||| ||

7

26

||||

4

27

||||

4

28

||

2

Total

39

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195

Chapter 7 — Displaying and interpreting single data sets

Exercise 7A 1

The age of players in a football team is recorded in a frequency table. a Copy and complete the table. b What was the most common age? c How many players are in the team?

Score

Tally

20

|||

21

|||| ||

22

6

23

||||

24 2

The number of times a fire engine is called out on a given day was recorded in a frequency table. a Copy and complete the table. b What was the most common number of calls? c How many days was the fire engine called out four times? d How many days was the fire engine called out fewer than three times?

2

Number of calls

Tally

0 |||| |||| ||

1 2

8

3

|||| 2

5

4

|

The number of brothers and sisters for 30 students is recorded below. Construct a frequency table for this data. 2

1

1

5

3

4

1

0

2

1

5

2

6

1

0

3

1

0

4

3

3

4

2

3

0

2

0

1

0

2

The shoe size of 20 seventeen year olds is recorded below. Construct a frequency table. 11 9

5

Freq. 7

4

3

Freq.

7 9

8 8

10

12

10

10

9

8

8 9

7 8

8 7

11 8

The assessment result for 30 students is recorded below. Construct a frequency table for this data. 96

97

97

95

92

94

97

98

91

97

95

91

96

97

94

92

92

98

94

93

93

94

95

93

92

91

98

96

92

91

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Preliminary Mathematics General

Development 6

A die was rolled and the results listed below. 1

1

5

2

2

5

1

3

6

1

4

3

1

5

5

4

6

3

3

4

2

1

1

6

5

4

2

6

6

1

4

4

1

6

1 6 5 2 4 6 2 4 2 2 3 6 4 2 Construct a frequency table using a tally column. How many times was the die rolled? How many results are higher than 2? What was the most common number rolled? Do you think the die is biased? Give a reason for your answer.

4

1

4 a b c d e

7

David recorded the following times, in seconds, for the 50 m freestyle. 32

34

37

35

35

37

34

33

38

34

36

33

34

37

37

37

38

33

33

36

35

34

34

38

32

36

32

38

38

34

36

36

34

38

32

34

38

37

35

36

38

35

32

35

35

33

38

36

35

36

32

a b c d e

8 

Construct a frequency table using a tally column. How many times were recorded? How many times were below 35 seconds? What was the most common time for the 50 m freestyle? What percentage of times are 37 seconds or more?

Count the number of letters in each word of the paragraph below. A frequency table is a listing of the outcomes and how often (frequency) each outcome occurs. The outcomes are often listed under a heading called ‘score’. The tally of the frequency and the final count are listed in separate columns. Frequency tables are also called a frequency distribution. a Create a frequency table for the length of words used in the above paragraph. b Using the frequency table, what is the most frequent word length in the English language? c Comment on the fairness of the conclusion made in part b.

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197

7.2 Grouped frequency tables

7.2

Data with a large range of values are often grouped into a small number of convenient intervals, called class intervals. When choosing class intervals ensure: • Every data value is in an interval. • Intervals do not overlap. • No gaps exist between the intervals. The choice of intervals can vary, but generally a division of 5 to 15 groups is preferred. It is usual to choose an interval that is easy to read such as 5 units, 10 units, etc. Grouped frequency table 1 2 3

Classes or groups are listed in the first column in ascending order. Tally column shows the number of times a score occurs in a class (groups of 5s). Frequency column is total count of scores in each class.

Example 3

Constructing a grouped frequency table

Twenty-six people were asked to record how many cups of coffee they drank in a particular week. The results are listed below. 0

33

6

14

0

32

0

25

10

0

2

9

23

0

34

5

17

3

0

23

1

32

0

8

0

2

Solution 1

2

3

4 5

Draw a table with 4 columns and label them class, class centre, tally and frequency. The data ranges from 0 to 34. A class interval width of 5 results in 7 classes. Record a mark in the tally column for each data value in the class interval. Count the tally marks in the frequency column. Add the frequency column to find the total number of scores.

Class

Class centre

Tally

Freq.

0–4

2

|||| |||| ||

12

5–9

7

||||

4

10–14

12

||

2

15–19

17

|

1

20–24

22

||

2

25–29

27

|

1

30–34

32

||||

4

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Total

26

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198

Preliminary Mathematics General

Exercise 7B 1

2

The age of the people living in Matilda Rd was recorded. a Copy and complete the table opposite. b How many people are younger than 20? c Which class occurred the most number of times? d How many people are living in this road?

A grouped frequency table is shown below. a Copy and complete the grouped frequency table. b How many class intervals have been used? c Which class occurred the least? d What is the total number of scores?

Class

Class Centre

Class

Freq.

5–19

10

20–34

8

35–49

6

50–64

4

Class Centre

3–7 10

Total

28

Tally

Frequency

||||

4

|||| |

6

13–17

15

2

18–22

20

|||| ||

25

|

1

28–32 33–37

3

3 35

|||| |||

40

||||

5

The heights of the 30 players in a netball club are recorded below. Construct a grouped frequency table using class intervals (170–174, 175–179, 180–184, 185–189, …).

174

184

183

179

180

181

189

188

194

189

184

182

183

189

193

194

185

178

173

183

188

183

182

184

189

194

188

192

190

180

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199

Development 4

A drink vending machine records the number of bottles sold each day. 13

22

26

34

21

33

32

31

33

26

30

33

37

27

36

28

35

24

31

3

35

39

26

13

34

21

29

36

24

33

25

9

39

19

29

36

38

29

38

38

37

38

31

37

37

28

40

34

29

18

35

22

20

35

25

31

39

39

18

36

38

35

8

29

35

20

34

30

37

33

27

32

32

36

16

39

30

14

29

20

22

12

24

17

21

18

17

38

28

25

a b c

5

The players’ scores after the second round of a golf tournament are recorded below. 162

163

175

161

166

163

167

151

150

176

159

173

162

155

149

171

181

163

154

165

145

177

184

171

154

166

168

158

136

156

161

162

169

162

160

150

174

176

146

137

a b c

6 

Decide on appropriate classes for a frequency table. Calculate the class centres for these classes. Construct a grouped frequency table using these class intervals.

Decide on appropriate classes for a frequency table. Calculate the class centres for these classes. Construct a grouped frequency table using these class intervals.

Jordan surveyed his friends to check the number of emails they saved on their computers. The numbers he found were 22, 9, 51, 6, 30, 18, 30, 4, 10, 5, 19, 23, 37, 17, 18, 12, 10, 24, 28, 25, 60, 45, 19, 17, 11, 8, 16, 1, 24 and 3. a Decide on appropriate classes for a frequency table. b Calculate the class centres for these classes. c Construct a grouped frequency table using these class intervals. d What percentage of friends had fewer than 20 emails saved on their computer?

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Preliminary Mathematics General

7.3 Cumulative frequency

7.3

Cumulative frequency is the frequency of the score plus the frequency of all the scores less than that score. A cumulative frequency column is often inserted next to the frequency column in a frequency table.

Score

Frequency

Cumulative Frequency

18

1

1

19

5

6

20

3

9

21

7

16

Cumulative frequency The frequency of the score plus the frequency of all the scores less than that score.

Example 4

Calculating the cumulative frequency

The frequency table opposite shows the temperatures for 17 days. Complete the cumulative frequency column.

Solution 1 2

3

The lowest score is 18 and it has a frequency of 1. Cumulative frequency of 18 is 1. Add the frequency of the score to the cumulative frequency of the previous score. Cumulative frequency of 19 = 2 + 1 = 3. Repeat this process.

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Score

Frequency

18

1

19

2

20

3

21

3

22

5

23

3

Cumulative Frequency

Score

Freq.

Cml. Freq.

18

1

1

19

2

2+1=3

20

3

3+3=6

21

3

6+3=9

22

5

9 + 5 = 14

23

3

14 + 3 = 17

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Chapter 7 — Displaying and interpreting single data sets

Finding cumulative frequency using a graphics calculator

Example 5

Score

Frequency

1

5

2

9

3

4

4

6

a b c d

201

Cumulative Frequency

Calculate the cumulative frequency using a graphics calculator. What is the total number of scores? How many scores are less than or equal to 2? How many scores are greater than 3? Solution 1 2 3 4 5 6 7

Select the STAT menu. Enter the scores into List 1. Enter the frequencies into List 2. Highlight the words List 3 using the arrow keys. Press OPTN and select LIST. Press F6 twice and select Cum1. Press F6 and select List. Type the number 2.

8

Press EXE and the cumulative frequencies will appear in List 3.

9

List 3 is the cumulative frequency. The total number of scores is the last number in this column. Cumulative frequency of 2 is 14 (List 3). This is the total number of scores less than or equal to 2. The only score greater than 3 is 4. The frequency of 4 is found in the List 2 column.

10 11

a

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b

Total number of scores is 24.

c

Total number of scores less than or equal to 2 is 14. There are 6 scores greater than 3.

d

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Preliminary Mathematics General

Exercise 7C 1

2

3

The frequency table shows the results of a test. a Copy and complete the table. b How many students scored 8? c How many students scored more than 5? d How many students scored less than 6? e How many students completed the test?

The number of times an ambulance was called out each day is recorded in a frequency table. a Copy and complete the table. b How many days was the ambulance called out 21 times? c How many days was the ambulance called out less than 25? d How many days was the ambulance called out more than 23?

Score

Freq.

4

4

5

6

6

7

7

10

8

5

Number of calls

Freq.

20

4

21

3

22

10

23

12

24

6

25

5

Cml. Freq.

Cml. Freq.

The results of a survey are listed below.

a b c d e

13

11

11

10

10

10

9

10

9

11

9

10

12

11

13

10

9

12

11

13

10

11

10

10

10

13

8

8

10

9

Copy and complete the table How many people completed the survey? How many people scored 10 in the survey? What is the difference between the highest and lowest scores? Which score had the highest frequency?

Score

Tally

Freq.

Cml. Freq.

8 9 10 11 12 13

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203

Development 4 

The ages of children at the local park are shown below.  9

8

4

6

10

10

8

6

 5

4

5

6

8

4

9

10

5

8

8

9

10

4

6

10

9

9

7

7

6

5

a b c d e f g h i j k l m

Construct a frequency table to represent this data, including a tally column. Add a cumulative frequency column. How many children are there altogether? How many children are 8 years old? How many children are 6 years old? How many children are older than 5? How many children are younger than 9? What is the most common age? What is the least popular age? What fraction are 7 years old? What fraction are 4 years old? What percentage are older than 5? What percentage are younger than 6?

5

Use a graphics calculator to calculate the cumulative frequency for the data in question 3. Check that the results are the same as question 3.

6

Blake measured his time (in seconds) to run 400 metres hurdles throughout the year. 61

62

62

63

64

62

66

64

63

63

62

61

62

62

63

64

63

65

63

63

62

61

62

63

62

61

61

64

64

64

63

63

64

65

64

63

63

62

61

62

62

63

64

63

62

a b c d e f g h

Construct a frequency table to represent this data including a tally column. Add a cumulative frequency column to the frequency table. How many times were recorded? How many times did Blake run the 400 metres in 61 seconds? How many times are less than or equal to 63? What fraction of his 400 metre times is 62? What percentage of his 400 metre times is 65? Answer correct to two decimal places. What percentage of his times are less than or equal to 64? Answer correct to two decimal places.

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Preliminary Mathematics General

7.4 Range and interquartile range A measure of the spread is calculated to determine whether most of the values are clustered together or stretched out. The range and interquartile range are measures of spread. 7.4

Range The range is the difference between the highest and lowest scores. It is a simple way of measuring the spread of the data. Range Range = Highest score – Lowest score

Calculating the range

Example 6

The assessment results for two different tasks are shown below. Task A

Task B

3

5

10

13

14

14

17

22

24

24

27

27

28

33

38

40

40

41

43

44

45

45

46

50

52

52

55

55

58

95

10

15

19

20

24

27

31

31

35

38

40

49

51

51

54

55

58

62

62

68

68

71

72

76

78

79

79

86

88

90

Find the ranges for Task A and Task B. Solution 1 2 3 4 5 6

Write the formula for range. For Task A, substitute the highest score (95) and the lowest score (3). Evaluate. Write the formula for range. For Task B, substitute the highest score (90) and the lowest score (10). Evaluate.

Task A Range = Highest − Lowest = 95 − 3 = 92 Task B Range = Highest − Lowest = 90 − 10 = 80

In the above example, the range for Task A has been influenced by an outlier (95). The range for Task A is not a good indicator of the spread.

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205

Interquartile range The interquartile range does not rely on the extreme values like the range. The data is arranged in increasing order and divided into 4 equal parts or quartiles. The interquartile range (IQR) is the difference between the first quartile and third quartile. The first quartile cuts off the lowest 3 25% 14 and the third quartile cuts off the lowest 75% 4 .

()

()

Interquartile range IQR = Third quartile – First quartile = Q3 − Q1 1 Arrange the data in increasing order. 2 Divide the data into two equal-sized groups. If n is odd, omit the median. 3 Find Q1 the median of the first group. 4 Find Q3 the median of the second group. 5 Calculate the interquartile range (IQR) by subtracting Q1 from Q3.

Calculating the interquartile range

Example 7

The assessment results for two different tasks are shown below. Task A Task B

3

5

10 13 14 14 17 22 24 24

27

27

28

33

38

40 40 41 43 44 45 45 46 50 52

52

55

55

58

95

10 15 19 20 24 27 31 31 35 38

40

49

51

51

54

55 58 62 62 68 68 71 72 76 78

79

79

86

88

90

Find the interquartile ranges for Task A and Task B. What is shown by these results? Solution 1 2

3 4

5 6

7

Arrange the data in increasing order. Divide the data into two equal-sized groups. There are 30 scores in total, hence 15 scores in each group. Write the formula for interquartile range. For Task A, the median of the first group is 22 (or Q1) and the median of the second group is 46 (or Q3). Substitute into the formula and evaluate. For Task B, the median of the first group is 31 (or Q1) and the median of the second group is 72 (or Q3). Substitute into the formula and evaluate.

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Task A IQR = Q3 − Q1 = 46 − 22 = 24 Task B IQR = Q3 − Q1 = 72 − 31 = 41

Task A is more consistent than Task B (lower IQR).

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Finding the range and IQR using a graphics calculator

Example 8

Use a graphics calculator to calculate the range and interquartile range for Task A. Task A

3

5

10 13 14 14 17 22 24 24

27

27

28

33

38

40 40 41 43 44 45 45 46 50 52

52

55

55

58

95

Solution 1 2

3

4

5

Select the STAT menu. Enter the data into List1. Press EXE to enter each number. Select 1VAR to view the summary statistics. minX – lowest score Q1 = First quartile Q3 = Third quartile maxX = Highest score Substitute into the formula for range and evaluate. Substitute into the formula for IQR and evaluate.

Range = maxX − minX = 95 − 3 = 92 IQR = Q3 − Q1 = 46 − 22 = 24

Deciles, quartiles and percentiles Deciles, quartiles and percentiles are different ways of dividing data. The data must be sorted in order (ascending or descending) before it can be divided. Decile

Percentiles

Data is divided into 10 equal parts

Data is divided into 100 equal parts

1st decile cuts off the lowest 10%

25th percentile cuts off the lowest 25% − Q1

2nd decile cuts off the lowest 20%

50th percentile cuts off the lowest 50% − Q2

9th decile cuts off the lowest 90%

75th percentile cuts off the lowest 75% − Q3

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207

Exercise 7D 1

2

Find the range of each data set. a 13, 6, 0, 9, 6, 5, 6, 17, 1 c 9, 15, 9, 3, 6, 9, 13, 10, 7, 9 e 8, 12, 5, 5, 9, 10, 13, 3, 7 g 13, 6, 9, 9, 3, 9, 15, 7, 9, 10 Find the range of each data set. a 23, 32, 43, 23, 34, 22, 35, 28 c 8, 35, 8, 1, 5, 8, 33, 13, 3, 8 e 3, 22, 5, 5, 8, 13, 31, 3, 3

b d f h

b d f

22, 31, 28, 22, 43, 22 1, 1, 7, 9, 5, 9, 10 3, 0, 1, 2, 11, 9, 7, 7, 5 10, 18, 7, 2, 14, 9, 10

12, 43, 28, 21, 44, 22 3, 3, 3, 8, 5, 8, 13 4, 3, 3, 2, 13, 8, 3, 3, 5

3

Joshua scored the following marks in a spelling test: 4, 5, 5, 6, 7, 7, 8, 8, 9 and 9. a What is the first quartile? b What is the third quartile? c What is the second decile? d What is the seventh decile?

4

Find the interquartile range for each data set (odd number of scores). a 2, 7, 9, 10, 12, 14, 18 b 12, 16, 18, 23, 29 c 2, 2, 2, 3, 7, 13, 14, 14, 18, 24, 55 d 3, 5, 8, 9, 11, 13, 13, 13, 19 e 42, 45, 49, 50, 52, 54, 68, 68, 72 f 0, 1, 4, 6, 6, 6, 10, 13, 19

5

Find the interquartile range for each data set (even data set). a 21, 21, 21, 28, 31, 44 b 3, 5, 5, 7, 8, 9, 12, 13 c 4, 6, 8, 9, 9, 9, 9, 11, 13, 15 d 39, 39, 41, 44, 47, 49, 67, 68, 68, 69 e 3, 6, 8, 11, 12, 14, 22, 23 f 56, 58, 58, 61, 66, 67, 69, 70

6

Find the range and interquartile range for each data set. a 14, 18, 20, 25, 31 b 0, 2, 2, 4, 5, 6, 9, 10

7

There are 101 odd numbers from 1 to 201 (1, 3, 5, 7, …) a What is the 1st percentile? b What is the 10th percentile? c What is the 20th percentile? d What is the 100th percentile? e What is the 50th percentile? f What is the 49th percentile?

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Development 8

9

Find the interquartile range for each data set. a 18, 16, 19, 18, 16, 13, 12, 15 c 32, 29, 24, 26, 25, 28, 29, 24, 30, 22 e 43, 57, 57, 61, 31, 34, 38, 39, 41 g 27, 8, 11, 17, 13, 19, 28, 16

b c d

What is the range? What is the first quartile? What is the third quartile? What is the interquartile range?

The systolic blood pressure for a sample of 20 people is listed below: 100

193

170

138

106

120

109

159

190

144

105

125

180

210

124

148

161

203

192

131

a c e

11

19, 22, 17, 18, 23, 15, 15, 13 d 29, 37, 39, 57, 58, 34, 58, 59, 29, 31 f 40, 50, 46, 41, 46, 53, 59, 44, 46 h 64, 65, 53, 56, 61, 62, 51, 53

The number of service calls per day made by an air-conditioning technician is recorded below: 9, 2, 7, 9, 12, 5, 10, 12, 11, 9, 8, 11, 9, 5, 6, 7, 10 a

10

b

What is the fifth decile? What is the range? What is the third quartile?

b

What is the tenth decile? d What is the first quartile? f What is the interquartile range?

The maximum temperatures for the past 30 days are listed below. 26

33

32

24

36

32

34

29

30

33

37

27

28

29

31

32

33

36

33

34

25

22

23

31

30

34

25

28

27

29

Arrange the data and calculate the following. a First quartile c Interquartile range e 5th decile

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b

Third quartile d Range f 100th percentile

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209

7.5 Frequency and cumulative frequency graphs Frequency histogram and polygon

Frequency Polygon

Frequency histogram is a column graph of a frequency table.

Frequency polygon is a line graph of a frequency table. Frequency

Frequency Histogram

Frequency

7.5

A frequency histogram is a graph of a frequency table in which equal intervals of the scores (or classes) are marked on the horizontal axis and the frequencies associated with these intervals are indicated by vertical rectangles. A frequency polygon is a line graph of the frequency table and can be constructed by joining the midpoints at the tops of the rectangles of a frequency histogram.

3 2 1

3 2 1 1

Score

Example 9

2

3 4 Score

5

6

Constructing a histogram using a graphics calculator

Use a graphics calculator to display the following data as a frequency histogram. 23

22

18

17

13

15

23

18

17

22

18

12

15

13

15

12

14

13

14

17

15

25

19

18

20

16

14

Solution 1 2

Select the STAT menu. Enter the data into List1. Press EXE to enter each number.

3

To construct a histogram select GPH1 and SET. Choose Graph Type : Hist and XList1 : List1.

4

To draw the histogram press GPH1 and EXE.

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Example 10

Constructing a grouped frequency histogram

The weight of 25 students was recorded and displayed in a grouped frequency table. Construct a grouped frequency histogram and polygon. Class

Class Centre

Frequency

30–39

34.5

0

40–49

44.5

5

50–59

54.5

7

60–69

64.5

9

70–79

74.5

4

80–89

84.5

0

2

3

Draw the horizontal axis with each class (or class centre) the same distance apart. Draw a vertical axis using a scale that will cater for the lowest to highest frequency. Label the horizontal and vertical axis.

Frequency histogram 4

Draw a rectangle for each class to the matching frequency. The class is in the centre of the rectangle.

Frequency polygon 5

8 6 4 2 30–39 40–49 50–59 60–69 70–79 80–89 Weight (kg)

Number of students

1

Number of students

Solution

8

6 4 2

Draw a line for each class to the matching frequency.

30–39 40–49 50–59 60–69 70–79 80–89 Weight (kg)

Cumulative frequency graphs A cumulative frequency histogram is constructed using equal intervals of the scores (or classes) on the horizontal axis and the cumulative frequencies associated with these intervals indicated by vertical rectangles. A cumulative frequency polygon or ogive is a line graph constructed by joining the top right-hand corner of the rectangles in a cumulative frequency histogram.

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Chapter 7 — Displaying and interpreting single data sets

Example 11 a b

211

Constructing cumulative frequency graphs

Construct a cumulative frequency histogram and polygon or ogive. Estimate the median using the ogive.

Class

Class Centre

Frequency

Cumulative Frequency

30–39

34.5

0

0

40–49

44.5

5

5

50–59

54.5

7

12

60–69

64.5

9

21

70–79

74.5

4

25

80–89

84.5

0

25

Solution

2

Draw the horizontal axis with each class (or class centre) the same distance apart. Draw a vertical axis using a scale that will cater for the lowest to highest frequency.

a

Number of students

1

25 20 15 10 5 34.5 44.5 54.5 64.5 74.5 84.5 Weight (kg)

3

4

Draw a rectangle for each class to the matching cumulative frequency. The class is in the centre of the rectangle. Draw a line for each class to the top right-hand corner of the rectangle.

Number of students

Cumulative frequency histogram 25 20 15 10 5 34.5 44.5 54.5 64.5 74.5 84.5 Weight (kg)

5 6

7

There are 25 students the median student is 12.5. Draw a horizontal line from 12.5 until it intersects the ogive. Draw a vertical line from this point to the horizontal axis.

Number of students

Cumulative frequency polygon

Estimate the median value.

b

25 20 15 10 5 34.5 44.5 54.5 64.5 74.5 84.5 Weight (kg)

The median is about 60.

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Exercise 7E 1

2

The frequency table shows the results of a mathematics quiz. Use this data to construct a: a Frequency histogram b Frequency polygon

Freq.

4

|||

3

5

||||

5

6

|||| ||

7

7

|||| ||||

10

8

|||

3

3

2

2

6

4

5

2

1

3

2

6

3

7

2

1

4

2

1

5

4

4

5

3

4

1

3

1

2

1

3

Construct a frequency table for this data. Use the frequency table to construct a frequency histogram. Use the frequency table to construct a frequency polygon.

b c

The number of magazines purchased in a month by 20 different people was: 6, 5, 8, 4, 4, 5, 4, 7, 5, 6, 7, 4, 5, 5, 4, 7, 6, 5, 4, 4 Construct a frequency table for the data. Use the frequency table to construct a frequency histogram. Use the frequency table to construct a frequency polygon.

a b c

4

Tally

The numbers of brothers and sisters reported by each of the 30 students is as follows.

a

3

Score

The maximum temperatures for several capital cities around the world are as follows.

a b c

22

23

23

24

18

18

19

19

19

20

21

20

21

20

21

17

24

23

17

22

24

24

17

17

Construct a frequency table for this data. Construct a frequency histogram for this data. Construct a frequency polygon for this data.

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5

Jade owns five clothing stores that sell jackets. She recorded the total number of jackets sold each day for the month of April. This data is shown below. 61

66

67

67

60

63

67

63

65

61

63

63

67

67

60

64

62

65

65

67

66

61

62

66

64

67

62

65

65

67

a b c d e

Construct a frequency table with a cumulative frequency column. Construct a cumulative frequency histogram for this data. Construct a cumulative frequency polygon for this data. On how many days were fewer than or equal to 63 jackets sold? On how many days were fewer than or equal to 66 jackets sold?

The marks for a university exam are shown in the cumulative frequency polygon. Cumulative frequency

6

213

500 400 300 200 100 10 20 30 40 50 60 70 80 90 100 Mark

a c e g i k l

What was the frequency of 10? b What was the frequency of 20? What was the frequency of 30? d What was the frequency of 40? What was the frequency of 50? f What was the frequency of 60? What was the frequency of 70? h What was the frequency of 80? What was the frequency of 90? j What was the frequency of 100? How many students completed this university exam? Construct a frequency table from the cumulative frequency polygon.

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Development 7

The numbers of words in each of the first 30 sentences of a book were recorded. 22

20

22

23

23

24

24

25

25

21

21

20

20

22

26

22

20

24

26

20

21

21

23

22

23

22

20

20

21

21

a b c d e f g

8 

Construct a frequency table with a cumulative frequency column. Use the frequency table to construct a frequency histogram. Use the frequency table to construct a frequency polygon. Construct a cumulative frequency histogram and polygon for this data. Use the cumulative frequency graphs to estimate the median. Use the cumulative frequency graphs to estimate the first and third quartile. Use the cumulative frequency graph to estimate the interquartile range.

The percentage of female births, correct to the nearest whole number, is shown below. These birth percentages have been taken from 30 different hospitals. 38

56

57

59

58

60

43

52

49

61

47

38

41

50

51

55

45

50

49

53

54

48

51

43

55

53

42

42

44

46

a b c d e f g h i

Decide on appropriate classes for a frequency table. Construct a grouped frequency table using these class intervals. Add a cumulative frequency column. Construct a frequency histogram for this data. Construct a frequency polygon for this data. Construct a cumulative frequency histogram and polygon for this data. Use the cumulative frequency graphs to estimate the median. Use the cumulative frequency graphs to estimate the upper and lower quartile. Use the cumulative frequency graph to estimate the interquartile range.

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215

7.6 Box-and-whisker plots

7.6

A box-and-whisker plot or boxplot is a graph that uses five important statistics – lower extreme (or lowest value), lower quartile (or first quartile), median, upper quartile (or third quartile) and the higher extreme (or highest value). These statistics are referred to as a five number summary. A box-and-whisker plot is constructed from a scale of data values. The box is between the two quartiles with a dividing line for the median and the whiskers are drawn to the two extremes. Box-and-whisker plotthe – uses the five-number Box-and-whisker plot – uses five-number summary. summary. Lower extreme

Higher extreme

Example 12

Upper quartile

Median

Lower quartile

Constructing a box-and-whisker plot

The final results of a survey are shown below. 88

67

92

50

73

75

51

76

74

76

65

83

83

90

73

60

81

95

89

76

82

61

57

64

58

78

Construct a box-and-whisker plot, given the five-number summary: Minimum = 50, Q1 = 64, Median – 75.5, Q3 = 83 and Maximum = 95 Solution 1 2 3 4

Draw a labelled and scaled number line that covers the full range of values. Draw a box starting at Q1 = 64 and ending at Q3 = 83. Mark the median value with a vertical line at 75.5. Draw in the whiskers, the lines joining the midpoint of the ends of the box to the minimum and maximum values.

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Survey

50

60

70

80

90

100

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Using box-and-whisker plots to compare data Box-and-whisker plots are useful for comparing two or more sets of data collected on the same variable, such as the assessment results for two different groups of students. Two box-and-whisker plots are drawn on the same axis. This allows the median and the spread to be easily identified and compared. Example 13

Comparing data with box-and-whisker plots

Robert recorded the distances he ran during weeks 1 and 2 of his holidays. This data is shown in the box-and-whisker plot below. a What is the lower extreme for week 1? b What is the upper quartile for week 2? c What is the median for week 1? d What was the lower quartile for week 2? e Compare and contrast the two sets of data.

Week 1 Week 2 10

12

14

16 18 20 Distance (km)

22

24

Solution 1 2 3 4 5

Read the value from the graph. Read the value from the graph. Read the value from the graph. Read the value from the graph. Spread of the data for Week 1 is larger than the spread for Week 2 (box widths and extreme values are larger)

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a b c d e

Lower extreme for week 1 is 10 km Upper quartile for week 2 is 20 km Median for week 1 is 14 km Lower quartile for week 2 is 16 km Robert was more consistent and generally improved his running in Week 2.

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217

Constructing a boxplot using a graphics calculator

Example 14

The ages of people who have read a fantasy fiction book are listed below. a Enter this data into a graphics calculator using a list. b Display the data as a boxplot using the graphics calculator. c What is the median of this data? d What is the lower extreme? e What is the upper quartile?

23

22

18

17

13

15

23

18

17

22

18

12

15

13

15

12

14

13

14

17

15

25

19

18

20

16

14

Solution 1 2

Select the STAT menu. Enter the data into List1. Press EXE to enter each number.

3

To construct a box-and-whisker plot, select GPH1 and SET. Choose Graph Type : MedBox and XList1 : List1.

4

To draw the box-and-whisker plot, press GPH1 and EXE.

5

To obtain the five-number summary, select CALC, then 1VAR and scroll down through the data.

6

Read Med to obtain the median, minX to obtain the lower extreme and Q3 to obtain the upper quartile.

a

b

c

d e

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Median is 17 Lower extreme 12 Upper quartile 19

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Exercise 7F 1

Using the box-and-whisker plot, find: a Lower extreme b Higher extreme c Lower quartile d Upper quartile

0

50

100

150

200

2

Construct a box-and-whisker plot from each of the following five-number summaries. a Min = 1, Q1 = 4, Median – 7, Q3 = 22 and Max = 25 b Min = 5, Q1 = 6, Median – 14, Q3 = 26 and Max = 30 c Min = 1, Q1 = 2, Median – 6, Q3 = 10 and Max = 12

3

Given that the five-number summary for a set of data is 6, 17, 27, 39, 48, find the: a Median b Range c Interquartile range

4

Nira and her class visited two parks and measured the heights of the trees in metres. In East Park there were 30 trees and in West Park there were 36 trees. The data sets were displayed in two box-and-whisker plots shown below.

East Park West Park 0

a b c d

5

10 15 Height (m)

20

What is the median height of the trees in East Park? What is the median height of the trees in West Park? What was the height of the tallest tree? Where is it located? Which park has the largest interquartile range? What is its value?

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Development 5

The height (in centimetres) of students in a year 11 class is listed below. 156 149 180 154 159 157 143 154 143 123 150 145 132 140 140 168 167 135 154 133 167 123 176 157 163 157 160 165 a c e g

6 

The assessment results for two students are listed below. Holly

80

85

94

35

77

75

40

92

45

54

66

30

Grace

60

65

70

62

75

80

75

72

78

85

69

77

a c e g

7

8

What is the lower extreme? b What is the upper extreme? What is the median? d What is the lower quartile? What is the upper quartile? f Construct a box-and-whisker plot. Describe the data in terms of shape, centre and spread.

What is Holly’s lower extreme? b What is Holly’s higher extreme? What is Grace’s median? d What is Grace’s lower quartile? What is Grace’s upper quartile? f Construct a box-and-whisker plot. Describe the data in terms of shape, centre and spread.

Use a graphics calculator to check your results to questions 5, 6 and 7.

The golf scores of two friends are listed below. Lachlan

74

70

76

76

72

74

70

75

77

68

73

75

Andrew

75

76

75

69

70

78

79

81

63

72

72

73

a b c d e f g h

What is Andrew’s lower extreme? What is Lachlan’s median? What is Lachlan’s higher extreme? What is Andrew’s median? What is Lachlan’s lower quartile? What is Andrew’s upper quartile? Construct a box-and-whisker plot for both sets of data. Who is the better golfer? Give a reason for your answer.

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7.7 Sector and divided bar graphs Sector graph 7.7

A sector graph or pie chart presents data as sectors of a circle (‘slices’ of a ‘pie’). A sector graph shows the relationship or proportions of parts to a whole. Sector graphs appeal to people because they are easy to read and attractive. Sector graph A sector graph presents data as sectors of a circle. The steps to construct a sector graph are: 1 Draw a circle and mark the centre. 2 Multiply the proportion of the whole by 360 to determine the sector angle. 3 Use a protractor to draw the angle with the vertex at the centre of the circle. 4 Repeat steps 1 and 2 until all the sectors or parts have been drawn. 5 Label all sectors.

Example 15

Constructing a sector graph

Connor earns $600 and spends $240 on rent, $180 on food, $120 on petrol and saves $60. Construct a sector graph to represent this data. Solution 1 2

Draw a circle and mark the centre. Calculate the angles for each sector. Rent =

240 × 360 = 144° 600

Food =

180 × 360 = 108° 600

Petrol =

120 × 360 = 72° 600

Rent Food Petrol Save

60 × 360 = 36° 600 Place the centre of the protractor on the centre of the circle and mark angles 144°, 108°, 72° and 36°. Label each sector or create a legend. Save =

3

4

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Divided bar graph A divided bar graph is an alternative to the sector graph. It shows the relationship or proportions of parts to a whole. The bars or rectangles are drawn to scale. The total length of the divided bar graph matches the whole. A scale is selected that allows easy calculations. For example, if the whole is 500 kg, a suitable scale is 1 cm to 100 kg. Then the total length of the divided bar graph with this scale is 5 cm.

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

A

B

Divided bar graph A divided bar graph presents data as bars or rectangles. Steps to construct a divided bar graph are: 1 Draw a rectangle using an appropriate scale. 2 Multiply the proportion of the whole by the total bar length. 3 Use a ruler and draw the bar. 4 Repeat steps 1 and 2 until all the bars or parts have been drawn. 5 Label all bars or create a legend. Example 16

Constructing a divided bar graph

Connor earns $600 and spends $240 on rent, $180 on food, $120 on petrol and saves $60. Construct a divided bar graph to represent this data. Solution 1

2

Draw a rectangle using an appropriate scale. Connor’s total earnings are $600 so appropriate length is 6 cm. Calculate the size for each bar. Rent =

Draw a rectangle that is 6 cm in length and 1 cm in width.

240 × 6 cm = 2.4 cm 600

Food =

180 × 6 cm = 1.8 cm 600

Petrol =

120 × 6 cm = 1.2 cm 600

Rent

Food

Petrol Save

60 × 6 cm = 0.6 cm 600 Use a ruler and draw the bar. Label each bar or create a legend. Save =

3 4

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Exercise 7G 1

On a sector graph, how many degrees would represent each of these numbers if the total is 180? a 90 b 120 c 30 d 45

2

Convert the following percentages to angles at the centre of a circle. a 40% b 75% c 15% d 90%

3

Alyssa is drawing a sector graph. It will show how 200 students in year 11 selected the school captain. Forty of the year 11 students selected Bailey. What angle should Alyssa use for the sector that represents the students who selected Bailey as the school captain?

4

Examine the spreadsheet below. What is the angle used for the following sectors? a Food b Glass c Metals d Paper e Plastic

5

A survey was conducted on pet ownership. The results of the survey were 40% owned dogs, 25% cats, 20% had no pets and 15% another type of pet (other). Construct a sector graph.

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223

Development 6

Motor vehicle thefts have been recorded for each day of the week and expressed as a percentage: Monday – 13%, Tuesday – 13%, Wednesday – 12%, Thursday – 14%, Friday – 17%, Saturday – 16% and Sunday – 15%. a Construct a sector graph to represent this data. b Calculate the number of motor vehicles stolen on Saturday if the total number of vehicles stolen for the week was 300. c If there were 60 motor vehicles stolen on Wednesday, how many motor vehicles were stolen on Sunday?

7 

The type of drink and the number sold in the school canteen was recorded: Juice – 60, Milk – 40, Soft drink – 120 and Water – 20. a Construct a sector graph to represent this data. b Construct a divided bar graph to represent this data.

8

A new Sims computer game has been released. The projected sales in different regions are shown below. The numbers shown represent thousands of games.

Pacific a b c

9

Europe

USA

Asia

South America

Canada

16 48 64 36 24 Construct a sector graph to represent this data. What is the percentage of sales in Asia? What is the total percentage of sales outside the USA?

12

A survey of 80 students showing their preferred sport as a percentage is shown below.

a b c

Tennis

Netball

Football

Swimming

Golf

Cricket

10

15

20

25

15

15

Construct a divided bar graph to represent this data. How many students preferred swimming? How many students preferred tennis?

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7.8 Radar charts A radar chart (or spider chart) is used to compare the performance of one or more entities. Data on temperature, rainfall, humidity and sales are commonly presented using radar charts. For example, the radar chart below compares the temperature at Sydney to the temperature at Perth. Radar charts may also have multiple axes along which data can be plotted. For example, you could use a one radar chart to analyse the temperature and the rainfall of a particular city. Drawing a radar chart 1 2 3 4

Determine the data to be presented as sectors. Draw the sectors. Choose an appropriate scale for the data. Draw the scale beginning at the centre. Draw line segments for each scale to create the spider web. Plot the points and join them with a straight line. Create a legend if necessary.

Example 17

Reading a radar chart

The average monthly temperatures for Sydney and Perth are shown below. a What is the average monthly temperature for Perth in December? Nov b What is the difference in the average monthly temperatures for July between Sydney and Perth? Oct c Which month is the average monthly temperature of Sydney and Perth the same?

Dec

35 30 25 20 15 10 5 0

Jan Feb Mar

Apr

May

Sep

Aug

Jun Jul

Solution 1 2 3

Read the temperature value for Perth in December. Read the temperature values for Sydney and Perth in July. Subtract the values. Read when Sydney and Perth have the same temperature.

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a

Perth’s temperature is 35°.

b

Difference = 25 – 15 = 10° February

c

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Chapter 7 — Displaying and interpreting single data sets

Exercise 7H 1

2

25

The wind speed (km/h) at Wollongong was measured for 10 days and recorded in the radar chart. a Find the wind speed on these days. i Day 1 ii Day 3 iii Day 5 iv Day 7 b Which day had the lowest wind speed? c What days had the highest wind speed? d What was the greatest wind speed? e What was the least wind speed? f What is the difference between the wind speed on day 8 and day 10?

The temperature in Newcastle was measured every four hours. The results are shown opposite. a b c d e

f

10

225

1 2

20 15 10

9

3

5 0

8

4

5

7 6

3 a.m.

7 a.m.

11 a.m.

3 p.m.

7 p.m.

11 p.m.

15

20

25

25

20

18

Copy the blank radar chart. Plot each of the data points onto the radar chart. Join the points with a straight line to create the radar chart. Make another copy of the blank radar chart. The temperatures on the following day were 5 degrees warmer. Plot these data points onto the radar chart. Join the points with a straight line to create the radar chart.

25

3 am

20 11 pm

15 10

7 am

5 0

7 pm

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11 am

3 pm

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Development 3

4 

The radar chart shows the average monthly temperatures for Liverpool and Cronulla. a What is the average monthly temperature for Liverpool in: i January ii May iii June iv August b In which months did Liverpool and Cronulla have the same average monthly temperature? c What was the average monthly temperature for Cronulla in August and July? d Which month had the largest difference in temperature between Liverpool and Cronulla?

Dec

Jan Feb

20 15 10 5 0

Nov

Oct

Mar

Apr

Sep

May Aug

Jun Jul

Liverpool

Cronulla

The daily rainfall (mm) for two towns is shown in the following table. Day

1

2

3

4

5

6

7

8

9

A

12

8

2

6

0

10

8

4

2

B

10

8

4

0

2

12

6

2

0

Determine the number of sectors and an appropriate scale for a radar chart. Show the above information on the radar chart. Create a legend for the radar chart.

a b c

5

30 25

The average monthly sales (in millions) of a company over a 12-month period are shown in the table. Jan 1 a b c d

Feb Mar 2

2.5

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

3

2

4

3

3.5

4

4.5

5

6

Draw the sectors for a radar chart. Choose an appropriate scale and create the spider web. Plot each of the data points onto the radar chart and join the points. Briefly comment on any trends that can be seen.

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Chapter 7 — Displaying and interpreting single data sets

7.9 Dot plots and stem-and-leaf plots Dot plots 7.9

The simplest display of numerical data is a dot plot. A dot plot consists of a number line with each data point marked by a dot. When several data points have the same value, the points are stacked on top of each other. Dot plots are a great way for displaying fairly small data sets where the data takes a limited number of values. Dot plot A number line with each data point marked by a dot. When several data points have the same value, the points are stacked on top of each other.

Example 18

Constructing a dot plot

The number of hours Philip spent watching television on the weekend is shown below. Construct a dot plot. 3

4

3

2

7

6

2

2

3

7

3

5

2

3

4

5

6

8

1

6

1

2

3

4

1

5

Solution 1

2 3 4

Draw a number line, scaled to all the data values. Label the line with the variable being displayed. The vertical axis indicates the frequency of data value. It may be omitted. Plot each data value by marking in a dot above the corresponding value on the number line. Count the number of dots and check that it matches the number of data values.

Hours watching television 6 5 4 3 2 1 1

2

3

4 5 Hours

6

7

8

Stem-and-leaf plots A stem-and-leaf plot or stem plot is used to present a small (less than 50 values) numerical data set. The tens digit of the data values becomes the ‘stem’ and is written in numerical order down the page. The ‘units’ digit becomes the ‘leaves’ and is written in numerical order across the page.

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Stem-and-leaf plot

Leaf

0 57 1 22557 2 125679

Stem

Example 19

Back-to-back stem-and-leaf plot

Leaf

9 0 8532 1 98442 2

Stem

0 12559 4578

Leaf

Constructing a stem-and-leaf plot

Use the following data set to construct a stem-and-leaf plot. 24

7

19

19

15

18

27

28

11

19

29

30

6

25

25

26

5

6

6

10

10

28

29

14

9

Solution 1

2

3

The data set has values from 5 to 30. This requires stems 0, 1, 2 and 3. Write these down from smallest to largest, followed by a vertical line. Attach the leaves. The first data value is ‘24’. It has a stem of ‘2’ and a leaf of ‘4’. Opposite the 2 in the stem, write the number 4. Complete all the values. Rewrite the leaves so that they are in increasing order.

Example 20

566679 001458999 455678899 0

0 1 2 3

Constructing a back-to-back stem-and-leaf plot

Use the following data set to construct a back-to-back stem-and-leaf plot. Girls

28

24

24

31

34

26

27

12

18

13

15

6

29

30

22

Boys

19

27

21

25

35

28

29

13

11

30

31

32

25

16

9

Solution 1

2 3 4

Girls The data set has values from 6 to 35. This requires 6 stems 0, 1, 2 and 3. Write these down from 8532 smallest to largest, followed by two vertical lines. Attach the leaves for the girls. The first data value 9 8 7 6 4 4 2 410 is ‘28’. It has a stem of ‘2’ and a leaf of ‘8’. Attach the leaves for the boys. The first data value is ‘19’. It has a stem of ‘1’ and a leaf of ‘9’. Rewrite the leaves so that they are in increasing order.

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0 1 2 3

Boys 9 1369 155789 0125

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Chapter 7 — Displaying and interpreting single data sets

229

Exercise 7I 1

The dot plot represents the number of customers per hour. Customers each hour 6 5 4 3 2 1 9

b c d

A group of 26 people were asked how many times in the last week they had shopped at a particular supermarket. Their responses were as follows: 1

3

3

2

1

1

4

5

4

2

1

3

4

6

2

1

1

1

3

3

2

2

1

5

6

1

Construct a dot plot of this data. How many people were at the supermarket four times last week? What is the difference between the highest and lowest visits to the supermarket?

a b c

3

16

What is the highest number of customers? What is the most common number of customers per hour? What is the least common number of customers? Calculate the total number of customers?

a

2

10 11 12 13 14 15 Number of customers

The goals scored in each match are listed below.

a b c

4

4

3

3

2

2

2

3

1

5

1

2

1

2

3

4

2

1

2

1

1

3

2

4

2

1

Construct a dot plot for this data set. How many matches resulted in 3 goals? How many matches scored 4 or 5 goals?

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Development 4 

The following table shows the number of nights spent away from home in the past year by a group of 15 Australian tourists and by a group of 15 New Zealand tourists. AUS

21

5

8

7

17

3

15

14

3

11

5

4

11

6

4

NZ

19

6

23

32

17

29

23

22

12

28

26

5

22

14

14

a b

5

The ages of patients admitted to a particular hospital during one week are given below. Male

72

56

57

77

63

71

57

54

63

72

59

56

57

67

75

Female

61

55

58

78

65

68

71

78

79

72

73

64

68

66

69

a b

6

Construct a back-to-back stem-and-leaf plot of these data sets. Compare the number of nights spent away by Australian and New Zealand tourists in terms of shape, centre and spread.

Construct a back-to-back stem-and-leaf plot of these data sets. Compare the ages at admission to the hospital for male and female patients in terms of shape, centre and spread.

The stem-and-leaf plot represents the results achieved by students in a test. 0 1 2 3 a b c d e

7

What is the highest score in this test? Which score occurred the most number of times? What is the range for this data? How many students completed the test? What is the five-number summary for this data set?

An investigator recorded the amounts of time for which 24 similar batteries lasted in a toy. Her results (in hours) were:

a 13.2

6 2 3 5 8 2 4 4 6 7 8 9 0 1 4

b c

41

25

37

46

17

4

33

31

28

34

19

26

40

24

31

27

30

22

33

20

21

27

30

26

Make a stem-and-leaf plot of these times. How many of the batteries lasted for more than 25 hours? What is the five-number summary for this data set?

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Chapter 7 — Displaying and interpreting single data sets

Frequency table – ungrouped and

1

grouped data

2 3

Study guide 7

Scores or classes are listed in ascending order. Tally column records the number of times the score occurred (groups of 5s). Frequency column is a count of each outcome or class.

Cumulative frequency

The frequency of the score plus the frequency of all the scores less than that score.

Range and interquartile range

• •

Frequency and cumulative

•

frequency histogram •

Frequency polygon and cumulative

•

frequency polygon •

Range = Highest score – Lowest score IQR = Third quartile – First quartile = Q3 – Q1 Frequency histogram is a column graph that uses the score or class as the horizontal axis and frequency as the vertical axis. Cumulative frequency histogram is a column graph that uses the score or class as the horizontal axis and cumulative frequency as the vertical axis. Frequency polygon is a line graph of a frequency table. It can be constructed by joining the midpoints of the histogram. Cumulative frequency polygon or ogive is a line graph constructed by joining the top right-hand corner of the rectangles in a cumulative frequency histogram.

Box-and-whisker plot

A graph that uses five-number summary – lower extreme, lower quartile, median, upper quartile and the higher extreme.

Sector and divided bar graph

• •

Sector graph or pie chart presents data as sectors of a circle. It shows the relationship or proportions of parts to a whole. A divided bar graph presents data as bars or rectangles.

Radar chart

Looks like a spider web and is used to compare the performance of one or more entities.

Stem-and-leaf plot

Used to present a small (less than 50 values) numerical data set. The ‘tens’ digit of the data values becomes the ‘stem’ and is written in numerical order down the page. The ‘units’ digit becomes the ‘leaves’ and is written in numerical order across the page.

Dot plot

A number line with each data point marked by a dot.

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Review

Chapter summary – Displaying and interpreting single data sets

231

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

What is the frequency of 22? Score

Frequency

Cumulative Frequency

21

5

5

22

A

13

23

7

24

3

5

B

23 8

C

13

D

22

2

What is the cumulative frequency of 23 from the above frequency table? A 7 B 13 C 16 D 20

3

Find the interquartile range of this data: 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10 A 4.5 B 5 C 5.5 D 7

4

What is the total number of people who recorded marks? Cumulative frequency

Review

232

200 150 100 50 10 20 30 40 50 Mark

A

50

B

100

C

150

D

200

5 What is the frequency of 40 from the graph in question 4? A 6

0

B

10

C

40

D

100

C

85

D

90

What is upper quartile?

50 A

50

60

70

80 B

65

90

7

What is the lower extreme from the box-and-whisker plot in question 6? A 50 B 65 C 85 D 90

8

A radar chart is constructed with 10 sectors. What is the size of the angle in each sector? A 10 B 18 C 30 D 36

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Review

Sample HSC – Short-answer questions 1

The time (in hours) spent completing an assessment task is listed below. Class

Class centre Frequency

4–8

5

9–13 14–18

8

19–23

4 Total

a b 2

Copy and complete the table. How many students spent greater than 13 hours?

Alyssa recorded the following times (in minutes) running a cross country course. 51

53

57

55

58

57

53

55

54

53

56

55

53

57

51

57

54

58

55

56

51

53

53

54

52

56

52

54

54

53

56

56

53

54

52

53

54

57

58

56

54

55

52

55

55

58

54

56

55

56

52

a b c d e f g h

3

23

Construct a frequency table using a tally column. How many times were recorded? How many times were below 55 minutes? Add a cumulative frequency column. Construct a frequency histogram. Construct a cumulative frequency histogram and a cumulative frequency polygon. Use the cumulative frequency polygon to estimate the median. Use the cumulative frequency polygon to estimate the first and third quartile.

The height (in centimetres) of 20 people is listed below: 189

193

196

238

206

174

209

159

190

244

199

225

196

209

224

168

161

203

192

231

a c

What is the fifth decile? What is the twenty-fifth percentile?

b

What is the tenth decile? d What is the seventy-fifth percentile?

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4

Preliminary Mathematics General

Two students had 12 guesses to the number of jelly beans in a jar. Samuel

42

46

51

43

70

72

67

47

55

41

66

57

Nikolas

49

61

72

52

74

80

67

71

68

55

60

77

a b c d e f

What is Samuel’s lower extreme? What is Samuel’s upper extreme? What is Nikolas’s median? What is Nikolas’s lower quartile? What is Samuel’s upper quartile? Construct a box-and-whisker plot.

5

Sienna earns $1800 and spends $600 on rent, $300 on food, $150 on petrol and saves $750. Construct a sector graph to represent this data.

6

A company made the following profits (in millions) during the past 6 months.

a b c

Jan

Feb

Mar

Apr

May

Jun

20

40

35

30

25

45

How many sectors are required for a radar chart? What is the size of each sector? Choose an appropriate scale and create the spider web. Plot the data and complete the radar chart.

7

The stem-and-leaf plot represents the results achieved by students in a test. a What is the range of these results? b What is the first quartile? c What is the third quartile? d What is the five-number summary for this data set?

8

Use the dot plot to answer the questions below. a What is the range of scores? b What is the first quartile? c What is the third quartile?

0 1 2 3

9 1 2 4 9 1 3 5 7 8 8 0 2 3 5

5 4 3 2 1 18 19 20 21 22 23 24 25 26

Challenge questions 7

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C H A P T E R

8

Applications of perimeter, area and volume Syllabus topic — MM2 Applications of perimeter, area and volume Find unknown sides using Pythagoras’ theorem Calculate the perimeter of simple figures Calculate the area of composite shapes Calculate the area from a field diagram Calculate the volume of prisms and cylinders Relate capacity to volume

8.1 Pythagoras’ theorem Pythagoras’ theorem links the sides of a right-angled triangle. In a right-angled triangle the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side.

Hypotenuse (longest side)

8.1 Pythagoras’ theorem Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. (Hypotenuse)2

=

(side)2

+ (other

side)2

h

a

b

h2

=

a2

+ b2

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Preliminary Mathematics General

Pythagoras’ theorem is used to find a missing side of a right-angled triangle if two of the sides are given. It can also be used to prove that a triangle is right angled. Example 1

Finding the length of the hypotenuse

Find the length of the hypotenuse, correct to two decimal places. h cm

5 cm

9 cm

Solution 1 2 3 4 5

Write Pythagoras’ theorem. Substitute the length of the sides. Calculate the value of h2. Take the square root to find h. Express answer correct to two decimal places.

Example 2

h2 = a2 + b2 = 92 + 52 2 2 h = 9 +5 = 10.30 cm

Finding the length of a shorter side

A rectangle has a breadth of 5 mm and a diagonal measuring 12 mm. What is the length of the rectangle, correct to one decimal place?

12 mm

5 mm

Solution 1 2

Draw a triangle and label the information. Mark the unknown length as x.

x mm

12 mm

3 4 5 6 7 8

Write Pythagoras’ theorem. Substitute the length of the sides. Make x2 the subject. Take the square root to find x. Express answer to correct one decimal place. Write the answer in words.

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5 mm

h2 = a2 + b2 122 = x2 + 52 x2 = 122 − 52 2 2 x = 12 − 5

= 10.9 mm The length of the rectangle is 10.9 mm.

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Chapter 8 — Applications of perimeter, area and volume

Exercise 8A 1

Find the length of the hypotenuse, correct to one decimal place. a

b

6 cm

c

5 cm

h cm 24 mm

12 cm

h mm

h cm

8 cm

10 mm d

e

f

h cm

2.5 cm h mm

54 cm

20 mm

63.2 cm

4.2 cm

h cm

10 mm

2

Find the value of x, correct to two decimal places. a

15 cm

b

c

15 cm

x cm 12 cm x mm

21 cm

x cm

13 mm

6 mm d

e

2.3 cm

f

x cm x cm 24 mm

32 mm

4.8 cm

14.1 cm

9.5 cm

x mm

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Preliminary Mathematics General

Development 3

Calculate the length of the side marked with the pronumeral. (Answer to the nearest millimetre.) a b c y mm 16 mm 30 mm

42 mm

35 mm

63 mm

x mm

28 mm

a mm d



10 mm

d mm 27 mm

4

e



33 mm

52 mm

f

m mm

8 mm

12 mm

b mm

Find, correct to one decimal place, the length of the diagonal of a rectangle with dimensions 7.5 metres by 5.0 metres.

5.0 m 7.5 m

5 

Find the value of the pronumerals, correct to two decimal places. a b 90 cm

7 cm

72 cm

4 cm

x cm 6

Calculate the length of x, correct to one decimal place. a b 7 cm 6 cm

x cm

x cm

y

4 cm

y cm

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6 cm

25 cm

18 cm

x cm 14 cm

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Chapter 8 — Applications of perimeter, area and volume

239

8.2 Perimeter Perimeter is the total length of the outside edges of a shape. It is the distance of the boundary.

8.2

Perimeter formulas Name

Shape a

Triangle

Quadrilateral

Perimeter b

P=a+b+c

c b

a

P=a+b+c+d

c

d

P = 4s

s

Square

b

Rectangle

P = 2(l + b)

l

Circle

Example 3

Circumference C = 2πr C = πd

r

Finding the perimeter of a rectangle

Find the perimeter of the following rectangle. 3m 8m Solution 1 2 3 4

The shape is a rectangle, so use the formula P = 2(l + b). Substitute the values for l and b (l = 8 and b = 3). Evaluate. Write the answer in words.

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P = 2(l + b) = 2 × (8 + 3) = 22 m Perimeter of the rectangle is 22 m.

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Preliminary Mathematics General

Example 4

Finding the perimeter of a triangle

Find the perimeter of the following triangle. Answer correct to one decimal place.

3.7 cm 4.2 cm

Solution 1

Find the length of the hypotenuse or h.

h2 = 4.22 + 3.72

2

Write Pythagoras’ theorem. Substitute the length of the sides. Evaluate the value of h. Add the lengths of sides to find the perimeter. Express answer correct to 1 decimal place. Write the answer in words.

h = 4.2 2 + 3.72 ≈ 5.6 cm

3 4 5 6 7

Example 5

P = 3.7 + 4.2 + 5.6 = 13.5 cm Perimeter of the triangle is 13.5 cm.

Finding the circumference of a circle

Find the perimeter of a circle with a radius of 9 mm. Answer correct to two decimal places. 9 mm Solution 1 2 3 4

The shape is a circle, use the formula C = 2π r. Substitute the value for r (r = 9). Evaluate. Write the answer in words.

Example 6

C = 2π r = 2× π × 9 = 56.55 mm Perimeter of the circle is 56.55 mm.

Finding the perimeter of a semicircle

Find the perimeter of a semicircle with a diameter of 4 m. Answer correct to two decimal places. 4m Solution 1 2 3 4 5 6

The shape is a semicircle, use the formula C = π d ÷ 2. Substitute the value for d (d = 4) to find the curved distance. Evaluate. Add the curved distance to the diameter. Evaluate. Write the answer in words.

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πd 2 π ×4 = 2

C=

= 6.28 m P = 6.28 + 4 = 10.28 m Perimeter is 10.28 m. Cambridge University Press

Chapter 8 — Applications of perimeter, area and volume

241

Perimeter of composite shapes A composite shape is made up of two or more plane shapes. The perimeter of a composite shape is calculated by adding the distances that make up the boundary of the shape. Perimeter of a composite shape 1 2 3 4 5

Sides with the same markings are of equal length. Unknown side lengths of some sides are determined by using the given lengths of the other sides. Pythagoras’ theorem is used to find unknown side lengths involving a right triangle. Distances that are part of a circle are found using C = 2π r. Add the distances that make up the boundary of the shape to calculate the perimeter.

Finding the perimeter of composite shapes

Example 7

Find the perimeter of each of these shapes. a

b

2 cm

5m

6 cm 5 cm

5m

8 cm Solution 1 2 3 4

1 2 3 4 5 6

Find the unknown side lengths using the measurements given in the question. Add the lengths of all the edges to find the perimeter. Evaluate. Write the answer in words.

a

Use the formula C = 2π r ÷ 4 for the curved distance. Substitute the value for r (r = 5). Evaluate. Add the curved distance to other edges. Evaluate. Write the answer in words.

b

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2 cm 6 + 5 = 11 cm

6 cm

8 − 2 = 6 cm 5 cm

8 cm

P = 2 + 6 + 6 + 5 + 8 + 11 = 38 cm Perimeter is 38 cm. C=

2π r 2 × π × 5 = = 7.85 m 4 4

P = 7.85 + 5 + 5 + 5 + 5 = 27.85 m Perimeter is 27.85 m.

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242

Preliminary Mathematics General

Exercise 8B 1

Find the perimeter of each quadrilateral. Answer correct to one decimal place. a

b

c

7.2 m 20 m

13.4 m

5.4 cm

2

Find the perimeter of a square with a side length of 12.3 m. Answer correct to one decimal place.

3

Find the perimeter of each right triangle. Answer correct to one decimal place. a

b

15 m

c

8m

7 mm

5 cm

2 cm

9.5 m

8.5 mm

4

Find the perimeter of a right triangle with a base of 10.25 cm and a height of 15.15 cm. Answer correct to two decimal places.

5

Find the perimeter of each circle. Answer correct to one decimal place. a

b

c

3m 2 cm

6

14 mm

What is the circumference of each circle? Answer correct to one decimal place. a Radius of 4 cm b Radius of 19 m c Radius of 34 mm d Diameter of 50 mm e Diameter of 22 m f Diameter of 6 cm

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Chapter 8 — Applications of perimeter, area and volume

243

Development 7

Find the perimeter of each semicircle. Answer correct to one decimal place. a b c 2.1 m

5m

16 m 8

Find the perimeter of each shape. Answer correct to two decimal places. a b c 3 mm

5 cm

7m 9

Find the perimeter of each composite shape. Answer to the nearest whole number. a b c 8 cm 6m 2m

4m

8m

1m

4m

2m 2m

10 m d

1m



10 cm

e



f

4m 4m

6m 4m

3m

5m

6m

10

11

An annulus consists of two circles with the same centre. Find the perimeter of an annulus if the inner diameter is 3 cm and the outer diameter is 6 cm. Answer correct to the nearest centimetre. A rectangle ABCD has length AB = 12 cm and a width of BC = 6 cm. a Find the value of x. b Calculate the perimeter of quadrilateral AECF. (Answer correct to two decimal places).

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6 cm

3 cm

A

E

B

x cm D

F

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C

244

Preliminary Mathematics General

8.3 Area

8.3

The area of a shape is the amount of surface enclosed by the boundaries of the shape. It is measured by counting the number of squares that fit inside the shape. When calculating area, the answer will be in square units. 100 mm2 = 1 cm2 10 000 m2 = 1 ha 1 ha = 10 000 m2

10 000 cm2 = 1 m2 1 000 000 m2 = 1 km2

To calculate the area of the most common shapes, we use a formula. These formulas are listed below.

Area formulas Name

Shape

Area

h

Triangle

A = 12 bbh

b

A = s2

s

Square

b

Rectangle

A = lb

l h

Parallelogram

A = bh

b a

A = 12 ( a + b ) h

h

Trapezium b

Rhombus

Circle

x

y

r

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A = 12 xxy

A = π r2

Cambridge University Press

Chapter 8 — Applications of perimeter, area and volume

Example 8

Finding the area of a triangle

Find the area of the following triangle.

5.5 m 8.1 m

Solution 1 2 3 4

The shape is a triangle, so use the formula A = 12 bbh . Substitute the values for b and h (b = 8.1 and h = 5.5). Evaluate. Write the answer using the correct units.

Example 9

245

1 bbh 2 1 = × 8.1 × 5.5 2 = 22.275 m 2

A=

Finding the area of a trapezium

Find the area of the following shape.

2.9 cm 3 cm

Solution 1

2 3 4

5.1 cm

The shape is a trapezium, so use the formula 1 A = (a + b)h . 2 Substitute the values for a, b and h. Evaluate. Write the answer using the correct units.

Example 10

1 (a + b)h 2 1 = ( 2.9 + 5.1)3 2 = 12 cm 2 The area of the shape is 12 cm2. A=

Finding the area of a parallelogram

Find the area of the following shape. 4 mm Solution 1 2 3 4

The shape is a trapezium, so use the formula A = bh. Substitute the values for b and h (b = 6.5 and h = 4). Evaluate. Write the answer using the correct units.

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6.5 mm

A = bbh = 6.5 × 4 = 26 mm 2 The area of the shape is 26 mm2.

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Preliminary Mathematics General

Example 11

Finding the area of a circle

Find the area of a circle with a radius of 5 metres. Give your answer correct to one decimal place.

5m

Solution 1

The shape is a circle, so use the formula A = π r2.

2

Substitute the value for r (r = 5). Evaluate correct to one decimal place. Write the answer using the correct units.

3 4

A = πr2 = π × 52 = 78.5 m 2 The area of the circle is 78.5 m2.

Area of composite shapes A composite shape is made up of two or more plane shapes. The area of a composite shape is calculated by adding or subtracting the areas of simple shapes. Area of composite shapes • •

Composite shapes are made up of more than one simple shape. Area of composite shapes can be found by adding or subtracting the areas of simple shapes.

Example 12

Finding the area of a composite shape

Find the area of the composite shape. Answer correct to one decimal place.

12 cm 10 cm

Solution 1 2 3 4

Divide the shape into a rectangle and a semicircle. Use the formula A = lb for the rectangle. Substitute and evaluate. 1 Use the formula A = π r 2 for the semicircle. 2

5

Substitute and evaluate.

6

Add the area of the rectangle to the semicircle. Evaluate. Write the answer using the correct units.

7 8

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A = lb = 12 × 10 = 120 cm 2 1 A = πr2 2 1 = × π × 52 2 ≈ 39.3 cm 2 A = 120 + 39.3 = 159.3 cm2 The area of the shape is 159.3 cm2. Cambridge University Press

247

Chapter 8 — Applications of perimeter, area and volume

Exercise 8C 1

Find the area of each triangle. Answer correct to one decimal place. a

c

b

23 m

6 mm

13 m 4 cm

6.5 mm

2 cm

f

e

d

13 mm

7.6 m 15.5 mm

9.5 m

8.5 m

19 m

2

Find the area of each shape. Answer correct to one decimal place. a

c

b

9m

6.4 m 11.2 m

6.1 cm d

e

7 mm

4m 7m

22 m f

3.8 cm 4 cm

10 mm 6.7 cm

3

Find the area of a triangle with a base of 8.25 cm and a height of 10.15 cm. Answer correct to the nearest square centimetre.

4

Find the area of a square with a side length of 105.1 m. Answer correct to the nearest square metre.

5

Find the area of a circle with a radius of 7 cm. Answer correct to the nearest square centimetre.

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248

Preliminary Mathematics General

Development 6

Find the area of each shape. Answer correct to one decimal place. a

c

b

9.1 cm

2.3 km 5 km

d

5.7 cm

4.8 cm

Diagonals are 4.4 mm and 6.8 mm

9.1 cm

e

f

6 mm

8 mm

17 m

8m

4.1 cm 12.5 mm

15 m

9.8 cm

7

Jasmine is planning to build a circular pond. The radius of the pond is 1.5 m. What is the area of the pond, correct to the nearest square metre?

8

A 25 m swimming pool increases in depth from 1.3 m at the shallow end to 2.6 m at the deep end. Calculate the area of the side wall of the pool. Answer correct to the nearest square metre.

25 m 1.3 m

2.6 m

9

Philip wants to tile a rectangular area measuring 2.2 m by 3 m in his backyard. The tiles he wishes to use are 50 cm by 50 cm. How many tiles will he need? Give your answer as a whole number.

10

Find the area of each composite shape. a

b

4 cm 6 cm

30 cm

32 cm

3 cm 8 cm

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c

15 cm 4 cm 20 cm

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Chapter 8 — Applications of perimeter, area and volume

11

12

13

An annulus consists of two circles with the same centre. Find the area of an annulus if the inner diameter is 6 cm and the outer diameter is 10 cm. Answer correct to the nearest square centimetre.

6 cm

249

10 cm

A metal parallelogram has two identical squares removed from its shape. The two squares have a side length of 2 cm. Find the shaded area. Answer correct to the nearest square centimetre.

5 cm

7 cm

10 cm

A lawn is to be laid around a rectangular garden bed. a What is the amount of lawn required? b Find the cost of the new lawn if the required turf costs $20 per square metre.

13 m 19 m

20 m

25 m

14

The cross-section of an ice-cream cone is shown opposite. a What is the radius of the semicircle? b What is the height of the triangle? c Calculate the area of the region. Answer correct to one decimal place.

16 cm 10 cm

15

What is the area of a quadrant if it has a radius of 8 mm? Answer correct to two decimal places.

16

Decking for a house consists of a square and a triangle. The square has a side length of 8 metres and the triangle is isosceles. a Use Pythagoras’ theorem to find the value of x. b Calculate the area of the shaded region.

17

14 cm

xm 8m xm 8m

A metal worker cut circles with a diameter of 2 cm from a rectangular sheet of tin 4 cm by 8 cm. a What is the area of the rectangular sheet? b How many circles can be cut from the rectangular sheet? c What is the area of the remaining metal after the circles have been removed from the rectangular sheet? Answer correct to two decimal places.

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250

Preliminary Mathematics General

8.4 Field diagrams Field diagrams are used to calculate the area of irregularly shaped blocks of land. Measuring and recording the data in a field diagram is called surveying. One type of survey is called the traverse or offset survey.

Traverse or offset survey This type of survey involves measuring distances along a suitable diagonal or traverse. The perpendicular distances from the traverse to the vertices of the shape are called the offsets. When conducting a traverse survey, measurements are taken of the traverse and the offsets. These measurements are recorded in a field book entry. Offset Offset

Traverse

The field book entry records the distances along the traverse between two vertical lines. The distances along the offsets are recorded on either side of these measurements. Field Book Entry

Field Diagram

D 163 C 65 110 75 28 B 0 A

D

C

65

53 35

28

B

75 A

The measurements along the traverse are the distances starting from the bottom. For example, the distance 110 to the offset at point C is the distance from point A. This results in a distance of 35 between offset B and C. Conducting a traverse survey 1 2 3 4

Choose a suitable diagonal or traverse. Place a tape measure along the traverse. Starting from the bottom, measure the offsets using a taut string. Record the measurements in a field book entry.

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Chapter 8 — Applications of perimeter, area and volume

Example 13

Calculating area from a field diagram

A field diagram from a traverse survey is shown opposite. Measurements are in metres. a Find the area of the quadrilateral ABCD. Answer correct to one decimal place. b Find the distance AB. Answer correct to two decimal places.

251

D 65

C

53 35

28

B

75 A

Solution 1

Divide the quadrilateral into two triangles – ∆ADB and ∆ADC.

2

Find the area of each triangle using the formula A = 12 bbh.

3

Substitute the values for b and h. Evaluate. Express answer correct to one decimal place and with the correct units.

4 5

6

Add the area of the two triangles.

7

Write Pythagoras’ theorem by substituting the length of the sides. Take the square root to find AB. Evaluate. Express answer correct to two decimal places.

8 9 10

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a

Area of quadrilateral ABCD: For ∆ADB, b = 163 and h = 28 1 bbh 2 1 = × 163 × 28 2 = 2282 m 2

A=

For ∆ADC, b = 163 and h = 65 1 bbh 2 1 = × 163 × 65 2 = 5297.5 m 2 Total area = 2282 + 5297.5 = 7579.5 m2 A=

b

Distance AB AB 2 = 752 + 282 AB = 752 + 282 = 80.056 230 24 = 80.06 m Distance AB is 80.06 m.

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252

Preliminary Mathematics General

Exercise 8D 1

Find the area of the following fields using each field diagram. Units are in metres. a

b

35

c

28

60

80

20

30

60

80

50

D

d

50 50

e

f

D

D

18 C

60

50 40

C 45

60

48

50

20

A

2

C

110

B

45

25

26 12 32

B

A

B

24 A

Find the area of the following fields using each field book entry. Units are in metres. a

B 60 C 50 25 0 A

b

D 75 50 20 B C 15 40 0 A

c

D 150 C 70 110 75 30 B 0 A

d

E 140 100 32 D C 55 90 30 32 B 0 A

e

E 90 C 36 80 75 10 D B 36 20 0 A

f

E 40 30 10 D C 40 20 10 10 B 0 A

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Chapter 8 — Applications of perimeter, area and volume

Development 3

4

253

D

The diagram on the right shows a block of land that has been surveyed. All measurements are in metres. a Find the area of the quadrilateral ABCD. Answer correct to one decimal place. b What is the length of AB? Answer correct to the nearest metre.

35 25 C

32

14 13 A

The diagram below shows a block of land that has been surveyed. All measurements are in metres. E C

11 44 G

D

54 67 46

F 15

B

A a b c d e f

5 

Find the area of the triangle ABF. Answer correct to one decimal place. Find the area of the triangle ACE. Answer correct to one decimal place. Find the area of the triangle DGE. Answer correct to one decimal place. Find the area of the trapezium BFGD. Answer correct to one decimal place. What is the total area of the block of land? Answer correct to one decimal place. Find the distance AB. Answer correct to two decimal places.

The field book entry below shows a block of land that has been surveyed. All measurements are in metres. E 71 58 43 D C 23 34 25 14 B 0 A a b

Find the area of ABDEC. Answer correct to one decimal place. What is the length of AC? Answer correct to the nearest metre.

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B

254

Preliminary Mathematics General

8.5 Volume of prisms and cylinders Volume is the amount of space occupied by a three-dimensional object. It is measured by counting the number of cubes that fit inside the solid. When calculating volume, the answer will be in cubic units. 8.5

1000 mm3 = 1 cm3 1 000 000 cm3 = 1 m3 1 000 000 000 m3 = 1 km3 To calculate the volume of the most common solids, we use a formula. Some of these formulas are listed below. The volume of a prism is found by using its cross-sectional area. Prisms are three-dimensional objects that have a uniform cross-section along their entire length.

Volume formulas Name

Solid

Cube

V = Ah = (s2) × s = s3

s s

s

Rectangular prism

Volume

b h

V = Ah = lb × h = lbh

l

h

Triangular prism A=

1 2

bh

V=A Ah 1  =  bh × h 2 

r

Cylinder

h

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V = Ah = (πr2) × h = πr2h

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255

Chapter 8 — Applications of perimeter, area and volume

Example 14

Finding the volume of a right prism

A rectangular prism has a length of 8 cm, a breadth of 2 cm and a height of 4 cm. Find the volume of this rectangular prism. Answer in cubic centimetres.

2 cm

4 cm 8 cm

Solution 1 2 3 4 5

Use the volume formula for a right prism V = Ah. Determine the shape of the base and the formula to calculate the area of the base A = lb. Substitute the values into the formula. Evaluate. Give the answer to the correct units.

Example 15

V = Ah = lbh =8×2×4 = 64 cm3

Finding the volume of a cylinder

A cylinder has a radius of 8 mm and a height of 12 mm. Find the volume of the cylinder. Answer correct to two decimal places in cubic millimetres.

12 mm 8 mm

Solution 1 2 3 4

Use the volume formula for a cylinder V = πr2h. V = πr2h = π × 82 × 12 Substitute the r = 8 and h = 12 into the formula. = 2412.743 158 mm3 Evaluate. Write the answer correct to two decimal places and ≈ 2412.74 mm3 with the correct units.

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256

Preliminary Mathematics General

Exercise 8E 1

Find the volume of the following prisms where A is the area of the base. a

b

A = 8 m2

c

40 m

12 m

16 m

A = 110 m2

A = 7 m2

2

What is the volume of a rectangular prism with a base area of 15 mm2 and a height of 11 mm?

3

Find the volume of a triangular prism with a height of 15 m and a base area of 50 m2.

4

Find the volume of the following solids. Answer to the nearest whole number. a

b

c

18 cm

10 mm

3m 18 cm

4m

9m

10 mm 6 mm

18 cm 8 mm

d

e

f

14 mm

7 mm

6 mm

6m

15 m 10 m

20 mm

5

What is the volume of a rectangular prism with dimensions 4.5 cm by 6.5 cm by 10.5 cm? Answer correct to one decimal place.

6

A closed cylindrical plastic container is 20 cm high and its circular end surfaces each have a radius of 5 cm. What is its volume correct to two decimal places?

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Chapter 8 — Applications of perimeter, area and volume

257

Development 7

8

A water tank is in the shape of a closed cylinder with a radius of 10 m and height of 8 m. a What is the area of the top circular face of the water tank? Answer correct to one decimal place. b Determine the volume of the water tank. Answer correct to one decimal place.

10 m 8m

A hollow container is in the shape of a rectangular prism as shown.

2m

6m

6m

10 m

10 m a b c 9

What is the volume of the container if it was solid? What is the area of the shaded base? What is volume of the hollow container?

A step is shown opposite. a What is the area of the shaded base? b Determine the volume of the step.

5m

12 m 2m 1m

10 m

10

What is the volume, correct to one decimal place, of a cylindrical paint tin with a height of 30 cm and a diameter of 25 cm?

11

Find the volume of an equilateral triangular prism with side lengths 3 cm and a depth of 10 cm. Answer correct to three decimal places.

12

A vase with a volume of 200 cm3 is packed into the cardboard box shown below. The space around the vase is filled with foam to protect from breaking. The parcel is sealed and posted.

6 cm 10 cm a b

8 cm

What is the volume of the foam? What is the area of cardboard on the surface of the box?

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258

Preliminary Mathematics General

8.6 Capacity The capacity of a container is the amount of liquid it can hold. Some solids have both a volume and a capacity. For example, a can of soft drink is a cylinder that has a volume (V = π r2h) and a capacity (360 mL). The base unit for capacity is a litre (L). Three commonly used units for capacity are a megalitre, (ML), kilolitre (kL) and millilitre (mL).

Capacity 1 ML = 1000 kL 1 ML = 1 000 000 L 1 kL = 1000 L 1 L = 1000 mL

Example 16

1 cm3 = 1 mL 1 cm3 = 0.001 L 1000 cm3 = 1 L

1 m3 = 1 000 000 cm3 1 m3 = 1 000 000 mL 1 m3 = 1000 L 1 m3 = 1 kL

Finding the capacity

The container shown opposite is filled with water. a Find the volume of the container in cubic centimetres. b Find the capacity of the container in litres.

30 cm 40 cm

70 cm Solution 1 2 3 4 5 6 7

Use the volume formula for a right prism V = Ah. Determine the shape of the base and the formula to calculate the area of the base A = lb. Substitute the values into the formula. Evaluate. Give answer to the correct units.

a

To change cm3 to L multiply by 0.001. (1 cm3 = 0.001 L) Alternative method is to convert to mL. (1 cm3 = 1 mL)

b

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V = Ah

= lbh = 70 × 40 × 30 = 84 000 cm3 Capacity = 84 000 × 0.001 L = 84 L Capacity = 84 000 × 1 mL = 84 000 mL = 84 L

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Chapter 8 — Applications of perimeter, area and volume

259

Exercise 8F 1

A can of soft drink has a capacity of 375 mL. How many cans of soft drink would it take to fill a 1.2 L bottle? How much would remain?

2

A medicine bottle has a capacity of 0.3 L. a What is the capacity in millilitres? b How many tablespoons (15 mL) does the bottle contain? c How many teaspoons (5 mL) does the bottle contain? d The correct dosage is 10 mL, 3 times a day. How many doses does the bottle contain?

3

Complete the following. a 4 cm 3 = mL c

70 cm 3 =

e

900 cm 3 =

g

43 m 3 =

i

103 m 3 =

k

5 m3 =

mL mL kL kL kL

b

2000 cm 3 =

d

34 000 cm 3 =

f

500 cm 3 =

h

30 m 3 =

L L L

L

j

7 m3 =

L

l

8 m3 =

mL

4

What is the capacity of a rectangular prism whose base area is 20 cm2 and height is 10 cm? Answer correct to the nearest millilitre.

5

Find the capacity of a triangular prism with a height of 18 m and a base area of 40 m2. Answer in litres, correct to two significant figures.

6

Find the capacity of a rectangular pyramid whose base area is 12 cm2 and height is 15 cm. Answer correct to the nearest millilitre.

7

Find the capacity of a cylindrical plastic container 16 cm high and with circular end surfaces of radius 8 cm. Answer correct to the nearest litre.

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Preliminary Mathematics General

Development 8

Find the capacity of the following solids in millilitres, correct to two decimal places. a

b

6 mm

4m 5 mm

3m

8m

c

18 mm 10 mm

3 mm

4 mm

d

e

10 cm

7m

9m

10 m

6 cm

3m

9

Find the capacity of a cube whose side length is 75 mm. Answer in millilitres, correct to two decimal places.

10

A water tank is the shape of a cylinder with a radius of 2 m and height of 2.5 m. a What is the area of the top circular face of the 2m water tank? Answer correct to one decimal place. b Determine the volume of the water tank in cubic metres. Answer correct to one decimal place. c What is the capacity of the tank, to the nearest kilolitre?

11

14.4

f

A = 21 m2

A swimming pool is the shape of a rectangular prism as shown opposite. The swimming pool is filled 25 cm from the top. a What is the volume of water in cubic metres? b How much water does the swimming pool contain, to the nearest kilolitre?

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2.5 m

1.7 m

15 m

10 m

Cambridge University Press

Chapter 8 — Applications of perimeter, area and volume

Pythagoras’ theorem

Study guide 8

Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. h2 = a 2 + b 2

Perimeter

Perimeter is the total length of the outside edges of a shape. It is the distance of the boundary. Rectangle P = 2(l + b) Circle C = 2π r or C = π d

Area

Triangle

A = 12 bbh

Square

A = s2

Rectangle

A = lb

Parallelogram

A = bh

Trapezium

A = 12 ( a + b ) h

Rhombus

A = 12 xxy

Circle

A = πr 2

Area of composite shapes

• •

Composite shapes are made up of more than one simple shape. Area of composite shapes can be found by adding or subtracting the areas of simple shapes.

Field diagrams

Field diagrams are used to calculate the area of irregularly shaped blocks of land. A traverse survey measures distances along a suitable diagonal or traverse.

Volume of prisms and

Cube Rectangular prism

V = Ah = (s2) × s = s3 V = Ah = lb × h = lbh

Triangular prism

V = Ah Ah =

Cylinder

V = Ah = (πr2) × h = πr2h

cylinders

Capacity

( bbhh) × h 1 2

The amount of liquid a container can hold. Base unit is a litre.

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Review

Chapter summary – Applications of perimeter, area and volume

261

Review

262

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

2

3

What is the length of the hypotenuse? A 400 cm B 20 cm C 28 cm D 7.46 cm

16 cm

12 cm

What is the perimeter of a quadrant with a radius of 5 mm? A 3.9 mm B 7.9 mm C 13.9 mm What is the perimeter of the composite shape? A 25 m B 29 m C 30 m D 32 m

D

17.9 mm

9m 4m 7m 5m

4

What is the area of the composite shape in question 3? A 25 m2 B 51 m2 C 63 m2 D 75 m2

5

What is the area of a triangle with a base of 5 m and a perpendicular height of 8 m? A 13 m2 B 20 m2 C 40 m2 D 80 m2

6

What is the area of the trapezium? A 42 cm2 B 63 cm2 C 96 cm2 D 126 cm2

6 cm 8 cm

7 cm 12 cm

7

What is the area of ABCD using the field book entry? A 40 B 450 C 600 D 1200

D C 10

40 30 10 0 20 B A

8

What is the volume of a cylinder with a height of 6 cm and radius of 2 cm? A 75 cm3 B 113 cm3 C 302 cm3 D 452 cm3

9

A cubic water tank has a side length of 6 m. What is the capacity of the tank? A 36 kL B 216 kL C 360 kL D 216 000 kL

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263

Chapter 8 — Applications of perimeter, area and volume

1

Find the value of x, correct to two decimal places. a b x mm



xm

c

27 cm

21 m 28 m

47 cm

21 mm

14 mm

x cm 2

Calculate the length of x, correct to the nearest millimetre.

18 mm 6 mm

x 10 mm 3

Find the perimeter of each shape. Answer correct to one decimal place. a b c 5 cm

d

10 cm

12 m

11 cm

6.7 cm



e



f

7m

6 mm

9m 4m 11 m

4

Find the area of each shape. Answer correct to one decimal place. a

c

b

3.5 m

6m 9m

10 m 11.2 cm

d

e

5.6 cm

f

2m

4 mm 3m

8.4 cm

7 mm

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Review

Sample HSC – Short-answer questions

Review

264

Preliminary Mathematics General

Chapter summary – Earning Money area measuring 3.6 m by 3.2 m in his backyard. 5 Mahendra is planning to tile a rectangular The tiles he wishes to use are 40 cm by 40 cm. a What is the area of the rectangle? b What is the area (in m2) of the tiles? c How many tiles will he need? 6

The diagram on the right shows a block of land that has been surveyed. All measurements are in metres. a Find the area of the quadrilateral PQRS. Answer correct to one decimal place. b What is the length of PQ? Answer correct to the nearest metre.

S 40 R

58 52 46 18

Q

P 7

Find the volume of the following solids. a

c

b

11 mm

4 mm 11 mm 5 mm 10 m

3 mm

4 mm

11 mm 4.5 m 3.2 m 8

9

A closed cylindrical container is 16 mm long and its end surfaces have a radius of 8 mm. a What is the area of the circular ends? Answer correct to the nearest square millimetre. b What is the volume of the container? Answer correct to the nearest cubic millimetre.

16 mm 8 mm

Find the capacity of a triangular prism with a height of 50 cm and a base area of 120 cm2. Answer in litres.

Challenge questions 8

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C H A P T E R

9

Relative frequency and probability Syllabus topic — PB1 Relative frequency and probability Calculate and use relative frequencies to estimate probabilities Understand the definition of probability Calculate probabilities using fractions, decimals and percentages Demonstrate the range of possible probabilities Identify and use the complement of an event

9.1 Relative frequency

9.1

Relative frequency is calculated when an experiment is performed. The frequency of an event is the number of times the event occurred in the experiment. Relative frequency is the frequency of the event divided by the total number of frequencies. It is also known as experimental probability as it estimates the chances of something happening or the probability of an event. Relative frequency is expressed using fractions, decimals and percentages. Relative frequency Relative frequency is an estimate for the probability of an event. Relative frequency =

Frequency of an event Total number of frequencies

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Preliminary Mathematics General

Finding the relative frequency

Example 1

An experiment of tossing two coins was completed and the number of heads recorded in the frequency table shown below. Number of heads Frequency 0

100

1

192

2

108

Relative frequency

Find the relative frequency of obtaining the following number of heads: a 0 b 1 c 2 Solution 1

Add the frequency column to determine the total number of frequencies.

2

Write the formula for relative frequency. Substitute the frequency and total number of frequencies into the formula. Simplify the fraction if possible or express as a decimal. Write answer in words.

3 4 5

1 2 3 4

1 2 3 4

a

Total Frequencies = 100 + 192 + 108 = 400

Freq of Event Total Freq 100 = 400 1 = or 0.25 or 25% 4 Relative frequency of 0 heads is 0.25. Rel. Freq. =

Write the formula for relative frequency. Substitute the frequency and total number of frequencies into the formula. Simplify the fraction if possible or express as a decimal. Write answer in words.

b

Write the formula for relative frequency. Substitute the frequency and total number of frequencies into the formula. Simplify the fraction if possible or express as a decimal. Write answer in words.

c

Freq of Event Total Freq 192 = 400 12 = or 0.48 or 48% 25

Rel. Freq. =

Relative frequency of 1 head is 0.48. Freq of Event Total Freq 108 = 400 27 = or 0.27 or 27% 100

Rel. Freq. =

Relative frequency of 2 heads is 0.27.

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Chapter 9 — Relative frequency and probability

Example 2

267

Performing a simulation using a graphics calculator

Perform a simulation to model the rolling of dice. Use the simulation to complete 100 trials and present these results in a histogram.

Solution 1 2

3 4

Select the TABLE menu. Enter the function Int (6Ran# + 1). • Int is found by pressing the OPTN key followed by NUM. • Ran# is found by pressing the OPTN key followed by PROB. Select SET to specify the range and simulate the 100 trials. Select TABL to view the results of the 100 trials. Note: a new simulation is performed each time you move between the function and the table.

To perform statistics on the results they must be in a list. 5 Select OPTN and LMEM to copy the results in X and Y1 to LIST1 and LIST2. 6

Select the STAT menu.

7

To construct a histogram select GPH, GPH1 and SET. Choose Graph Type : Hist and XList1 : List2.

8

To draw the histogram press GPH1 and EXE.

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Preliminary Mathematics General

Exercise 9A 1

2

3

4

A frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Express as a fraction in simplest form. a A b B c C d D

A frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer correct to three decimal places. a HH b HT c TH d TT

A frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer as a percentage correct to one decimal place. a Black b Yellow c Red d Blue e Green f White

A frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer as a percentage, correct to the nearest whole number. a 30 b 31 c 32 d 33 e 34

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Letter

Frequency

A

12

B

9

C

15

D

6

Outcome Frequency HH

8

HT

20

TH

28

TT

12

Colour

Frequency

Black

105

Yellow

210

Red

145

Blue

170

Green

215

White

155

Score

Frequency

30

4

31

6

32

2

33

3

34

5

Relative freq.

Relative freq.

Relative freq.

Relative freq.

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5

Calculate the relative frequency for each of these numbers if the total frequency is 48. Write your answer as a fraction in simplest terms. a 16 b 40 c 24 d 6

6

Calculate the relative frequency for each of these numbers if the total frequency is 40. Write your answer as a percentage. a 4 b 30 c 15 d 32

7

A retail store sold 512 televisions last year. There were 32 faulty televisions returned last year. What is the relative frequency of a faulty television last year? Answer as a percentage, correct to two decimal places.

8

A pistol shooter at the Olympic Games hits the target 24 out of 25 attempts. What is the relative frequency of him hitting the target? Give answer as a decimal, correct to two decimal places.

9

The birth statistics in a local community were 142 girls and 126 boys. What is the relative frequency for a girl? Answer as a fraction in lowest terms.

10

Create the spreadsheet below using the frequency table in question 4.

9A

a b c

Cell B10 has a formula that adds cells B5 to B10. Enter this formula. The formula for cell C5 is ‘=B5/$B$10’. It is the formula for relative frequency. Fill down the contents of C6 to C9 using this formula. Cells D5 to D9 have the same formulas as cells C5 to C9. Enter these formulas and format the cells to a percentage.

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Development 11

A frequency distribution table is shown below.

a b c

Score

Frequency

Relative freq.

3

x

0.20

4

6

0.30

5

5

0.25

6

5

y

What is the value of x? What is the value of y? What is the total number of scores?

12

Perform an experiment by rolling a die 120 times. a Use a frequency table to record the results of the experiment. b Calculate the relative frequency of each outcome. c What result would you have predicted for each outcome? d Compare your results to those of the other students in your class.

13

Perform an experiment by dropping a drawing pin 100 times. Record whether it landed point up or point down. a Use a frequency table to record the results of the experiment. b Calculate the relative frequency of each outcome. c What result would you have predicted for each outcome? d Compare your results to those of the other students in your class.

14

Perform an experiment by tossing two coins 80 times. a Use a frequency table to record the results of the experiment. b Calculate the relative frequency of each outcome. c What result would you have predicted for each outcome? d Compare your results to those of the other students in your class.

15

Perform a simulation using a graphics calculator to model the random selection of choosing one card from four cards labelled 1, 2, 3 or 4. Use the simulation to complete 100 trials and present the results in a histogram.

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9.2 Multistage events Multistage event consists of two or more events. For example, tossing a coin and throwing a die or selecting 3 cards from a pack of cards. The fundamental counting principle is used to determine the total number of outcomes for a multistage event. 9.2

Fundamental counting principle The fundamental counting principle states that if we have p outcomes for first event and q outcomes for the second event, then the total number of outcomes for both events is p × q. It simply involves multiplying the number of outcomes for each event together. Consider the multistage event of having two babies and the sex of each baby. The first baby has two outcomes (boy or girl) and the second baby has two outcomes (boy or girl). The total number of outcomes for both events is 2 × 2 = 4 (BB, BG, GB or GG). Fundamental counting principle Number of outcomes (two events) = p × q p – Number of outcomes of the first event. q – Number of outcomes of the second event.

Example 3

Determining the number of arrangements

David, Ella and Fran are required to stand in a row for selection to a committee. a How many different arrangements are possible? b List all the possible outcomes.

Solution 1

2

3

4

The first event is the first person in the row. There are 3 possible outcomes (D, E or F). The second event is the second person in the row. There are 2 possible outcomes. The third event is the third person in the row. There is only 1 possible outcome. Multiply the number of outcomes for each event to determine the number of arrangements.

a b

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Number of arrangements = 3 × 2 × 1 =6 Possible outcomes = {DEF, DFE, EDF, EFD, FDE, FED}

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Preliminary Mathematics General

Exercise 9B 1

Jasmine places three different types of apples in a row on the counter. The types of apples are red delicious, golden delicious and granny smith. a How many different arrangements are possible? b List all the possible arrangements.

2

Four cards each with a different suit (diamond, heart, spade or club) are placed in a row on the table. a How many different arrangements are possible? b List all the possible arrangements.

3

The letters of the word KINGSFORD are to be rearranged. a How many different arrangements are possible? b How many different arrangements are possible if the letters FORD are removed. c How many different arrangements are possible if the letters KINGS are removed.

4

How many ways can Aaron, Bailey, Connor, Daniel and Eddie stand in a queue?

5

There are 10 horses in a race. a How many different ways can the horses finish? b How many ways can first, second and third place be filled?

6

Two dice are rolled. How many different outcomes are possible?

7

A fair coin is tossed three times. a How many different outcomes are possible? b List all the possible outcomes. c If the coin is tossed again, how many different outcomes are now possible?

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Development 8

Lucy, Madison and Nikki are nominated for school captain and vice captain. What are all the possible combinations?

9

Adam goes to a shop that sells blue, red, pink and green pens. He decides to buy two pens, each one a different colour. a How many different arrangements are possible? b List all the possible arrangements.

10

A restaurant menu offers a choice of 4 entrees, 5 main courses and 3 desserts.

a b

How many combinations of meal (entree, main, dessert) are possible? The restaurant adds another dessert. How many combinations of meals are now possible?

11

A box contains five discs labelled ‘M’, ‘N’, ‘O’, ‘P’ and ‘Q’. a A disc is chosen and removed from the box at random. A second disc is then chosen and removed from the box. How many different choices are possible? b A third disc is then chosen and removed from the box. How many different choices for the three discs are possible?

12

A golf team has 4 players to be selected from a squad of 7 players. How many different teams are possible?

13

The letters from the word CARLTON are being used to form other words. a How many two-letter arrangements are possible? b How many three-letter arrangements are possible?

14

Jade has 5 shirts, 6 skirts and 3 pairs of shoes. a How many different combinations are possible? b Jade buys two more shirts. How many different combinations are now possible?

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9.3 Systematic lists A systematic list is essential for finding the sample space for a multistage event. It is an orderly method of determining all the possible outcomes. Tables are used to generate a systematic list. 9.3

Tables A table is an arrangement of information in rows and columns. The table below shows all the possible outcomes for a tossing two coins. There are two events – tossing the first coin and tossing the second coin. The outcomes of the first event are listed down the first column (Head or Tail). The outcomes of the second event are listed across the top row (Head or Tail). Each cell in the table is an outcome. There are 4 possible outcomes. Head

Tail

Head

HH

HT

Tail

TH

TT

Sample space = {HH, HT, TH, TT} Fundamental counting principle verifies the result from the table. There are two events each with two outcomes (head or tail). Number of outcomes = 2 × 2 = 4. Example 4

Using a table for a multistage event

Two red cards (R1, R2) and one black card (B1) are placed in a box. Two cards are selected at random with replacement. Use a table to list the sample space. Solution 1

2

3

4

List the outcomes of the first event (first card) down the first column. There are 3 outcomes: R1, R2 and B1. List the outcomes of the second event (second card) across the top row. There are 3 outcomes: R1, R2 and B1. Write the outcome in each cell using the intersection of the row and column. List the sample space.

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R1

R2

B1

R1

R1 R1 R1 R2 R1 B1

R2

R2 R1 R2 R2 R2 B1

B1

B1 R1 B1 R2 B1 B1

Sample space = {R1R1, R1R2, R1B1, R2R1, R2R2, R2B1, B1R1, B1R2, B1B1}

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Tree diagrams A tree diagram shows each event as a branch of the tree. The tree diagram below shows all the possible outcomes for a tossing two coins. The outcomes of the first event are listed (H or T) with two branches. The outcomes of the second event are listed (H or T) with two branches on each of the outcomes from the first event. The sample space is HH, HT, TH and TT. 1st

2nd H

HH

T

HT

H

TH

T

TT

H

T

Tree diagrams • Draw a tree diagram with each event as a new branch of the tree. • Always draw large clear tree diagrams and list the sample space on the right-hand side.

Example 5

Drawing a tree diagram

A coin is tossed and a die is rolled. a Construct a tree diagram of these two events. b List the sample space. Solution 1 2 3 4

5

Draw the first branch for first event – tossing a coin. Tossing a coin has two outcomes (head or tail) so there are two branches. Draw the second branch for the second event – rolling a die. Rolling a die has six outcomes (1, 2, 3, 4, 5 or 6) so there are six branches. Draw six branches for each of the two outcomes from the first event. Use the branches of the tree to list the sample space. Write the outcomes down the right-hand side (sample space).

Coin

H

T

Die 1

H1

2

H2

3

H3

4

H4

5

H5

6

H6

1

T1

2

T2

3

T3

4

T4

5

T5

6 T6

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Exercise 9C 1

2

Emily and Bailey are planning to have two children. a Use a table to list the number of elements in the sample space. Consider the sex of each child as an event. b Verify the total number of outcomes using the fundamental counting principle.

Boy

Girl

Boy Girl

Two fair dice are thrown and their sum recorded. a Use a table to list the all the possible outcomes. +

1

2

3

4

5

6

1 2 3 4 5 6 b 3

Verify the total number of outcomes using the fundamental counting principle.

A menu has three entrees (E1, E2 and E3) and four mains (M1, M2, M3 and M4). a Use a table to list the all the possible outcomes. M1 M2 M3 M4 E1 E2 E3 b

Verify the total number of outcomes using the fundamental counting principle.

4

Three people (A, B and C) applied for the manager’s position and two people (D and E) applied for the assistant manager’s position. a Use a table to list the all the possible outcomes. b Verify the total number of outcomes using the fundamental counting principle.

5

One bag contains two discs labelled ‘X’ and ‘Y’. A second bag contains four discs labelled ‘D’, ‘E’, ‘F’ and ‘G’. A disc is chosen from each bag at random. Use a table to determine the number of elements in the sample space.

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Three yellow cards (Y1, Y2, Y3) and one green card (G1) are placed in a box. Two cards are selected at random with replacement. a Use a table to list the number of elements in the sample space. b Verify the total number of outcomes using the fundamental counting principle.

7

Three cards (king, queen and jack) are placed face down on a table. One card is selected at random and the result recorded. This card is returned to the table. A second card is then selected at random. a Use a table to list the number of elements in the sample space. b Verify the total number of outcomes using the fundamental counting principle.

8

Two coins are tossed and the results recorded. a List the sample space using a tree diagram. b How many possible outcomes? c Use the fundamental counting principle to confirm your answer to part b.

9

A survey has two questions whose answers are ‘Yes’ or ‘No’. Construct a tree diagram to list the sample space.

10

There are three questions in a True or False test. 1st

2nd

Y1

Y2

Y3

277

G1

Y1 Y2 Y3 G1

1st

2nd

H

T

3rd

T

F

a b c

List the sample space using a tree diagram. How many possible outcomes are there? Use the fundamental counting principle to confirm your answer to part b.

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Development 11

A two-digit number is formed using the digits 1, 2 and 3. The same number cannot be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit. a List the sample space from the tree diagram. Tens 1

2

3 b c

Units 2 3 1 3 1 2

How many possible outcomes are there? Use the fundamental counting principle to confirm your answer to part b.

12

A spinner has equal amounts of red and green sections. This spinner is spun twice. a Use a tree diagram to list the total possible outcomes. b Verify the total number of outcomes using the fundamental counting principle.

13

Ebony tosses a coin and spins a spinner, which has red, amber and green sections. a Use a tree diagram to list the sample space. b Verify the total number of outcomes using the fundamental counting principle.

14

Four cards (ace, king, queen and jack) are placed face down on a table. One card is selected at random and the result recorded. This card is not returned to the table. A second card is then selected at random. a Use a tree diagram to list the total possible outcomes. b Verify the total number of outcomes using the fundamental counting principle.

15

There are four candidates for the leader and deputy leader. The four candidates are Angus, Bridget, Connor and Danielle. a Construct a tree diagram with the leader as the first event and the deputy leader as the second event. Use a tree diagram to list the sample space. b Verify the total number of outcomes using the fundamental counting principle.

16

A two-digit number is formed using the digits 3, 5 and 7. The same number can be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit. Use a tree diagram to list the sample space.

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9.4 Definition of probability

9.4

Probability is the chance of something happening. To accurately calculate the probability a more formal definition is used. When a random experiment is performed the outcome or result is called the event. For example, tossing a coin is an experiment and a head is the event. The event is denoted by the letter E and P(E ) refers to the probability of event E. The probability of the event is calculated by dividing the number of favourable outcomes by the total number of outcomes. It is expressed using fractions, decimals and percentages. Probability Number of favourable outtccomes Total number of outcomes n( E ) P( E ) = n( S )

Probability (Event) =

Example 6

Calculating the probability

A coin is chosen at random from 7 one dollar coins and 3 two dollar coins. Calculate the probability that the coin is a: a one dollar coin. b two dollar coin. Solution 1 2

3 4 5 1 2

3 4 5

Write the formula for probability. Number of favourable outcomes (or $1 coins) is 7. The total number of outcomes or coins is 10. Substitute into the formula. Simplify the fraction if possible. Express as a decimal or percentage if required. Write the formula for probability. Number of favourable outcomes (or $2 coins) is 3. The total number of outcomes or coins is 10. Substitute into the formula. Simplify the fraction if possible. Express as a decimal or percentage if required.

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a

n($1) n( s ) 7 = 10

P($1) =

= 0.7 or = 70%

b

n($2 ) n( s ) 3 = 10

P($2 ) =

= 0.3 or = 30%

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Equally likely outcomes Equally likely outcomes occur when there is no obvious reason for one outcome to occur more often than any other, for example, selecting a ball at random from a bag containing red, blue and white ball. Each of the balls is equally likely to be chosen. Winning a bike race is an example of an event where the outcomes are not equally likely. Some riders have more talent and some riders are better prepared. If one person is a better rider, their chance of winning the race is greater.

A deck of playing cards A normal deck of playing cards has 52 cards. There are four suits called clubs, spades, hearts and diamonds. In each suit there are 13 cards from ace to king. There are 3 picture cards in each suit (jack, queen and king).

Example 7

Calculating the probability from playing cards

What is the probability of choosing the following cards from a normal pack of cards? a Red four b Diamond c Picture card Solution 1 2 3 4 1 2

3 4 1 2

3 4

n( Red 4 ) n( s ) 2 = 52 1 = 26

Write the formula for probability. Number of favourable outcomes (or red 4s) is 2. The total number of outcomes is 52. Substitute into the formula and simplify the fraction. Simplify the fraction.

a

P( Red 4 ) =

Write the formula for probability. Number of favourable outcomes (or diamonds) is 13. The total number of outcomes is 52. Substitute into the formula. Simplify the fraction.

b

P( Diamond ) =

Write the formula for probability. Number of favourable outcomes (or picture cards) is 12. The total number of outcomes is 52. Substitute into the formula. Simplify the fraction.

c

P( Picture ) =

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n( Diamond ) n( s ) 13 = 52 1 = 4

n( Picture ) n( s ) 12 = 52 3 = 13

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Exercise 9D 1

What is the probability of the following experiments? a A card dealt from a normal deck of cards is a diamond b A day selected at random from the week is a weekend c A head results when a coin is tossed d A letter from the alphabet is a vowel e A two results when a die is rolled f A six is chosen from {2, 4, 6, 8, 10}.

2

A bag contains 5 blue and 3 red balls. Find the probability of selecting at random: a a blue ball. b a red ball. c not a red ball.

3

Aaron chooses one ball at random from his golf bag. The table below shows the type and quantity of golf balls in his bag. Type of golf ball Quantity B51 Impact

3

Maxfli

4

Pinnacle

13

Find the probability of him choosing: a a Maxfli c a B51 Impact

b d

a Pinnacle not a Maxfli

4

An unbiased coin is tossed three times. On the first two tosses the result is tails. What is the probability that the result of the third toss will be a tail?

5

In Amber Ave there are 3 high school students, 4 primary school students and 5 preschool students. One student from Amber Ave is chosen at random. What is the probability that a primary school student is chosen?

6

A box contains 3 blue, 4 green and 2 white counters. Find the chance of drawing at random one counter which is: a blue b green c white d not blue

7

A card is chosen at random from a standard deck of 52 playing cards. Find the probability of choosing: a the seven of clubs b a spade c a red card d a red picture card e a nine f the six of hearts g an even number h a picture card i a black ace

8

The weather on a particular day is described as either wet or dry. Therefore there is an even chance of a wet day. Do you agree with this statement? Give a reason.

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Development 9

A die with 16 faces marked 1 to 16 is rolled. Find the probability that the number is: a an odd number b neither a 1 nor a 2 c a multiple of 3 d greater than 12 e less than or equal to 15 f a square number

10

A wheel contains 8 evenly spaced numbers labelled 1 to 8. The wheel is spun until it stops at a number. It is given that the wheel is equally likely to stop at any number. Find the probability that the wheel stops at: a a 7 b a number greater than 5 c an odd number d a 1 or 2 e a number less than 10 f a number divisible by 3

11

In poker, a player is dealt five cards. Lucy is dealt four cards from a normal deck: two aces and two kings. What is the probability that the next card is: a another ace? b another king? c not an ace? d not a king?

12

Two cards are drawn at random from a normal deck of cards. What is the probability that the second card is: a a two if the first card was a two? b an ace if the first card was an ace? c the six of clubs if first card was a ten? d a two if the first card was a king? e a diamond if the first card was a diamond? f a picture card if the first card was a picture card?

13

A four-digit number is formed from the digits 2, 3, 4 and 5 without replacement. What is the probability that the number: a starts with the digit 4? b is greater than 3000? c ends with a 2 or a 3? d is 2345?

14

‘Six students enter a swimming race. The chance of a particular student winning is Is this statement true or false? Give reasons to support your opinion.

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1 6

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9.5 Range of probabilities

9.5

The probability of an event that is impossible is 0 and the probability of an event that is certain is 1. Probability is always within this range or from 0 to 1. It is not possible to have the probability of an event as 2. The range of probability is expressed as 0 ≤ P(E) ≤ 1 or P(E) ≥ 0 and P(E) ≤ 1. It is also important to realise that the probability of every event in an experiment will sum to 1.

1

Certain

0.75 0.5

Even chance

0.25 0

Impossible

Range of probability Probability of an event is between 0 and 1 or 0 ≤ P(E) ≤ 1. P(A) + P(B) + … = 1 A, B, … are all the possible outcomes or events.

Example 8

Using the range of probability

A box contains red, yellow and blue cards. The probability of selecting a red card is

3 5

and the probability of selecting a yellow card is

1 10

. What is the probability of selecting a

blue card?

Solution 1 2

Write the formula for the range of probability. Substitute into the formula the probabilities of

(

the other events P( R) = 3 4 5

3 5

1

)

and P(Y ) = 10 .

Solve the equation by making P(B) the subject of the equation. Simplify the fraction if possible. Write the answer in words.

P( R) + P(Y ) + P( B) = 1 3 1 + + P( B) = 1 5 10 3 1 P( B) = 1 − − 5 10 3 = 10 Probability of a blue card is

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Exercise 9E 1

A hat contains tickets labelled as ‘A’, ‘B’ and ‘C’. The probability of selecting ticket A is 3 7 and the probability of selecting ticket B is 15 . 10 a What is the value of P(A)? b What is the value of P(B)? c What is the probability of selecting a ticket with the letter ‘D’? d What is the probability of selecting tickets A, B or C? e What is the probability of selecting ticket C?

2

A bag contains black, yellow and white cards. The probability of drawing a black card is 57% and the probability of drawing a yellow card is 8%. What is the value of the following expressed as a fraction in simplest form: a P(Black)? b P(Yellow)? c P(White)?

3

In a particular event the probability of Blake winning a gold medal is 83 and a silver medal is 1 . There is no bronze medal. 4

a b

What are Blake’s chances of winning a gold or a silver medal? What are Blake’s chances of not winning any medals?

4

Some picture cards from a deck of cards are placed face down on the table. The probability of drawing a king is 0.25 and a queen is 0.60. What is the value of the following expressed as a decimal: a P(king)? b P(jack)? c P(jack) + P(king)? d P(king) + P(queen) + P(jack)?

5

There are four outcomes of an experiment. Three of the outcomes have probabilities of 20%, 25% and 40% respectively. What is the probability of the fourth outcome?

6

A biased die is rolled. The probability of obtaining an even number is 0.4 and the probability of a 1 or a 3 is 0.3. Find the value of the following probabilities. a P(1, 2, 3, 4, 5, 6) b P(2, 4, 5, 6) c P(Odd)

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Development 7

A disc is chosen at random from a bag containing five different colours: black, green, 1 2 pink, red and white. If P( B) = 5 , P(G ) = 13 , P( P ) = 29 and P( R) = 61 , find the probability of the following outcomes. a Black or green disc b Pink or red disc c Black or red disc d Black, green or pink disc e Black, green or red disc f Black, green, pink or red disc

8

A card is chosen at random from some playing cards. The probability of a spade is 0.24, the probability of a club is 0.27 and the probability of a heart is 0.23. Find the probability of the following outcomes. a Black card b Red card c Club or a heart d Spade or a heart e Diamond f Diamond or a club

9

Julia and Natasha are playing a game where a standard six-sided die is rolled. Julia wins if an even number is rolled. Natasha wins if a number greater than three is rolled. What is the probability that the number rolled is neither even nor greater than three?

10

A bag contains white, green and red marbles. The probability of selecting a white marble is

2 7

and the probability of selecting a green marble is

1 8

. What is the probability of

selecting a red marble?

11

The numbers 1 to 20 are written on separate cards. One card is chosen at random. What is the probability that the card chosen is a prime number or is divisible by 3?

12

One letter is selected at random from a word containing the letters TAMPR. It is given 1 . that P(T ) = 1 , P( A) = 2 , P( M ) = 1 and P( P ) = 10 5 5 10

a

b

c

Find the probability of the following outcomes. i Letters T or A ii Letters T or P iii Letters M or P iv Letters A, M or P v Letter T, A, M or P vi Letter R The word contains 10 letters. From the letters TAMPR how many of the following letters are in the word? i T ii A iii M iv P v R What is the word? (Hint: Australian place)

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9.6 Complementary events

9.6

The complement of an event E is the event not including E. For example, when throwing a die the complement of 2 are the events 1, 3, 4, 5 and 6. The complement of an event E is denoted by E . An event and its complement represent all the possible outcomes and are certain to occur. Hence the probability of an event and its complement will sum to be 1. Complementary events P( E ) + P( E ) = 1 or P( E ) = 1 − P( E ) E – Event or outcome. E – Complement of event E or the outcomes not including event E.

Example 9

Using the complementary event

Lisa selects a card at random from a normal pack. Find the probability of obtaining the following outcomes. a Not a ten b Not a black jack

Solution 2

Write the formula for the complement. Substitute into the formula the probability 4 (ten)) = 52 for a ten ( or P (ten ).

3

Evaluate.

4

Simplify the fraction.

1 2

Write the formula for the complement. Substitute into the formula the probability 2 for a black jack ( or P (bl (black ack jack jack)) = 52 ).

3

Evaluate.

4

Simplify the fraction.

1

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a

P (ten ) = 1 − P (ten) 4 = 1− 52 48 = 52 12 = 13

b

P (black jack ) = 1 − P (black jack) 2 = 1− 52 50 = 52 25 = 26

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Exercise 9F 1

What is the event that is the complement of the following events? a Selecting a black card from a normal pack of cards b Winning first prize in Lotto c Throwing an even number when a die is rolled d Obtaining a tail when a coin is tossed e Drawing a spade from a normal pack of playing cards f Choosing a green ball from a bag containing a blue, a red and a green ball

2

Find the value of P( E ) given the following information about event E. 1 P( E ) = a b P(E) = 0.9 c P(E) = 62% d P(E) = 1 : 4 5 3 e P( E ) = f P(E) = 0.45 g P(E) = 37.5% h P(E) = 3 : 7 11 The chances of the Sydney Swans winning the premiership are given as 29%. What are the chances that the Sydney Swans will not win the premiership? a Express your answer as a decimal. b Express your answer as a fraction.

3

4

The probability of obtaining a three on a biased die is 0.6. What is the probability of not obtaining a three?

5

The probability of a rainy day in March is day in March does not have rain?

6

The probability of drawing a red marble from a bag is drawing a red marble? Express your answer as a: a fraction b decimal

11 15

. What is the probability that a particular

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5 8

. What is the probability of not c

percentage

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Development 7

A ball is chosen at random from a bag containing four different colours: brown, orange, 2 purple and yellow. If P(O ) = 11 , P( P ) = 29 and P(Y ) = 14 , find the probability of the following outcomes. a Not a yellow ball b Not an orange ball c Not a purple ball d Orange or a purple ball e Yellow or a purple ball f Not a brown ball g A brown ball h Not an orange or a yellow ball

8

Samuel selects a card at random from a normal pack. Find the probability of obtaining the following outcomes. a Not a queen b Not a red ace

9

What is the probability that a person selected at random will: a not be born on Saturday? b not be born on a weekend?

10

14.2

A 12-sided die has 12 faces marked 1 to 12. The die is biased. If P(8) = 0.1, P(2) = 0.15 and P(( 3) = 0.91 , find: a

P(( 8 )

b

P(( 2 )

c

d

P(8) + P P(( 8 )

e

P(3) P(2) + P P(( 2 )

f

P( 3) + P P(( 3)

g

P(2) + P(8)

h

P(2) + P P((8)

11

One card is selected at random from a non-standard pack of playing cards. If P(ace) = 8%, P(king) = 7% and P(queen) = 10%, find the probability of the following outcomes. a Not an ace b Not a king c Not a queen d King or a queen e Ace or a queen f Not an ace, king or queen

12

The probability of selecting a card labelled with a ‘T’ from 32 cards is given as P(( T) = 163 . a b

What is the probability of not selecting a card labelled with a ‘T’? How many of the 32 cards were labelled with a ‘T’?

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Chapter 9 — Relative frequency and probability

Relative frequency

• •

Study guide 9

Relative frequency is an estimate for the probability of an event. Frequency of an event Relative frequency = Totaall number of frequencies

Equally likely outcomes

Outcomes have an equal chance of occurring.

Multistage events

Two or more events.

principle

Number of outcomes (two events) = p × q. p – Number of outcomes of the first event. q – Number of outcomes of the second event.

Systematic lists

Orderly method of determining all the possible outcomes.

Fundamental counting

Definition of probability

• •

Range of probability

• • • • •

Complementary events

• • • • •

Probability is the chance of something happening. Outcome or result of a random experiment is called an event. Number of favourable outtccomes Probability (Event) = Total number of outcomes n( E ) P( E ) = n( S ) Probability of an event that is impossible is 0. Probability of an event that is certain is 1. Probability of an event is between 0 and 1 or 0 ≤ P(E) ≤ 1. P(A) + P(B) + … = 1 A, B, … are all the possible outcomes or events. Complement of an event E is the event not including E. Probability of an event and its complement will sum to be 1. P( E ) + P( E ) = 1 or P( E ) = 1 − P( E ) E – Event or outcome. E – Complement of event E or the outcomes not including event E.

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Review

Chapter summary – Relative frequency and probability

289

Review

290

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

The frequency of an event is 6 and the total number of frequencies is 20. What is the relative frequency? A 14% B 26% C 30% D 70%

2

A local community has 220 motorbikes and 740 cars. What is the relative frequency for a car? Answer as a fraction in lowest terms. 11 220 37 740 A B C D 48 960 48 960

3

How many possible outcomes are there when four coins are tossed? A 4 B 8 C 16 D 32

4

How many different ways can the letters of the word FORBES be arranged in a row? A 6 B 21 C 24 D 720

5

One card is selected from a normal deck of cards. What is the probability that it is a diamond? 1 1 1 3 A B C D 52 13 4 4

6

A three-digit number is formed from the digits 5, 7, 8 and 9. What is the probability that the number will be odd? A 0.25 B 0.50 C 0.75 D 0.80

7

One card is selected from cards labelled 1, 2, 3, 4 and 5. What is the probability of an even number and/or a number divisible by 5? A 10% B 50% C 60% D 100%

8

A bag contains black, white and grey balls. The probability of selecting a black ball is 0.3 and a grey ball is 0.6. What is the probability of selecting a white ball? A 0.1 B 0.36 C 0.63 D 0.9

9

A letter is chosen at random from the word NEWCASTLE. What is the probability that the letter will not be a vowel? 1 2 1 2 A B C D 9 9 3 3

10

What is the value of P( E ) given that P(E) = 0.32? A 0.32 B 0.64 C 0.68

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D

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1

A class frequency table shows the scores in a test. What is the relative frequency for the following outcomes? Answer correct to two decimal places. a 50–59 b 60–69 c 70–79 d 80–89

Score

Frequency

50–59

5

60–69

6

70–79

8

80–89

6

Relative freq.

2

The local football club sold 350 raffle tickets to raise money for some equipment. Liam sold 60 of these tickets. What is the relative frequency of Liam’s tickets? Answer as a percentage correct to the nearest whole number.

3

Last year Oscar bought a packet of biscuits every week and found 30 of these packets contained broken biscuits. What is the relative frequency of this event? Answer as a decimal correct to two decimal places.

4

There were 23 people who applied for a particular job. Are the chances of each person getting the job equally likely? Why?

5

A paper bag contains 3 green, 4 brown and 5 yellow beads. To win a game, Greg needs to draw two green beads from the bag. How many elements are in the sample space?

6

List all the possible ways to answer the first three questions of a true or false test.

7

A poker machine has three reels, with 12 symbols on each wheel. How many arrangements are possible when the poker machine is spun?

8

A PIN has four digits. How many possible PINs are there?

9

A raffle ticket is drawn from a box containing 50 raffle tickets numbered from 1 to 50. Find the probability of the following outcomes. a The number 50 b Even number c Less than 20 d Greater than 30 e Divisible by 5 f Square number

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Sample HSC – Short-answer questions

Review

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Chapter – Earning Money 10 Whatsummary is the probability of choosing a black card from a standard deck of cards? 11

Four kings are taken from a standard deck of cards and placed face down on a table. One card is selected at random. What is the probability of selecting: a the king of clubs? b a black king? c a picture card?

12

A house contains 4 girls, 3 boys and 2 adults. If one person is chosen at random, what is the probability that the person: a is a girl? b is a boy? c is a girl or a boy?

13

An eight-sided die has the numbers 1 to 8 on it. What is the probability of rolling the following outcomes? a Number 2 b Either a 3 or a 5 c Number 9 d Divisible by 3 e Odd number f Prime number

14

There are five students in a group whose names are Adam, Sarah, Max, Hayley and David. If one name is chosen at random, find the probability of selecting a name: a with 5 letters. b with the letter ‘a’. c with exactly one vowel.

15

A fair coin is tossed three times. The probability of throwing three tails is 0.125, two tails is 0.375 and one tail is 0.375. What is the probability of the following outcomes? a No tails b Three or two tails c At least one tail d Not throwing a head e Not throwing exactly two tails f Throwing one head

16

There are three outcomes of a rugby league game: win, lose or draw. If P(( Win ) = 75 and P(( Lose) = 15 , find the probability of the

following. a Winning or losing the match b Drawing the match c Not winning the match d Not losing the match

17

Caitlin selects a card at random from a standard deck of cards. Find the probability of obtaining the following outcomes. a Not an ace b Not a heart c Not a red six

18

The probability of drawing a blue card from a bag is

5 16

. What is the probability of not

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C H A P T E R

10

Taxation Syllabus topic — FM3 Taxation Calculate the allowable deductions from gross income Calculate the taxable income Calculate the Medicare levy Determine PAYE tax payable or refund owing Calculate the GST payable Create graphs that describe different tax rates

10.1 Allowable deductions Allowable deductions are deductions allowed by the Australian Taxation Office (ATO). Details of allowable deductions are given on the ATO website (www.ato.gov.au).

Allowable deductions include: • Work-related expenses – costs incurred while performing your job • Self-education expenses – costs of education related to your work • Travel expenses – costs of travel directly connected with your work • Car expenses – costs of using your car related to your work • Clothing expenses – costs of work clothing and laundry • Tools – cost of work tools • Gifts and donations – gifts made to an eligible organisation. 293 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Allowable deductions Deductions allowed by the Australian Taxation Office include work-related, self-education, travel, car, clothing, tools and donations.

Example 1

Calculating allowable deductions

Riley works as an information technology consultant. He is entitled to the following tax deductions: • equipment costs of $1260 • car expenses of $1060 • professional learning of $985 • union fees of $650 • charity donations of $250 • tax agent fee of $212. What is Riley’s total allowable tax deduction? Solution 1 2 3 4

Write the quantity (tax deduction) to be calculated. Add all the allowable deductions. Evaluate and write using correct units. Write the answer in words.

Example 2

Tax deduction = 1260 + 1060 + 985 + 650 + 250 + 212 = $4417 Riley has an allowable tax deduction of $4417.

Calculating allowable deductions

Ava has used her own car for a total of 7900 km on work related travel this financial year. Calculate her tax deduction if she is entitled to claim 70 cents per kilometre. Solution 1 2 3 4

Write the quantity (tax deduction) to be calculated. Multiply the kilometres travelled by the rate per kilometre. Evaluate and write using correct units. Write the answer in words.

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Tax deduction = 7900 × 0.70 = $5530

Ava has an allowable tax deduction of $5530.

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295

Exercise 10A 1

Stephanie works as a waitress and is entitled to an allowance for the cost of her work clothing and laundry. Her clothing expenses are listed below. Calculate the total cost for each item of clothing and her allowable deduction for clothing. Work clothing

2

Quantity Unit cost Total cost

Blue shirt

3

$55.00

a

Black trouser

3

$110.00

b

Belt

1

$45.00

c

Tie

2

$34.00

d

Dry-cleaning

4

$32.00

e

Zara is a child care worker who is entitled to the following tax deductions: • $420 for union fees • tax agent fee of $125 • charity donations of $160 • self-education fee of $840 • stationery costs of $46. What is Zara’s total tax deduction?

3

Chris is entitled to the following tax deductions: training courses of $1460, motor vehicle expenses of $1420, stationery costs of $760, union fees of $480, charity donations of $310 and accountant fee of $184. What is Chris’s total allowable tax deduction?

4

Car expenses are claimed according to the engine capacity of the motor vehicle. The rate per kilometre for travel in a private vehicle is listed below. Engine capacity

Allowable deduction

0 to 1600 cc

58 c/km

1600 cc to 2600 cc

69 c/km

Greater than 2600 cc

70 c/km

Calculate the tax deduction for the following car expenses related to work. a 3560 km in a Toyota Corolla with an engine capacity of 1.8 L (1788 cc) b 1280 km in a Ford Falcon with an engine capacity of 4.0 L (3984 cc) c 4580 km in a Holden Barina with an engine capacity of 1.6 L (1598 cc) d 2340 km in a Honda Civic with an engine capacity of 1.3 L (1339 cc) e 3105 km in a Nissan Pulsar with an engine capacity of 1.8 L (1769 cc) f 2818 km in a Mazda CX-9 with an engine capacity of 3.7 L (3726 cc)

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Development 5

Harrison buys a new van costing $42 560 for his business. He is entitled to claim as a tax deduction 12% of the cost of the vehicle if the motor vehicle travels more than 5000 business kilometres in a year. What is Harrison’s allowable deduction for the van if he travelled 16 230 kilometres for business?

6

Joel is a wheat farmer. He has capital equipment on the farm worth $240 000. The ATO allows a tax deduction for the depreciation of capital equipment based on a percentage of the current value. What is the tax deduction using the following rates of depreciation? a 10% p.a. b 20% p.a. c 30% p.a. d 40% p.a.

7

Mitchell has an investment property that contains furnishings valued at $12 600. The furnishings are an allowable deduction with a rate of depreciation of 15% p.a. How much can be claimed for depreciation over the year?

8

Xiang is a teacher who bought a $2350 laptop for school use. The laptop is an allowable deduction with a rate of depreciation of 33% p.a. of the current value. How much can be claimed for depreciation in each of the following years? a First year b Second year c Third year

9 

Dylan is the owner of a newspaper shop in a shopping centre. He pays rent of $860 per week, has an electricity bill of $280 per quarter and a telephone bill of $110 per month. These expenses are work related so he is entitled to a tax deduction. What is Dylan’s total allowable tax deduction?

10

Chelsea has a small office in her home to run a business. The office in her home is 8% of the area of the house. The tax office allows 8% of the household bills as a tax deduction if it is a work-related expense. Calculate the allowable tax deduction for the financial year on the following household bills. a Electricity bill of $360 per quarter b Telephone bill of $70 per month c House insurance of $684 per year d Rent of $440 per fortnight

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10.2 Taxable income Each year people who earn an income are required to complete a tax return. A tax return is a form that states a person’s income, the amount of tax paid and any allowable tax deductions. Most people have PAYG tax deducted from their wage or salary throughout the year. This PAYG tax can be greater or less than the required amount of tax to be paid. Tax is calculated on the taxable income. The taxable income is the gross income minus any allowable deductions. The gross income is the total amount of money earned from all sources. It includes interest, profits from shares or any payment received throughout the year. Taxable income Taxable income = Gross income – allowable deductions Example 3

Calculating taxable income

Anthony is a businessman who earns a gross salary of $93 250 per year. His accountant completed his tax return and calculated $2890 in allowable deductions. What is Anthony’s taxable income?

Solution 1 2 3

Write the quantity (taxable income) to be calculated. Subtract the deductions from the gross income. Evaluate and write using the correct units.

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Taxable income = 93 250 – 2890 = $90 360

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Example 4

Calculating taxable income

Emily is a journalist with a gross annual salary of $87 620. She also made $5680 from her share portfolio and received $7320 from royalties. If Emily has tax deductions totaling $6472, calculate her taxable income. Solution 1 2 3 4

Calculate the gross income by adding all income. Write the quantity (taxable income) to be calculated. Subtract the deductions from the gross income Evaluate.

Example 5

Gross income = 87 620 + 5680 + 7320 = $100 620

Taxable income = 100 620 − 6472 = $94 148

Calculating taxable income

Nicole is a scientist who earns a gross weekly pay of $1624. She has allowable tax deductions of $8 per week for dry-cleaning, $60 for work-related travel per year, $460 per year for union fees and she made donations to charities of $620 throughout the year. a What is Nicole’s gross yearly salary? b What is Nicole’s total allowable tax deduction? c Calculate Nicole’s taxable income. Solution 1 2 3 4 5 6 7 8 9

Write the quantity (salary) to be calculated. Multiply the weekly pay by 52. Evaluate.

a

Salary = 1624 × 52 = $84 448

Write the quantity (tax deduction) to be calculated. Add all the allowable deductions. Evaluate.

b

Tax deduction = (8 × 52) + 60 + 460 + 620 = $1556

Write the quantity (taxable income) to be calculated. Subtract the deductions from the gross income. Evaluate.

c

Taxable income = 84 448 − 1556 = $82 892

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Exercise 10B 1

Benjamin has a gross income of $84 000. What is Benjamin’s taxable income given the following allowable deductions? a $5120 b $9571 c $4720 d $24 104 e $8205 f $17 594 g $12 520 h $23 890 i $34 560

2

Chris earns a gross salary of $67 840 per year. His tax deductions total $3462. Calculate Chris’s taxable income.

3

Jessica earned a gross income of $75 480 in the last financial year. Allowable deductions

Amount

Work-related expenses

$1260

Self-education expenses

$680

Travel expenses

$940

Clothing expenses

$320

a b

The table above is a summary of her allowable deductions. What is her total allowable deduction? Calculate Jessica’s taxable income in the last financial year.

4

Eliza earned $88 784 from her employer in the last financial year. She also earned bank interest of $380. Eliza spent $240 on books, $520 on stationery and $380 on a printer, all of which are needed for her work. a What is Eliza’s gross income? b What are Eliza’s total allowable deductions? c Calculate Eliza’s taxable income.

5

Daniel is a police officer who earned a gross income of $63 620. He claimed a tax deduction for a utility belt ($160), a pair of safety glasses ($390), bullet proof vest ($1240) and handcuffs ($420). a What is Daniel’s total allowable deduction? b Calculate Daniel’s taxable income.

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Development 6

Nicholas earns a gross weekly pay of $1120. He has tax deductions of $1460. a What is Nicholas’s gross yearly salary? b What is Nicholas’s taxable income?

7

Abbey is a real estate salesperson who earns a commission of 2% on all sales. During the year Abbey sold real estate to the value of $3 232 100. a What is Abbey’s gross annual income? b Abbey has calculated her tax deductions to be $4320. What is her taxable income?

8

Isabelle works for a travel agency and earns a gross fortnightly pay of $2780. She pays PAYG tax of $602 per fortnight and has tax deductions of $7 per week for dry-cleaning, $80 for work-related travel per year and $380 per year for union fees.

a b c d

9 

What is Isabelle’s gross yearly salary? How much tax is deducted each week? What is Isabelle’s total allowable tax deduction? Calculate Isabelle’s taxable income.

Oscar is a tradesman who receives a yearly gross salary of $92 200. He also works part-time at TAFE for a wage of $135 per week. Oscar received $360 in share dividends. a What is Oscar’s gross annual income? b Oscar is entitled to tax deduction for travelling between his two places of employment. Oscar has calculated that he travelled 340 km and the allowable deduction is 69 c/km. What is Oscar’s travel expense? c In addition to the above travel expenses Oscar is entitled to the following tax deductions: $530 for union fees, tax agent fee of $180, charity donations of $280 and equipment costs of $750. Calculate Oscar’s total allowable deduction. d What is Oscar’s taxable income?

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10.3 Medicare levy The Medicare levy is an additional charge to support Australia’s universal health care system. It ensures that all Australians have access to free or low-cost medical and hospital care. The Medicare levy is calculated at a rate of 1.5% of taxable income. Medicare levy Additional charge for health services. It is calculated at 1.5% of the taxable income.

Example 6

Calculating the Medicare levy

The Medical levy is 1.5% of the taxable income. What is the Medicare levy if the taxable income is $90 600? Solution 1 2 3 4

Write the Medicare rate of the taxable income. Express 1.5% as a decimal 0.015. Evaluate using correct units. Write the answer in words.

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Medicare levy = 1.5% of $90 600 = 0.015 × 90 600 = $1359 Medicare levy payable is $1359.

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Exercise 10C 1

The Medicare levy is 1.5% of the taxable income. What is the Medicare levy on the following taxable incomes? Answer correct to the nearest dollar. a $23 000 b $88 541 c $40 600 d $46 906 e $67 800 f $200 592 g $170 300 h $15 790 i $90 640

2

Liam has a taxable income of $56 400. He is required to pay $11 520 in tax plus a Medicare levy of 1.5% of his taxable income. a How much is Liam’s Medicare levy? b Calculate the total amount of tax due including the Medicare levy.

3

Mia works for a superannuation fund and received a taxable income of $124 800. She is required to pay $36 520 in income tax. a Medicare levy is 1.5% of the taxable income. How much is Mia’s Medicare levy? b Calculate the total amount of tax due including the Medicare levy.

4

The government is planning to change the rate of Australia’s Medicare levy. Calculate the Medicare levy payable on $120 000 for the following rates. a 0.50% b 0.75% c 1.00% d 1.25% e 1.50% f 1.75% g 2.00% h 2.25% i 2.50%

5

Sami has a taxable income of $88 900. He is required to pay $22 160 in tax plus a Medicare levy of 2.0% of his taxable income. a How much is Sami’s Medicare levy? b Calculate the total amount of tax due including the Medicare levy.

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Development 6

The rate of the Medicare levy is 1.5% of the taxable income. What is the taxable income given the following Medicare levies? a $300.00 b $675.00 c $2250.00 d $1689.30 e $888.09 f $543.21 g $2105.52 h $1131.18 i $367.74

7 

Create the spreadsheet below. Formulas have been entered in cells B5 to B10.

10C

a b c d

Cell B5 has a formula (=A5*0.015) that calculates the Medicare levy. The rate of the Medicare levy is 1.5% of the taxable income. Enter this formula. The formulas in cells B6 to B10 are similar to the formula in B5. Fill down the contents of B6 to B9 using the formula in cell B5. Use the spreadsheet to calculate the Medicare levy payable on a taxable income of: i $10 000 ii $210 000 iii $49 740 The Australian Government has decided to decrease the Medicare levy from 1.5% to 1.25% of the taxable income. Modify the spreadsheet.

8

Oscar is a coach driver who paid a Medicare levy of $1193.40. What was Oscar’s taxable income if the rate of the Medicare levy is 1.5% of the taxable income? Answer correct to the nearest dollar.

9

Laura paid a Medicare levy of $1303.40. What was Laura’s taxable income if the rate of the Medicare levy is 2.5% of the taxable income?

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Preliminary Mathematics General

10.4 Calculating tax The amount of taxation varies according to the amount of money you earn. A tax return must be lodged with the ATO each year. It is a statement of the income earned and the tax paid during the financial year from 1 July to 30 June. The ATO publishes a TaxPack to assist people in completing their tax return. The TaxPack provides information about the current income tax rates.

10.4

Personal income tax rates The personal income tax rates are regularly changed to take into account government policy and inflation. The tax rates are listed in the table below. (Note: these are not the current tax rates.) Taxable income

Tax payable

0–$6000

Nil

$6001–$30 000

Nil + 15 cents for each $1 over $6000

$30 001–$80 000

$3600 + 30 cents for each $1 over $30 000

$80 001–$180 000

$18 600 + 40 cents for each $1 over $80 000

$180 001 and over $58 600 + 45 cents for each $1 over $180 000 The tax payable is dependent on the taxable income. If the taxable income is $6000 or less then there is no tax payable. The tax rates then increase progressively. It starts out at 15 cents for every $1, for amounts between $6001 and $30 000.The highest tax rate is 45 cents for every $1 over $180 000. Most people have PAYG tax deducted from their wage or salary throughout the year. This PAYG tax can be greater or less than the required amount of tax to be paid. If a person pays more throughout the year than they are required to pay they will receive a tax refund. On the other hand, if a person pays less than the required amount of tax throughout the year they will have tax owing. Tax refund

Tax owing

Tax refund = Tax paid – Tax payable

Tax owing = Tax payable – Tax paid

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Chapter 10 — Taxation

Example 7

305

Calculating the tax payable

Manjula has a taxable income of $25 000. How much tax will she have to pay? Manjula has paid $4050 in PAYG tax. What is her refund? Solution 1 2 3 4 5 6 7 8 9

Look at the income tax rates table on page 304. Taxable income of $25 000 is between $6001 and $30 000 (row 2). Read the information in the tax payable column for this range (row 2). Write an expression for tax payable. The word ‘over’ implies ‘more than’. Evaluate using correct units. Write in words. Find the tax refund by subtracting the tax payable from the tax paid. Evaluate. Write in words.

Example 8

Tax payable = Nil + (25 000 − 6000) × 0.15 = $2850

Manjula needs to pay $2850 in tax. Tax refund = $4050 − $2850 = $1200 Manjula’s refund is $1200.

Calculating a tax refund

Joel has a taxable income of $200 000 and has paid $72 000 in tax instalments. How much does the tax department owe Joel at the end of the financial year? Solution 1 2 3 4 5

6

Look at the income tax rates Tax payable = 58 600 + (200 000 − 180 000) × 0.45 table on page 304. = $67 600 Taxable income of $200 000 is between $180 000 and over. Write an expression for tax payable. Evaluate. Find the tax refund by Tax refund = 72 000 − 67 600 subtracting the tax payable = $4400 from the tax paid. Evaluate. Tax department owes Joel $4400.

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Preliminary Mathematics General

Exercise 10D 1

The table below shows personal income tax rates. Taxable income

Tax payable

Tax bracket

0–$6000

Nil

A

$6001–$30 000

Nil + 15 cents for each $1 over $6000

B

$30 001–$80 000

$3600 + 30 cents for each $1 over $30 000

C

$80 001–$180 000

$18 600 + 40 cents for each $1 over $80 000

D

$180 001 and over

$58 600 + 45 cents for each $1 over $180 000

E

Which tax bracket (A, B, C, D or E) from the table applies to these taxable incomes? a $16 000 b $2500 c $75 000 d $122 500 e $230 000 f $80 000 g $30 001 h $180 000 i $4500 2

Calculate the tax payable on the following taxable incomes by completing the tax payable expression. a $16 000 Tax payable = Nil + (16 000 – 6000) × 0.15 = b

$32 500

Tax payable = 3600 + (32 500 – 30 000) × 0.30 =

c

$75 000

Tax payable = 3600 + (75 000 – 30 000) × 0.30 =

d

$122 600

Tax payable = 18 600 + (122 600 – 80 000) × 0.40 =

e

$230 000

Tax payable = 58 600 + (230 000 – 180 000) × 0.45 =

f

$80 000

Tax payable = 3600 + (80 000 – 30 000) × 0.30 =

3

Ava’s taxable income was $28 000. The first $6000 was tax-free and the balance was taxed at a marginal rate of 15%. Calculate the amount of tax payable.

4

Tyler’s taxable income was $111 000. The tax payable on the first $80 000 is $18 600 and the balance was taxed at a marginal rate of 40%. Calculate the amount of tax payable.

5

Use the tax table on page 304 to calculate the tax payable on the following taxable incomes. a $6001 b $30 001 c $80 001

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Development Use the tax table on page 304 to answer questions 6 to 11. 6

What is the tax payable on the following taxable incomes? a $10 890 b $73 966 c $37 814 d $115 900 e $196 430 f $53 410

7

Calculate the tax refund or tax owing for the following: a Taxable income of $34 850 with tax instalments paid of $4 678. b Taxable income of $129 864 with tax instalments paid of $31 272.

8

Ebony’s gross income is $32 540 with allowable deductions of $4120. a What is Ebony’s taxable income? b Calculate the amount of tax due. c Ebony has paid $95 per week in tax. How much refund should she receive for the year?

9

Charlie’s gross income is $55 730. His allowable deductions are $5230. a What is Charlie’s taxable income? b Calculate the amount of tax due. c Charlie has paid $160 per week in tax. How much tax has been paid for the year? d Will Charlie receive a refund or will he have to pay more tax? Justify your answer.

10

Ruby is a dentist with a taxable income of $145 684. a Find the tax payable on this amount. b What percentage of her income is paid as tax? Answer correct to one decimal place.

11

Nathan has a taxable income of $106 770. He has paid $980 per fortnight in tax. a How much tax has Nathan paid for the year? b Calculate the amount of tax payable by Nathan. c Will Nathan receive a refund or will he have to pay more tax? Justify your answer. d What percentage of his income is paid as tax? Correct to one decimal place.

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12

Create the spreadsheet below.

10D

a b

14.3

Formulas have been entered into cells C13, C14, C15, C16, E14, E16 and E18. These formulas are shown above. Enter all the formulas. Change the taxable income (cell B4) to the following amounts and observe the results. i $50 000 ii $10 000 iii $200 000

13

Emma received an income of $75 420 from her main job for the last financial year and paid $18 680 in tax instalments. In addition, Emma earned an income of $6890 from a part-time job and paid tax of $2980. a How much tax has Emma paid for the year? b Calculate the amount of tax payable by Emma. c Will Emma receive a refund or will she have to pay more tax? Justify your answer. d What percentage of her income is paid as tax? Answer correct to one decimal place.

14

Alexander earns an income of $42 000. He also has $15 000 in a bank account which earns interest at a rate of 8.5% p.a. Alexander has to pay tax on his total income. a How much interest does Alexander earn from his bank account this year? b Calculate the tax payable.

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309

10.5 Calculating GST GST 10.5

The Australian Government collects a tax when people purchase goods and services. The tax is called the GST (Goods and Services Tax) and is 10% of the purchase price of the item. There are exceptions for basic food items and some medical expenses. GST • • •

To calculate the GST, find 10% of the pre-tax price. To calculate the total cost of an item, add the GST to the pre-tax price. Alternatively, find 110% of the pre-tax price. To calculate the pre-tax price given the total cost of an item, divide the total cost by 110%.

Example 9

Finding the GST

John bought a ticket for $142 to see a concert at the Sydney Olympic Stadium. He was also required to pay a 10% GST. a How much GST is payable? b What was the total cost of his ticket including the GST? c What is the pre-tax price of a ticket if the final price of the ticket was $149.60? Solution 1 2 3 4 5 6 7 8 9

Write the quantity (GST) to be calculated. Multiply 0.10 by 142. Evaluate and write using correct units.

a

GST = 10% of $142 = 0.10 × 142 = $14.20

Write the quantity (total cost) to be calculated. Add the GST to the cost of the ticket. Evaluate and write using correct units.

b

Total cost = $142 + $14.20 = $156.20

Write the quantity (pre-tax price) to be calculated. Divide the final price by 1.10. Evaluate and write using correct units.

c

Pre-tax price = $149.60 ÷ 110% = $149.60 ÷ 1.10 = $136

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VAT In some countries the GST is called the VAT (Value Added Tax). The rate of the VAT ranges from 2% to 25%. The calculations for the VAT are similar to the calculations for the GST except the rate of taxation is different. VAT • • •

To calculate the VAT, find the VAT rate of the pre-tax price. To calculate the total cost of an item, add the VAT to the pre-tax price. Alternatively, find 100% + VAT rate of the pre-tax price. To calculate the pre-tax price given the total cost of an item, divide the total cost by 100% + VAT rate.

Example 10

Finding the VAT

Singapore has a Value Added Tax (VAT) levied at 5%. Olivia bought a microwave in Singapore for $275 plus a VAT of 5%.

a b c

How much VAT is payable? What was the total cost of her microwave including the VAT? What is the pre-tax price of the microwave if the final price of the microwave was $672? Solution 1 2 3 4 5 6 7 8 9

Write the quantity (VAT) to be calculated. Multiply 0.05 by 275. Evaluate and write using correct units.

a

VAT = 5% of $275 = 0.05 × 275 = $13.75

Write the quantity (total cost) to be calculated. Add the VAT to the cost of the ticket. Evaluate and write using correct units.

b

Total cost = $275 + $13.75 = $288.75

Write the quantity (pre-tax price) to be calculated. Divide the final price by 1.05. Evaluate and write using correct units.

c

Pre-tax price = $672 ÷ 105% = $672 ÷ 1.05 = $640

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Chapter 10 — Taxation

311

Exercise 10E 1

Calculate the GST payable on each of the following items. The GST rate is 10%. a Football at $36 b Shoes at $140 c Dinner at $170 d Bucket at $3.20 e Dress at $490 f Book at $42 g Belt at $42.90 h Ring at $2600 i Camera at $370

2

Blake received a $620 bill for electrical work and was required to pay a 10% GST. a How much GST is payable? b What was the total cost of the electrical work including the GST?

3

Isabelle received an invoice for her gym membership of $780. In addition she was required to pay a 10% GST. a How much GST is payable? b What was the total cost of her gym membership including the GST?

4

Great Britain has a Value Added Tax (VAT) similar to the GST. The VAT is 17 12 % on clothing. How much VAT is payable on the following items?

a c 5

Football jumper worth £150 Football shorts worth £20

b d

Football boots worth £80 Football socks worth £8

What is the VAT payable in the following countries on a car worth 42 000? Answer correct to the nearest dollar. a Argentina – 21% VAT b Canada – 7% VAT c China – 17% VAT d India – 12.5% VAT e Russia – 18% VAT f Singapore – 5% VAT g South Africa – 14% VAT h Switzerland – 6.5% VAT

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Development 6

The cost of the following items includes a 10% GST. What was the pre-GST charge? a Pen at $17.60 b Calculator at $24.20 c Chair at $99 d DVD at $38.50 e Plant at $15.40 f Watch at $198

7

New Zealand has a Value Added Tax (VAT) levied at 12.5%. a Ata bought a jacket in New Zealand for $480 plus the VAT. What price did she pay for the jacket? b A jacket costs $390 including the VAT. What was the price of the jacket before VAT is added? Answer correct to the nearest dollar.

8 

Create the spreadsheet below.

10E

a b

Cell C5 has a formula that calculates a 10% GST. Enter this formula. The formula in cell D5 adds the cost price and the GST. Enter this formula. Fill down the contents of B6 to B10 using this formula.

9

After the 10% GST was added, the price of a mobile phone was $362. What was the price without GST? Answer correct to the nearest cent.

10

What was the original cost of a notebook computer with a GST-included price of $1850? The rate of GST is 10%. Answer correct to the nearest cent.

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10.6 Graphing tax rates The personal income tax rates and a graph of these rates are shown below. (Note: these are not the current tax rates.) The taxable income is the horizontal axis and the tax payable is the vertical axis. Each change in the rate of tax results in a different line segment. Taxable income

Tax payable

0–$6000

Nil

$6001–$30 000

Nil + 15 cents for each $1 over $6000

$30 001–$80 000

$3600 + 30 cents for each $1 over $30 000

$80 001–$180 000

$18 600 + 40 cents for each $1 over $80 000

$180 000 and over

$58 600 + 45 cents for each $1 over $180 000

70 (180, 58.6)

Tax payable (thousands of dollars)

60 50 40 30 20

(80, 18.6)

10 (6, 0)

(30, 3.6) 20

40

60 80 100 120 140 Taxable income (thousands of dollars)

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160

180

200

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Preliminary Mathematics General

The personal income tax table on the previous page can be used to determine five points to be graphed. The points are (0, 0), (6, 0), (30, 3.6), (80, 18.6) and (180, 58.6). These values are in thousands of dollars: the first number is the taxable income and the second number is the tax payable. The point (180, 58.6) represents a taxable income of $180 000 and a tax payable of $58 600. The gradient of each line segment represents the rate of tax. For example, the gradient of the line between (80, 18.6) and (180, 58.6) is calculated below. Gradient =

Vertical r rtical rise Horizontal run

=

58.6 − 18.6 180 − 80

=

40 100

= 0.440 The gradient of 0.40 corresponds to the rate of tax of 40 cents for each $1. Graphing tax rates 1 2 3

Draw a number plane with the taxable income as the horizontal axis and the tax payable as the vertical axis. Use information in the tax rate table to determine the points. Plot the points. Join the points to make a straight line segment for each rate of tax.

Example 11

Finding information from tax rate graph

Amy pays $10 000 in tax. Use the graph on the previous page to determine her taxable income. Solution 1 2 3 4

Find 10 on the vertical axis. This represents a tax payable amount of $10 000. Use a ruler read a horizontal line from 10 until it reaches the line graph. Place a marker. Read vertically from the marker to the horizontal axis. Place a marker (about 50). Read and then approximate the value of the marker.

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Taxable income is approximately $50 000.

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Chapter 10 — Taxation

315

Exercise 10F The line graph below shows taxable income against tax payable.

Tax payable (thousands of dollars)

1

30 25 20 15 10 5 20

40 60 80 Taxable income (thousands of dollars)

100

Use the line graph to approximate the tax payable on the following taxable incomes. a $5000 b $90 000 c $15 000 d $70 000 e $50 000 f $80 000 2

Use the line graph in question 1 to approximate the taxable income on the following tax payable. a $12 000 b $30 000 c $6000 d $9000 e $25 000 f $15 000

3

Max paid $21 000 in tax. Use the line graph in question 1 to determine his taxable income.

4

Use the line graph in question 1 to determine the tax rate (or gradient) for the following amounts. a Taxable income between $0 and $10 000 b Taxable income between $10 000 and $30 000 c Taxable income between $30 000 and $60 000 d Taxable income between $60 000 and $100 000

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Development 5

The straight line on the graph represents a flat tax rate of 20%. The other line graph shows the current tax structure.

Tax payable (thousands of dollars)

60

50

40 30

20

10

20

a b c d

6 

40

60 80 100 120 140 Taxable income (thousands of dollars)

160

180

Calculate the tax payable on the flat tax rate for a taxable income of $40 000. What range of incomes would the flat rate of tax be a better system? Answer to the nearest ten thousand. What is the difference in tax payable between the two systems on a taxable income of $180 000? What is the difference in tax payable between the two systems on a taxable income of $80 000?

Draw a line graph of the tax rates shown in the table below. Use taxable income as the horizontal axis and tax payable as the vertical axis. Taxable income

Tax payable

0–$10 000

Nil

$10 001–$50 000

Nil + 15 cents for each $1 over $10 000

$50 001–$100 000

$7500 + 30 cents for each $1 over $50 000

$100 001–$200 000

$22 500 + 40 cents for each $1 over $100 000

$200 000 and over

$62 500 + 45 cents for each $1 over $200 000

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Chapter 10 — Taxation

Study guide 10

Allowable deductions

Deductions allowed by the Australian Taxation Office include work-related, self-education, travel, car, clothing, tools and donations.

Taxable income

• • •

Tax is calculated on the taxable income. Gross income is the total amount of money earned. Taxable income = Gross income – Allowable deductions.

Medicare levy

Additional charge for health services. Medicare is calculated at 1.5% of the taxable income.

Calculating tax

• • •

Calculating GST and VAT

• • •

To calculate the GST find 10% (or VAT rate) of the pre-tax price. To calculate the total cost of an item, add the GST (or VAT) to the pre-tax price. To calculate the pre-tax price given the total cost of an item, divide the total cost by 110% (or 100% + VAT rate).

1

Draw a number plane with the taxable income as the horizontal axis and the tax payable as the vertical axis. 2 Use information in the tax rate table to determine the points. Plot the points. 3 Join the points to make a straight line segment for each rate of tax. Tax payable (thousands of dollars)

Graphing tax rates

Personal income tax tables have an increasing rate of tax. Tax refund = Tax paid – Tax payable Tax owing = Tax payable – Tax paid

70 (180, 58.6)

60 50 40 30 20 10

(80, 18.6) (6, 0) (30, 3.6) 20 40 60 80 100 120 140 160 180 200 Taxable income (thousands of dollars)

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Review

Chapter summary – Taxation

317

Review

318

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

Alexei is a landscape gardener and entitled to the following tax deductions: union fees of $484, donations of $720, equipment cost of $860, and car expenses of $1455. What is Alexei’s total allowable tax deduction? A $484 B $860 C $1455 D $3519

2

Stephanie has an allowable deduction of $4690. What is her taxable income if her gross annual salary is $43 720? A $39 030 B $43 720 C $48 410 D $90 620

3

William has a taxable income of $53 684. What is his Medicare levy? (Assume the Medicare levy is calculated at a rate of 1.5% of the taxable income.) A $80.53 B $805.26 C $8052.60 D $80 526

4

The rate of the Medicare levy is 1.5% of the taxable income. What is the taxable income if the Medicare levy was $1208.85? Answer correct to the nearest dollar. A $18 B $1813 C $80 590 D $139 018

5

Charlotte is a librarian who has a taxable income of $69 410. The tax payable on the first $30 000 is $3600 and the balance was taxed at a marginal rate of 30%. How much does Charlotte have to pay in tax? A $3600 B $15 423 C $30 000 D $24 423



6

A house owner receives an electricity bill for $598, before a GST of 10% is added. How much is the GST? A $5.98 B $59.80 C $592.02 D $538.20

7

After the 16% VAT was added, the price of a DVD player was $278. What was the price without VAT? Answer to the nearest cent. A $44.48 B $239.66 C $239.67 D $322.48

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Chapter 10 — Taxation

319

1

Sienna owns a clothing store and is entitled to the following tax deductions: $570 for union fees, tax agent fee of $375, charity donations of $390, information technology costs of $3910 and printing costs of $528. What is Sienna’s total tax deduction?

2

What is the taxable income if the gross salary is $82 390 and the allowable tax deduction is $4870?

3

Ryan is a small business owner who bought a $3850 computer for business use. The computer is an allowable deduction with a rate of depreciation of 33% p.a. a How much can he claim for depreciation in the first year? b What is the depreciated value of the computer after the first year?

4

James works for a modelling agency and earns a gross fortnightly pay of $3720. He pays PAYG tax of $986 per fortnight and has tax deductions of $15 per week for dry-cleaning, $1450 for work-related travel and $1000 per year for charities. a What is James’s gross yearly salary? b How much tax is deducted each week? c What is James’s total allowable tax deduction? d Calculate James’s taxable income.

5

Jessica has a gross weekly pay of $1024. She received $490 in interest from a term deposit account. Jessica has an allowable deduction of $1380. a What is Jessica’s gross annual income? b Calculate Jessica’s taxable income. c The Medicare levy is 1.5% of the taxable income. How much will Jessica pay for the Medicare levy? d Each week $262.50 PAYG tax was deducted from Jessica’s pay. How much did Jessica pay in PAYG tax for the year?

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Review

Sample HSC – Short-answer questions

Preliminary Mathematics General

Chapter Earning Money Calculate the tax payable on this amount. Use the 6 Rubysummary has a taxable–income of $134 830. personal income tax rates on page 304. 7

Cooper is a retail store manager. After he received a $2300 annual pay rise, his salary became $73 450. How much of the $2300 pay rise was he required to pay in tax? Use the personal income tax rates on page 304.

8

Thomas received a bill for internet access of $286. In addition he was required to pay 10% GST. a How much GST is payable? b What was the total cost of internet access including the GST?

9

The straight line graph shows taxable income against tax payable.

60

50 Tax payable (thousands of dollars)

Review

320

40

30

20

10

20

a b c

40 60 80 100 Taxable income (thousands of dallars)

120

140

Use the graph to estimate the tax payable on a taxable income of $30 000. Use the graph to estimate the taxable income on a tax payable of $20 000. What is the rate of tax between a taxable income of $10 000 and $30 000? Challenge questions 10

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C H A P T E R

11

Summary statistics Syllabus topic — DS3 Summary statistics Calculate the median Calculate the mean, mode and median Determine the mean from grouped data Calculate the standard deviation Compare a summary of statistics from various samples

A statistic is any number that can be calculated from data. Summary statistics are special statistics that indicate certain features of the data. They are generally measures of the centre or the spread.

11.1 The median Median 11.1

The median is the middle score or value. To find the median, list all the scores in increasing order and select the middle one. For example, the median of 7, 5, 2, 4 and 9 is found by sorting the five scores and finding the middle score, or 5. 2

5

4

7

9

When there is an even number of scores, the median is the average of the two middle scores. For example, the median of 1, 7, 5, 2, 4 and 9 is found by sorting the six scores and finding the average of 4 and 5, or 4.5. 1

2

4

5

7

9

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Preliminary Mathematics General

Median 2

Arrange all the scores in increasing order. Count the total number of scores. This is represented by the letter n.

3

If there is an odd number of scores then the median is

4

If there is an even number of scores then the median is average of the the n + 1 score.

1

n+1 2

score. n 2

and

2

Example 1

Calculating the median

The table below shows the number of rainy days for the first six months.

a b c

J

F

M

A

M

J

12

15

13

9

8

10

Sort the data in ascending order. Calculate the median. Calculate the median using a graphics calculator. Solution 1 2 3

4 5 6

7

Write the scores in increasing order. Count the total number of scores (n = 6). There is an even number of scores so the median is the average of the 3rd (score 10) and the 4th scores (score 12). The average of 10 and 12 is 11.

a

Select the STAT menu. Enter the data into List1. Press EXE to enter each number.

c

b

8, 9, 10, 12, 13, 15 n=6, 2 2 =3

n +1= 6 +1 2 2 =4 10 + 12 Median = 2 = 11

Select 1VAR to view the summary statistics. The median is MED.

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323

Exercise 11A 1

There is an odd number of scores. Find the median of these numbers. a 3, 9, 10 b 3, 4, 6, 7, 9, 10, 15 c 23, 28, 29, 30, 34, 45, 46, 49, 50 d 1002, 1010, 1100, 1120, 1160 e 2, 2, 2, 2, 2, 2, 2, 9, 9 f 14, 15, 100, 101, 102 g 0, 0, 0, 1, 1, 1, 7, 8, 8, 8, 8 h 2, 7, 9, 10, 10, 14, 18

2

There is an even number of scores. Find the median of these numbers. a 2, 5, 6, 8, 8, 9 b 12, 14, 18, 22 c 20, 20, 20, 21, 22, 24 d 3, 4, 5, 9, 10, 14, 16, 18, 18, 18 e 100, 110, 130, 140 f 1, 1, 1, 1, 3, 3, 3, 5, 5, 5, 6, 6 g 10, 20, 22, 40, 60, 61, 70, 80 h 3, 5, 8, 10, 11, 14, 18, 19

3

The number of senior citizens entering a restaurant in the past 9 hours was 18, 17, 16, 17, 19, 13, 10, 16 and 15. a What is the smallest value? b What is the largest value? c What is the median?

4

Arrange these scores in order and find the median. a 13, 6, 0, 9, 6, 5, 6, 17, 1 b 22, 31, 28, 22, 43, 22 c 9, 15, 9, 3, 6, 9, 13, 10, 7, 9 d 1, 1, 7, 9, 5, 9, 10 e 8, 12, 5, 5, 9, 10, 13, 3, 7 f 3, 0, 1, 2, 11, 9, 7, 7, 5 g 13, 6, 9, 9, 3, 9, 15, 7, 9, 10, 4, 5 h 10, 18, 7, 2, 14, 9, 10, 8

5

Find the missing number in these data sets. a The scores 8, 9, 15, , 20, 27 and 30 have a median of 20. b The scores 1, 1, , 3, 5 and 8 have a median of 2. c The scores 9, 10, 12, 14, , 15, 17, 19 and 19 have a median of 14. d The scores 3, 4, 4, 4, , 7, 8 and 9 have a median of 5.

6

Find the median of each data set. a 23, 25, 29, 23

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b

7, 4, 5, 6, 15, 13, 11, 11, 9

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Development 7

Use the stem-and-leaf plot opposite to answer these questions. a How many scores are there? b What is the lowest score? c What is the median? d Remove 42 from the data. What is the new median?

8

Use the dot plot opposite to answer these questions. a What is the highest score? b What is the lowest score? c Calculate the median. d How many scores would need to be added to make the median 23? e What is the median if 26 is included in the data?

10

19 20 21 22 23 24 25

The table opposite shows the ages of players in the local football team. a What is the age of the oldest player? b What is the age of the youngest player? c What is the range of ages? d What is the median age? The frequency histogram opposite shows the number of jeans sold for each size. a What is the maximum size? b What is the minimum size? c What is the range in sizes? d Calculate the median size. e Construct a frequency table.

Age

Frequency

13

3

14

5

15

5

16

2

7 6 5 4 3 2 1 6

The cumulative frequency histogram and polygon opposite shows the results of a survey. a How many people participated in the survey? b What is the median?

8 10 12 14 16 Size of jeans

30 Cum freq

11

7 8 2 3 5 9 1 2 6 7 9 9 0 1 7 2 7

4 3 2 1

Frequency

9

0 1 2 3 4

25 20 15 10 5 19 20 21 22 23 24 25 26 27 Score

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11.2 Mean and mode Mean 11.2

The mean is a measure of the centre. It is calculated by summing all the scores and dividing by the number of scores. For example, consider the scores 1, 6, 3 and 2. The mean is 1+ 6 + 3+ 2 4

= 3. The mean of a set of data is what most people call the ‘average’. Mean

Mean =

Sum of scores Number of scores

or

x=

∑x n

or

x=

∑ fx ∑f

The mean formula contains symbols or statistical notation that enables complex formulas to be written in a compact form. The meaning of the symbols is as follows: • • • • •

∑ – ‘Sum of’ (Greek capital letter sigma) x – A score or data value x – Mean of a set of scores n – Total number of scores f – Frequency Example 2

Calculating the mean

The number of people who travelled from overseas to attend five meetings in Sydney was 2, 5, 6, 9 and 3. What is the mean of this data?

Solution 1 2 3

Write the formula x = ∑nx . Sum all of the scores and divide by the value of n. There are 5 scores so n = 5. Evaluate.

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∑x n 2 +5+6 + 9+ 3 = 5 =5

x=

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Calculating the mean using the graphics calculator

Example 3

The table below shows the number of sunny days for the first six months.

a b

J

F

M

A

M

J

15

13

12

10

9

8

What is the mean? Answer correct to two decimal places. Calculate the mean using a graphics calculator. Answer correct to two decimal places.

Solution 1

Write the mean formula x = ∑nx .

2

Sum all of the scores and divide by the value of n. There are 6 scores so n = 6.

3

Evaluate. Express answer correct to two decimal places.

4

5 6

7

Select the STAT menu. Enter the data into List1. Press EXE to enter each number.

a

∑x n 15 + 13 + 12 + 10 + 9 + 8 = 6 67 = 6 = 11.1666666 = 11.17

x=

b

Select 1VAR to view the summary statistics. The mean or x is the located at the top.

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Chapter 11 — Summary statistics

Mode The mode is the score that occurs the most. It is the score with the highest frequency. The mode is useful for categorical data which do not allow numerical calculations. For example, the data collected may be a colour. Modes may occur at the beginning or end of a range of values. Therefore, conclusions based only on the mode may be inaccurate. It is common for data to have several modes. For example, if there are two modes then data is referred to as bimodal. When data is grouped into classes, the class that occurs the most is called the modal class.

Mode 1 2

Determine the number of times each score occurs. Mode is the score that occurs the most number of times. If two or more scores occur the same number of times they are both regarded as the mode.

Example 4

Calculating the mode

The table opposite shows the number of sunny days for the first six months. a What is the mode? b Find the mode using a graphics calculator.

J

F

M

A

M

J

15

13

12

10

9

8

Solution 1 2 3 4 5

Each score occurs only once. Mode is each of the scores.

a

Select the STAT menu. Enter the data into List1. Press EXE to enter each number. Select 1VAR to view the summary statistics. The mode is MOD.

b

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Mode is 8, 9, 10, 12, 13 and 15

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Exercise 11B 1

Find the mean of each data set. a 6, 9, 9, 15, 9, 3, 7, 9, 13, 10 c 5, 5, 8, 12, 13, 3, 7, 9, 10 e 6, 5, 6, 17, 13, 6, 0, 9, 1 g 3, 5, 24, 19, 13, 13, 13, 9, 3, 8, 11

b d f h

22, 28, 22, 31, 43, 22 7, 5, 11, 3, 0, 1, 2, 9, 7 39, 35, 39, 41, 47, 49, 44 55, 24, 14, 18, 13, 3, 2, 2, 2, 7, 14

2

Find the mean of each data set. Answer correct to one decimal place. a 13, 14, 15 b 5, 6, 7, 8, 9 c 5, 7, 8, 9, 13, 15 d 6, 8, 11, 13 e 6, 7, 10, 11, 13, 19 f 1, 1, 2, 2, 2, 2, 3, 3, 3, 3 g 9, 9, 10, 10, 10, 11, 11, 11, 11 h 6, 6, 9, 9, 9, 10, 10, 10, 10

3

Twenty people measured their heart rate using a heart-rate monitor. The results were 64, 68, 64, 72, 75, 67, 91, 80, 77, 73, 68, 81, 73, 72, 60, 62, 74, 68, 55 and 62. a What is the sum of these heart rates? b Find the mean heart rate. Answer correct to two decimal places. c Another person with a heart rate of 63 is included in this data. What is the new mean? Answer correct to two decimal places.

4

Find the mode of each data set. a 18, 7, 9, 10, 2, 14, 10 c 13, 3, 2, 2, 7, 14, 55, 24, 14, 18, 2 e 24, 19, 3, 5, 13, 8, 11, 13, 13, 9, 3 g 13, 6, 9, 9, 3, 9, 15, 7, 9, 10

5

b d f h

6, 17, 6, 5, 9, 1, 13, 6, 0 22, 31, 22, 28, 43, 22 13, 3, 7, 9, 5, 5, 8, 12, 10 47, 49, 39, 35, 39, 41, 44

Eleven students were surveyed on the number of hours they used the internet in the past week. Their answers were: 3, 5, 15, 13, 12, 12, 9, 12, 13, 14 and 16. Find the mode number of hours of internet usage.

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Development 6

Nine students were surveyed on the number of hours they slept last night. Their answers were 10, 8, 7, 7, 4, 7, 8, 9 and 11. a Find the mean number of hours slept. b Find the mode number of hours slept. c A tenth student was then surveyed and the mean changed to 7.5. What was the number of hours slept by the tenth student?

7

Create the spreadsheet below.

11B

a b c d

Cell C5 has a formula that multiplies cells A5 to B5. Enter this formula. Enter formulas into cells C6 to C11 to complete the fx column. Enter a formula in cell B12 to sum the frequency column. Enter a formula in cell B13 to calculate the mean.

8 

The mean height of five basketball players at the start of the game is 1.92 m. During the game a player who is 1.80 m tall is injured and replaced by a player who is 1.98 m tall. What is the mean height of the five players now? Answer correct to two decimal places.

9

A score was added to the set of scores: 15, 18, 20, 22, 24 and 26. The new mean is equal to 20. What score was added?

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11.3 The mean from larger data sets Mean from ungrouped data 11.3

To calculate the mean from a frequency table, another column is added: the ‘frequency × score’ or fx column. See the example below. The sum of the fx column (or ∑ fx) gives the sum of all the scores and the sum of the f column (or ∑ f ) gives the number of scores. Mean Mean = or x =

Sum of scores Sum of ffxx column = Number of scores Sum off f column ∑ fx ∑ fx

Calculating the mean from a frequency table

Example 5

Find the mean from the following frequency distribution table. Score (x)

Frequency ( f )

18

1

19

5

20

3

21

7

Frequency × Score ( fx )

Solution 1

Complete the fx column by multiplying the score (x) by the frequency ( f ).

2

Sum the f column (∑f = 16).

3

Sum the fx column (∑ fx = 320).

4 5 6

∑ fx Write the formula x = f . ∑ Substitute the values for ∑fx and ∑ f. Evaluate.

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x

f

fx

18

1

18

19

5

95

20

3

60

21

7

147

16

320

∑ fx ∑f 320 = 16 = 20

x=

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Mean from grouped data To calculate the mean from a grouped frequency table, the class centre is used instead of the score (x). The ‘frequency × score’ (fx) column becomes the ‘frequency × class centre’ column. The same formula for the mean is used. When the mean is calculated from grouped data it is an approximation. Example 6

Finding the mean from a grouped frequency table

The result of a speed camera on Glen Rd is shown below in the grouped frequency table. The data represents the number of cars over the speed limit (km/h).

Class

Class centre (x)

Frequency ( f )

1–5

3

40

6–10

8

25

11–15

13

20

16–20

18

10

21–25

23

5

fx

What is the mean over the speed limit for this data? Solution 1 2 3

4 5 6

Complete the fx column by multiplying the class centre (x) by the frequency (f ). Sum the f column (∑ f = 100). Sum the fx column (∑ fx = 875).

∑ fx Write the formula x = f . ∑ Substitute the values for ∑ fx and ∑ f. Evaluate.

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x

f

fx

3

40

120

8

25

200

13

20

260

18

10

180

23

5

115

100

875

∑ fx ∑f 875 = 100 = 8.75

x=

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Exercise 11C 1

Complete the following tables and calculate the mean, correct to two decimal places. a

c

2

Freq ( f )

4

b

Score (x)

Freq ( f )

3

15

2

5

5

16

3

6

2

17

4

7

2

18

2

8

3

19

3

Score (x)

Freq ( f )

22

fx

d

fx

Score (x)

Freq ( f )

3

2

5

23

3

3

6

24

4

4

7

25

2

5

6

26

5

6

5

fx

fx

Find the mean of the data in the following table. Answer correct to two decimal places. a

b

3

Score (x)

Score

11

12

13

14

15

16

17

18

Frequency

2

1

3

5

4

6

8

4

Score

2

3

4

5

6

7

8

9

Frequency

6

9

12

10

15

12

14

15

Dylan selected 28 students at random and asked each of them how many text messages they sent from a mobile phone within the last day. The results are summarised in the opposite table. a Copy the table and insert an fx column. b Calculate the mean number of text messages. Answer correct to one decimal place.

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Text messages Frequency 0

4

1

3

2

5

3

5

4

7

5

4

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4

Find the mean of the data in the following table. Answer correct to two decimal places. Class

0–9

10–19

20–29

30–39

40–49

50–59

60–69

Class centre

4.5

14.5

24.5

34.5

44.5

54.5

64.5

6

8

4

7

2

4

5

Frequency 5

The grouped frequency table shows the number of passengers carried by an airliner. Class

a b c

6

333

Class centre (x)

Freq. (f )

15–19

20

20–24

29

25–29

22

30–34

18

35–39

32

40–44

18

45–49

23

f×x

Copy and complete the table by finding the class centre and fx column. How many passengers were carried? Find the mean of this data. Answer correct to the nearest whole number.

The grouped frequency table shows the weights of randomly selected packets of sugar. All the packets were labelled as 1 kg. Class

a b c

Class centre (x)

Freq. ( f )

980–984

2

985–989

16

990–994

130

995–999

352

1000–1004

353

1005–1009

128

1010–1014

19

1015–1019

1

f×x

Copy and complete the table by finding the class centre and the fx column. How many packets of sugar were selected? Find the mean of this data. Answer correct to one decimal place.

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Development 7

The heights in centimetres of 20 boys are recorded below. Height range

Class centre (x)

Frequency ( f )

140–149

2

150–159

8

f×x

6 170–179 180–189 a b c d e f

8 

1

What is the class for the middle height range? What is the frequency for the 170–179 class? Copy and complete the table by finding the class centre and the fx column. Find the mean of this data to the nearest whole number. What percentage of students is in the 150–159 class? What percentage of students is in the 180–189 class?

The number of breakdown calls received by the NRMA road service are shown below. 82

69

78

83

75

89

82

89

68

90

80

79

83

68

79

91

82

79

70

90

75

70

90

74

74

75

90

80

80

85

a b c d e f

g

Decide on appropriate classes for a grouped frequency table. Calculate the class centres for these classes. Construct a grouped frequency table using these class intervals. How many pieces of data have been collected? Find the mean of this data. Answer correct to one decimal place. Which class had the largest number of breakdown calls received by the NRMA? Which class had the smallest number of breakdown calls received by the NRMA?

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11.4 Standard deviation The standard deviation is a measure of the spread of data about the mean. It is an average of the squared deviations of each score from the mean. Standard deviation is calculated from the formula shown below. 11.4

σ=

∑ ( x − x )2 where σ is the standard deviation n

The calculations involved with this formula are complex and time consuming. Fortunately, the calculator is able to perform these calculations easily. Standard deviation Standard deviation measures the spread of data about the mean. σn– Population standard deviation.

Calculating the population standard deviation

Example 7

The results of six students in a Mathematics test are shown below. Test A a b

59

65

70

62

71

66

Find the population standard deviation of Test A, correct to two decimal places. Calculate the population standard deviation using the graphics calculator. Solution 1 2 3 4 5 6 7

Enter the statistics mode of the calculator. Clear the contents of the memory. Enter each score into the calculator. Select the σn key to view the result.

a

Select the STAT menu. Enter scores for Test A into List1. Press EXE to enter each number. Select CALC then 1VAR to view the statistics for Test A. The population standard deviation is xσn.

b

Population standard deviation is σn = 4.19

σn = 4.19

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Sample standard deviation There are slightly different methods for calculating the standard deviation. The population standard deviation (σn) is a better measure when we have all of the data or the entire population. However, when a sample is taken from a large population the sample standard deviation (σn−1) is a better measure. Sample standard deviation σn−1 – Sample standard deviation.

Example 8

Calculating the sample standard deviation

Hannah decorates and sells cupcakes at the local market. The frequency table below shows the number of cakes sold during the first three hours of the day. Score (x) Frequency ( f ) 1

13

2

19

3

14

4

10

5

9

Find the sample standard deviation. Answer correct to two decimal places. Check the result using the graphics calculator. Solution 1 2 3 4 5 6 7 8

Enter the statistics mode of the calculator. Clear the contents of the memory. Enter the data into the calculator by using score × frequency. Type 1 × 13 Data , 2 × 19 Data , … Select the σn−1 key to view the result. Select the STAT menu.

Sample standard deviation is σn−1 = 1.33

Enter scores into List1 and the frequencies into List2. Select SET to make list2 frequencies. Select CALC then 1VAR to view the statistics. The sample standard deviation is xσn−1.

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337

Exercise 11D 1

Find the population standard deviation of each data set, correct to one decimal place. a 4, 6, 8, 10, 12 b 23, 24, 25, 26, 27, 28, 29 c 0, 4, 8, 12, 16, 20, 24 d 3, 3, 5, 7, 9, 9 e 6, 6, 6, 6, 6, 6, 6, 6 f 124, 135, 145, 132, 130, 145, 156 g 4.3, 5.6, 3.4, 7.8, 2.3, 9.1 h 112, 4, 0, 100, 7, 98, 1

2

An amateur athletics championship was recently conducted. The following distances in metres were recorded in a long-jump championship. 4.9, 3.9, 4.0, 4.3, 4.6, 4.7, 4.3, 4.4, 4.1, 4.9, 3.9, 4.7, 4.8, 5.0 a Calculate the population standard deviation. Answer correct to the nearest hundredth. b Two athletes did not have their results recorded. Add 3.7 and 5.1 to the data. What is the new population standard deviation? Answer correct to the nearest hundredth.

3

Find the population standard deviation in each of the following tables. Answer correct to one decimal place. a

b

4

Score

4

5

6

7

8

9

10

11

Frequency

1

3

5

7

7

5

3

1

Score

12

13

14

15

16

17

18

19

Frequency

6

6

5

5

4

4

3

3

The tables below show the average number of cloudy days per month for two cities. Calculate the population standard deviation, correct to two decimal places. a

Score

Frequency

16

b

Score

Frequency

2

1

12

17

2

2

3

18

5

3

5

19

3

4

12

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Preliminary Mathematics General

Development 5

The results for Andrew and Kayla in their tests this term are listed below. Andrew

17

18

15

13

10

8

13

14

16

16

Kayla

16

10

15

11

3

15

9

7

16

19

a b c

What are the mean and population standard deviation for Andrew’s results? What are the mean and population standard deviation for Kayla’s results? Which student had the more consistent results throughout the term? Give a reason.

6

Find the sample standard deviation of each data set, correct to one decimal place. a 1, 3, 5, 7, 9, 11, 13 b 20, 21, 22, 31, 28, 22, 43, 22 c 13, 10, 7, 9, 9, 15, 9, 3, 6, 9 d 100, 100, 7, 9, 5, 9, 10 e 1.0, 1.3, 3.4, 7.5, 8.9 f 11, 9, 7, 7, 5, 3, 0, 1, 2

7 

A sample of the ratings for a television news program was 34, 28, 29, 36, 22, 26, 30, 28 and 31. All ratings are a percentage of the total audience. a What is the sample standard deviation, correct to two decimal places? b The best and the worst ratings were not included in this data. The best rating was 42 and the worst was 18. What is the new sample standard deviation? (Answer correct to two decimal places.)

8

The age of 30 customers entering a shopping centre is recorded below. Age range

Class centre (x)

Frequency ( f )

0–19

7

20–39

8

40–59

9

60–79

6

Use your calculator to estimate the sample standard deviation. Answer correct to one decimal place.

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11.5 Comparison of summary statistics Mean, median or mode 11.5

The selection and use of the appropriate statistic (mean, median, or mode) depends on the nature of the data and the relative merits of each measure. For example, if the data contains one or two extreme scores then the value of the mean will greatly increase or decrease. Advantages

Easy to calculate. Easy to understand. Depends on every score. Varies least from sample to sample.

Disadvantages

Distorted by outliers. Not suitable for categorical data.

Advantages

Easy to understand. Not affected by outliers.

Mean

Median Disadvantages

Not suitable for categorical data. May not be central. Varies more than the mean in a sample.

Advantages

Easy to determine. Most typical value. Not affected by outliers. Suitable for categorical data.

Disadvantages

May be more than one mode. May not be central. Often varies with sample.

Mode

Mean, median or mode Selection and use of the mean, median or mode depends on the nature of the data and the relative merits of each measure.

Outliers An outlier is a score that is separated from the majority of the data. For example, the data 0, 0, 0, 1, 1, 2, 45 has an outlier of 45. In small sets of data, the presence of an outlier will have a large effect on the mean, a smaller effect on the median and usually no effect on the mode.

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Example 9

Comparing the mean, median and mode

The data below is the number of soft drink cans recycled in each of the past nine days. 8, 3, 4, 8, 8, 8, 4, 5, 8 a Find the mean, median and mode for this data. Answer correct to one decimal place. b Which is the better measure for the centre of this data? Explain your answer.

Solution 1 2 3

x Write the mean formula x = ∑n .

a

Sum all the scores and divide by the value of n. There are 9 scores so n = 9. Evaluate. Write correct to one decimal place.

∑x n 56 = 9 = 6.2

Mean x =

5

Median is the middle score. Write the scores in increasing order.

Median 3, 4, 4, 5, 8, 8, 8, 8, 8

6

Count the total number of scores (n = 9).

n +1 = 9 +1 2 2 =5

7

There is an odd number of scores so the median is the 5th (or score 8).

Median is 8.

8

Mode is the score that occurs the most number of times (or score 8).

Mode is 8.

9

Look at the data. The median and mode are both 8 and it is not the centre of the data.

4

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b

The mean is central and typical of the data. The median and the mode are the end score (8). Even though the median and mode are easy to calculate, they are not a good measure.

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341

Exercise 11E 1

2

The students in a Year 11 class were given a short quiz. The test was marked out of ten and the results are listed opposite. Find the following measures, correct to one decimal place. a Mean b Median c Mode d Which is the better measure for the centre of this data? Explain your answer.

Score

Frequency

7

6

8

5

9

3

10

4

The area of each suburb of a city is listed below. 3.0 27.5 19.5 40.2 2.3 2.3 3.6 4.3 3.4 11.5 28.2 7.4 a Find the mean, median and mode areas. b Which is the better measure for the centre of this data? Explain your answer.

3

9.0

A local community were concerned about the number of people rescued from one of their beaches. The number of people rescued in past 13 days is recorded below. 0 a b

0

6

1

0

3

0

2

0

3

0

1

0

Find the mean, median and mode of this data. Which is the better measure for the centre of this data? Explain your answer.

4

The number of accidents in a particular workplace in one month was: 0, 3, 0, 0, 0, 6, 1, 0, 4, 3, 0, 1, 1 and 3. a Find the mean, median and mode. Answer correct to the nearest whole number. b Which is the best measure to use when summarising this data? Give a reason.

5

Consider the following set of scores: 12, 15, 16, 16, 18, 18, 19, 20, 20, 60. a Calculate the mean and median of the set of scores. b What is the effect on the mean and the median of removing the outlier?

6

A hospitality class has eight students. The class sat for a test and the results were: 99, 96, 92, 95, 96, 12, 96 and 95. a Find the mean, median and mode scores. Answer correct to the nearest whole number. b Molly scored 92. She told her father that her result was above the average. Do you agree with Molly’s statement? Give a reason for your answer.

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Development 7

Senior students were surveyed and the results of two samples are shown below. Sample A    Score 1 2 3 4 5 6 7

Sample B  

a

3

9

8

7

7

5

1

Score

1

2

3

4

5

6

7

Frequency

6

8

9

7

5

3

2

For sample A find, correct to one decimal place, the: i Median ii Mean iii Mode iv Standard deviation



For sample B find, correct to one decimal place, the: i Median ii Mean iii Mode iv Standard deviation

c

What are the similarities and differences between the two samples?

b

8

Frequency

Create the spreadsheet below.

11E

a b c d

9 

13.5

The formulas for cells D4:D11 are shown above. Enter these formulas. Compare the mean, median and mode for this data. Which measure does not provide an accurate measure of the centre? Explain your answer. Modify the data so that the mean, median and mode are the same value. Modify the data so that the mode is smaller than the mean and the median.

Real estate agents and the media use the median as a measure to compare house prices. Why is the median a better measure than the mean or the mode? 14.9

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Chapter 11 — Summary statistics

Median

Study guide 11

The median is the middle score or value. To find the median: 1 Arrange all the scores in increasing order. 2 Count the total number of scores (n). 3 If n is odd then the median is the n+1 score. 2

4

Mean

If n is even then the median is average of the n + 1 score. 2

n 2

and the

Sum of scores Number of scores ∑ x or x = ∑ fx x= ∑f n

Mean =

∑ – ‘Sum of’. x – A score or data value. x– – Mean of a set of scores. n – Total number of scores. f – Frequency. Mode

• •

Mode is the score that occurs the most number of times. Score with the highest frequency.

Standard deviation

Measures the spread of data about the mean. σn – Population standard deviation. σn−1– Sample standard deviation.

Comparison of summary statistics

Selection and use of the appropriate statistic (mean, median or mode) depends on the nature of the data and the relative merits of each measure.

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Review

Chapter summary – Summary statistics

343

Review

344

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

The scores 6, 7, 8, 9, , 12, 13, 15, 15 and 16 have a median of 12. What is the missing number? A 9 B 10 C 11 D 12

2

What is the median of 12, 20, 9, 4, 16, 11 and 13? A 4 B 11 C 12 D 13

3

What is the mean of the data in the table opposite? A 45.00 B 46.35 C 46.50 D 47.00

4

What is the median of the data in the table opposite? A 46.00 B 46.50 C 47.00 D 47.50

5

What is the mode of 7, 15, 5, 9, 10, 11, 12, 7 and 14? A 5 B 7 C 10 D 14

6

What is the class centre of 30–35? A 32 B 32.5 C 33

D

Score

Frequency

44

1

45

8

46

1

47

5

48

3

49

2

33.5

7

The mean of five scores is 4. What is the missing score if four of the scores are 2, 5, 5 and 7? A 1 B 2 C 4 D 5

8

What is the mode of the data in the table opposite? A 3 B 4 C 5 D 7

9

10

fx

What is the population standard deviation of the data in the table opposite? A 1.49 B 1.52 C 1.58 D 1.61

Score Frequency 1

3

2

4

3

2

4

6

5

7

6

3

What is the population standard deviation for 6, 10, 14, 18, 22, 26 and 30? A 6 B 8 C 10 D 12

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Review

Sample HSC – Short-answer questions 1

Arrange these scores in order and find the median. a 18, 11, 5, 14, 11, 10, 11, 22, 6 b 37, 46, 43, 37, 58, 37

2

Find the mean of each data set. a 11, 13, 9, 9, 12, 16, 17, 7, 14 b 14, 11, 12, 13, 18, 16, 22, 20, 18

3

Find the mode of each data set. a 14, 6, 8, 16, 16, 12, 27, 22, 16, 11, 6 b 35, 27, 30, 34, 25, 29, 31, 27, 32

4

The number of major road accidents on the Pacific Highway in the past 12 months is: 4, 8, 7, 9, 14, 4, 6, 5, 8, 4, 9 and 10. a What is the median? b What is the mode? c What is the mean?

5

A score of 20 is added to the table opposite. Calculate the value of the following measures, correct to the one decimal place. a Mean b Median c Mode

Score

Frequency

20

1

21

4

22

5

23

2

6

The number of teenagers who attended on each day of a young leaders’ conference was 4, 7, 8, 11 and 5. What is the mean of this data?

7

Find the population standard deviation of each data set. Answer correct to one decimal place. a 24, 30, 12, 18, 36 b 21, 22, 23, 18, 19, 20, 24

8

Find the population standard deviation in the following table. Answer correct to two decimal places. Score

20

21

22

23

24

25

26

27

Frequency

6

8

9

12

11

8

7

5

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Review

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9

Find the sample standard deviation of each data set. Answer correct to one decimal place. a 24, 18, 12, 22, 19, 16, 18, 18, 15, 18 b 10, 18, 50, 18, 50, 14, 20

10

Ten people were surveyed on the number of hours they watched television yesterday. Their answers were: 2, 5, 3, 0, 1, 5, 0, 0, 3 and 10. a What is the median? b What is the mean? c Find the mode. d Calculate the sample standard deviation. Answer correct to one decimal place. e Calculate the population standard deviation. Answer correct to one decimal place. f Another person was surveyed and the mean changed to 3. How many hours did this person watch television?

11

The numbers 3, 6, x, 9 and 11 have a mean of 8. What is the value of x?

12

The frequency table opposite shows the results of a test out of 24. a Copy and complete the fx column. b Calculate the class centres. c What is the modal class? d Find the mean of this data to the nearest whole number.

Class

Class centre

Freq. (f)

10–12

9

13–15

3

16–18

4

19–21

1

22–24

1

13

A science class has 20 students. The results of the first assessment task were: 67, 88, 69, 90, 75, 78, 81, 63, 90, 79, 89, 90, 80, 77, 32, 70, 69, 85, 91 and 24. a Find the mean, median and mode scores. Answer correct to one decimal place. b Which is the better measure for the centre of the data? Explain your answer.

14

Sales for a new book were: 2, 5, 0, 2, 0, 11, 3, 2, 6, 0, 0, 3, 3 and 5. Which is the best measure to summarise this data – mean, median or mode? Give a reason.

fx

Challenge questions 11

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C H A P T E R

12

Similarity and right-angled triangles Syllabus topic — MM3 Similarity and right-angled triangles Find scale factors of similar figures Solve problems involving similar figures Define the trigonometric ratios – sine, cosine and tangent Find unknown sides with trigonometry Find unknown angles using trigonometry Using trigonometry to solve a variety of problems

12.1 Similar figures and scale factors The pictures of the three pieces of cake are similar. Similar figures are exactly the same shape but they are different sizes. 12.1

When we enlarge or reduce a shape by a scale factor, the original and the image are similar. Similar shapes have corresponding angles equal and corresponding sides in same ratio or proportion. 20 mm A

40 mm B

5 mm

10 mm

For example, the above rectangles are similar. All the angles are 90°. The corresponding sides are in the same ratio 10 = 40 = 2 . The measurements in rectangle B are twice the 5 20 measurements in rectangle A. Rectangle B has been enlarged by a scale factor of 2.

(

)

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Similar figures • • • •

Similar figures are exactly the same shape but are a different size. Corresponding (or matching) angles of similar figures are equal. Corresponding (or matching) sides of similar figures are in the same ratio. Scale factor is the amount the first shape is enlarged or reduced to get the second shape.

Example 1

Calculating the scale factor

What is the scale factor for these two similar rectangles?

9

12

3

A

B

4

Solution 1

Look carefully at the similar figures.

2

Match the corresponding sides. (9 matches with 3 and 12 matches with 4.) Write the matching sides as a fraction (measurement in rectangle B divided by the matching measurement in rectangle A). Simplify the fraction by dividing both terms by the same number. This fraction is the scale factor.

3

4

Example 2

Rectangle B is smaller than rectangle A and is rotated. 3 4 or 9 12 1 = (or 1:3) 3

Scalee ffactor =

Rectangle B is rectangle A.

1 3

the size of

Using a scale factor

What is the length of the unknown side in the following pair of similar triangles?

48

8 10

x

Solution 1 2

3 4

Match the corresponding sides. (8 matches with 48 as it is opposite the same angle.) Write the matching sides as a fraction (second shape to the first shape). This fraction is the scale factor. Match the corresponding side for x (side marked with a 10 as it is opposite the same angle). Calculate x by multiplying 10 by 6 the scale factor (or 6).

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48 8 = 6 (or (or 6:1 6 :1 )

Scalee ffactor = x = 10 × 6 = 60

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Chapter 12 — Similarity and right-angled triangles

Exercise 12A 1

Match the following pairs of similar shapes. 124° 56° A 56° 124°

4

C

B

6

7

135°

71°

68°

4

E 99°

F

54°

D 56°

56°

101° G H

I

67°

101°

3 8

2

101° 67°

4

2

54°

99°

6

3 56°

68°

101°

124°

3.5

1 124°

71°

56°

5

56°

135°

56°

9

2

7

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2

What is the scale factor for the following pairs of similar figures? a    b 3

12

4

8

c



4

15

d

1

5 4

12

2 8

6

2 124° 56° 56° 124° 4

e

124° 56°

56° 3 124°



f

63°

72°

9

3

4 45°

72°

63°

45° 12

3

Use the scale factor to find the length of the unknown side in the following pairs of similar figures. a b 3 6 2

6

1

a

b

c

101°

10

101°

d c

67° 101° 10

6

68° 67°

101°

9

15

d

56°

68°

56°

56°

9 56°

45 4

Shapes A and B shown opposite are regular pentagons. a Why are these shapes similar? b What is the scale factor for these similar figures?

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6 A

4 B

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351

Development 5

Shape A and B consist of a rectangle and a triangle. Shape B is an enlargement of shape A. a What is the ratio of the: 3 i lengths of the rectangle B to A? 1.5 ii breadths of the rectangle B to A? 2 B b What is the scale factor? A 1 5

2.5

 6

A data projector is used to display a computer image measuring 12 cm by 15 cm onto a screen. The scale factor used by the data projector is 1:9. What are the dimensions of the screen?

7 

Consider these three triangles and write true or false to the following statements. 55° A

C

B

25° a

8

Diagrams not to scale

65°

Triangle A is similar to triangle B.

b

Triangle A is similar to triangle C.

Consider these two triangles. 18 cm

m 2 cm 6 cm a c

9

Why are these triangles similar? Find the value of m.

n

15 cm b d

What is the scale factor? Find the value of n.

Let BC = a, AC = b, BD = x, AD = y and DC = z. a Why is ∆ ADC similar to ∆ ABC? b Write an expression for the three corresponding sides of the similar triangles.

A

D

b

x

z C

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y

a Cambridge University Press

B

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Preliminary Mathematics General

12.2 Problems involving similar figures The lengths of the corresponding (or matching) sides in similar figures are in the same ratio or proportion. This property is used to calculate the length of an unknown side. 12.2

Finding an unknown side in similar figures 1 2

3 4

Determine the corresponding or matching sides in the similar figures. Write an equation using two fractions formed from the matching sides – a measurement from the second shape divided by a matching measurement from the first shape. This is the scale factor. Solve the equation. Check that the answer is reasonable and units are correct.

Example 3

Finding an unknown side in similar figures

What is the length of the unknown sides in the following pair of similar triangles? 1.5

x

y 1.875

20

12

Solution

Determine the matching sides in the similar triangles (x and 1.875, 12 and 1.5, 20 and y). Write an equation using matching sides involving x. Use x (second shape) and 1.875 (first shape) equal to 12 (second shape) and 1.5 (first shape). Solve the equation.

x 12  Second shape  = 1.875 1.5  First shape  12 x= × 1.875 1.5 = 15

4

Write an equation using matching sides involving y. Use 20 (second shape) and y (first shape) equal to 12 (second shape) and 1.5 (first shape).

5

Solve the equation.

20 12  Second shape  = y 1.5  First shape  12 y = 2200 × 1.5 20 × 1.5 y= 12 = 2.5

6

Check that the answers are reasonable.

x = 15 and y = 2.5

1 2

3

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Chapter 12 — Similarity and right-angled triangles

Similar figures are used to solve problems that require the length of an object. For example, we can calculate the height of a tree without physically measuring the height. A similar figure is drawn using a metre rule and the length of its shadow is measured (see example below). This is a very useful concept.

Solving a worded problem using similar figures 1 2 3 4 5 6

Read the question and underline the key terms. Draw similar figures and label the information from the question. Use a pronumeral (x) to represent an unknown side. Write an equation using two fractions formed from matching sides. Solve the equation. Check that the answer is reasonable and units are correct.

Example 4

Solving a problem involving similar figures

A tree casts a 5 m long shadow on the ground. At the same time a one metre rule casts a shadow with a length of 80 cm. What is the height of the tree? Answer in metres correct to two decimal places.

x

1m 5m

80 cm

Solution

Divide 80 cm by 100 to convert it to metres. Determine the matching sides in the similar triangles (1 and x, 0.8 and 5). Write an equation using matching sides involving x. Use 1 (second shape) and x (first shape) equal to 0.8 (second shape) and 5 (first shape). Solve the equation. Write the answer correct to two decimal places. Write the answer in words.

Let x be the height of the tree. 80 cm = 0.8 m

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1 2 3

4 5 6

1 0.8  Second shape  = x 5  First shape  0.8 x = 5 5 x= 0.8 = 6.25 25 m The height of the tree is 6.25 m.

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Preliminary Mathematics General

Exercise 12B 1

Find the length of the pronumeral for the following pairs of similar triangles. All measurements are in centimetres. a

b

x

4

1.5

9

8

x

12 6

c

x

d

7

5

21

6

7

10 x

2

Find the length of the unknown sides for the following pairs of similar triangles. All measurements are in centimetres. a

b

a

3

8

5

15

b

y

16

10

24 12

x

c

9 p

12 6

d

d

7 4

10

c

28

20

q

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Chapter 12 — Similarity and right-angled triangles

3

A tree casts a shadow 3.5 m long. At the same time a one metre ruler casts a shadow 0.5 m long. What is the height of the tree? Answer correct to one decimal place. 1m 0.5 m

3.5 m 4

A building casts a shadow 9 m long. At the same time a one metre ruler casts a shadow 0.75 m long. What is the height of the building? Answer correct to the nearest metre. 1m 0.75 m

9m

5

A stick 2 m high throws a shadow 1.5 m long. At the same time a tower throws a shadow 30 m long. How high is the tower? Answer correct to one decimal place.

Tower Stick

6

David is 1.8 m in height. When he is standing out in the sun his shadow is 2.4 m long. At the same time a block of units casts a shadow of 18 m. How tall is the block of units? Answer correct to the nearest metre.

7

Jessica found that her shadow was 3 m long when the shadow of a flagpole was 9 m long. If Jessica’s height is 1.5 m, what is the height of the flagpole? Answer to the nearest metre.

8

9

Flagpole Jessica

A 5.4 m high pole casts a shadow of 3.6 m in length. At the same time, the shadow of a building falls exactly over the pole and its shadow. The shadow cast by the building measures 14.4 m. How high is the building? Answer correct to one decimal place. A fence is 1.5 m in height and has a shadow of length 2.1 m. At the same time the shadow thrown by a light pole is 3.6 m. How high is the light pole? Answer correct to one decimal place.

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Pole

Light pole Fence

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Preliminary Mathematics General

Development 10

Find the value of the pronumerals in the following diagrams. a

12

8 x 3

6 y

b

7 3 9 z 11

Lucas and his younger brother Nathan are standing side by side. Nathan is 1.4 m tall and casts a shadow 3.5 m long. How tall is Lucas if his shadow is 5 m long? Answer correct to one decimal place.

12

Wollongong’s lighthouse casts a shadow of length 15 m. At the same time a one-metre beach umbrella casts a shadow whose length is 1.25 m. a What is the height of the lighthouse? Answer correct to nearest metre. b A nearby wall casts a shadow 5 m long. Calculate the height of this wall to the nearest metre.

13

A tree and a 1 m vertical stick cast their shadows at a particular time of the day. The shadow of the tree is 32 m and the shadow of the vertical stick is 4 m. a Draw two triangles to represent the above information. b Give a reason why the two triangles are similar. c Find the height of the tree correct to the nearest metre.

14

A 3.5 m ladder has a support 80 cm long placed 1.5 m from the top of the ladder. How far apart are the feet of the ladder? Answer in centimetres correct to the nearest whole number.

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80 cm

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357

12.3 Scale drawings A scale drawing is a drawing that represents the actual object. The scale factor of a scale drawing 12.3 is the ratio of the size of the drawing to the actual size of the object. For example, a map is a scale drawing. It is not the same size as the area it represents. The measurements have been reduced to make the map a convenient size. The scale of a drawing may be expressed with or without units. For example, a scale of 1 cm to 1 m means 1 cm on the scale drawing represents 1 m on the actual object. Alternatively, a scale of 1 : 100 means the actual distance is 100 times the length of 1 unit on the scale drawing.

Scale drawing Scale of a drawing = Drawing length : Actual length Scale is expressed in two ways: • Using units such as 1 cm to 1 m (or 1 cm = 1 m). • No units such as 1 : 100.

Example 5

Using a scale

A scale drawing has a scale of 1 : 50. a Find the actual length if the drawing length is 30 mm. Answer to the nearest centimetre. b Find the drawing length if the actual length is 4.5 m. Answer to the nearest millimetre. Solution 1 2 3 4

Multiply the drawing length by 50 to determine the actual length. Divide by 10 to change millimetres to centimetres. Divide the actual length by 50 to determine the drawing length. Multiply by 1000 to change metres to millimetres.

a

Actual length = 30 × 50 mm = 1500 mm = 150 cm

b

Drawing length = 4.5 ÷ 50 m = 0.09 m = 90 mm

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Preliminary Mathematics General

Exercise 12C 1

A scale drawing has a scale of 1 : 100. What is the actual length of these drawing lengths? Express your answer in metres. a 2 cm b 10 mm c 3.4 cm d 28 mm e 8.5 cm f 49 mm

2

A scale drawing has a scale of 1 : 25 000. What is the drawing length of these actual lengths? Express your answer in millimetres. a 2 km b 750 m c 4000 cm d 3.5 km e 50 000 mm f 1375 m

3

Express each of the following scales as a ratio in the form 1 : x. a 1 cm to 2 cm b 1 mm to 5 cm c 1 cm to 3 km

4

The scale on a map is 1 : 1000. Calculate the actual distances if these are the distances on the map. Express your answer in metres. a Road 20 cm b Shops 10 cm c Pathway 5 cm d Parking area 10 mm e Bridge 34 mm f Park 80 mm

5

The scale on a map is given as 1 cm = 15 km. What is the actual distance if the distance on the map is: a 2.5 cm? b 45 cm?

Woy Woy Palm Beach Penrith Parramatta

Mona Vale

SYDNEY Coogee

6

The scale on a map is 1 : 5000. Calculate the map distances if these are actual distances. Express your answer in millimetres. a 50 m b 80 m c 100 m d 120 m e 150 m f 240 m

7

The scale on a map is given as 1 mm = 50 m. If the distance between two points is 350 m, what is the map distance between these points?

8

A scale drawing has a scale of 1 : 75. Find the: a actual length if the drawing length is 15 mm. b drawing length if the actual length is 3 m.

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359

Development 9

The Parkes radio telescope dish has a diameter of 64 metres. The image opposite uses a photograph of the dish. a Determine a scale for the image. b Estimate the height of the top of the antenna above the ground.

10

The scale of a model is 2:150. Calculate the model lengths if these are actual lengths. Express your answer in millimetres. a 75 cm b 180 cm c 300 cm d 45 m e 6 m f 36 m

11

A scale drawing of the space shuttle is shown opposite. The actual length of the space shuttle is 47 metres. a What is the scale factor? b Calculate the length of the wing span. c Calculate the width of the shuttle. d What is the length of the nose of the shuttle?

12

The total length of the Sydney Harbour Bridge is 1150 metres. A scale model is built for a coffee table of length 1.2 metres using the picture below.

a b c d

What scale would be suitable? What is the maximum height of the bridge if the scale model has a height of 20 cm? Estimate the height of the bridge pillars. Estimate the length of the Sydney Opera House.

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Preliminary Mathematics General

12.4 Trigonometric ratios Trigonometric ratios are defined using the sides of a right-angled triangle. The hypotenuse is opposite the right angle, the opposite side is opposite the angle θ and the adjacent side is the remaining side. 12.4

Naming sides in a triangle hypotenuse

opposite

θ adjacent

The opposite and adjacent sides are located in relation to the position of angle θ. If θ was in the other angle, the sides would swap their labels. The letter θ is the Greek letter theta. It is commonly used to label an angle. Example 6

Naming the sides of a right-angled triangle

What are the values of the hypotenuse, the opposite side and the adjacent side in the triangle shown?

θ 3

4

Solution 1 2 3

5

Hypotenuse is opposite the right angle. Opposite side is opposite the angle θ. Adjacent side is beside the angle θ, but not the hypotenuse.

Hypotenuse is 5 (h = 5) Opposite side is 4 (o = 5) Adjacent side is 3 (a = 5)

The trigonometric ratios The trigonometric ratios sin θ, cos θ and tan θ are defined using the sides of a right-angled triangle. sin θ =

opposite hypotenuse

cos θ =

adjacent hypotenuse

tan θ =

opposite adjacent

sin θ =

o (SOH) h

cos θ =

a (CAH) h

tan θ =

o (TOA) a

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Trigonometric ratios The mnemonic ‘SOH CAH TOA’ is pronounced as a single word. SOH: Sine-Opposite-Hypotenuse CAH: Cosine-Adjacent-Hypotenuse TOA: Tangent-Opposite-Adjacent The order of the letters matches the ratio of the sides.

The meaning of the trigonometric ratios Consider the three triangles drawn below.

6 2 30°

4 1

3

2 30°

30°

The three triangles drawn above show the ratio of the opposite side to the hypotenuse as 0.5 1 2 3  2 , 4 or 6  . This is called the sine ratio. All right-angled triangles with an angle of 30° have a sine ratio of 0.5. If the angle is not 30° the ratio will be different, but any two right-angled triangles with the same angle will have the same value for their sine ratio. Similarly, the three triangles drawn below show the ratio of the opposite side to the adjacent 1 2 3 side as 1  , or  . This is called the tangent ratio. All right-angled triangles with an angle 3 1 2 of 45° have a tangent ratio of 1.

3 2 1 45° 1

45°

45° 2

3

The ratio of the opposite side to the hypotenuse (sine ratio), the ratio of the adjacent side to the hypotenuse (cosine ratio) and the ratio of the opposite side to the adjacent side (tangent ratio) will always be constant irrespective of the size of the right-angled triangle.

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Preliminary Mathematics General

Example 7

Finding the trigonometric ratio

Find the sine, cosine and tangent ratios for angle θ in the triangle shown.

15 θ 8

17 Solution 1 2 3

4 5

6 7

Name the sides of the right-angled triangle. Write the sine ratio (SOH). Substitute the values for the opposite side and the hypotenuse. Write the cosine ratio (CAH). Substitute the values for the adjacent side and the hypotenuse. Write the tangent ratio (TOA). Substitute the values for the adjacent side and the opposite side.

Example 8

o h 8 = 17 a cosθ = h 15 = 17 o tan θ = a 8 = 15 sin θ =

Finding a trigonometric ratio

Find sin θ in simplest form given tan θ =

6 . 8

Solution 1 2 3 4 5

6 7 8

Draw a triangle and label the opposite and adjacent sides. Find the hypotenuse using Pythagoras’ θ theorem. 8 Substitute the length of the sides into Pythagoras’ theorem. Take the square root to find the hypotenuse (h). h2 = 6 2 + 82 Evaluate. h = 6 2 + 82 = 10 o sin θ = Write the sine ratio (SOH). h Substitute the values for the opposite side and 6 the hypotenuse. = 10 Simplify the ratio. 3 = 5

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6

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Chapter 12 — Similarity and right-angled triangles

363

Exercise 12D 1

State the values of the hypotenuse, opposite side and adjacent side in each triangle. a

b

8 10

d

θ

θ

c

13

θ

θ

6

3

12

5 e

39

4

5 f

9

θ

30

18

36

15

12

15

24

θ

2

State the values of the hypotenuse, opposite side and adjacent side in each triangle. a

b

x

c

c θ

y

z

f

e a

θ d

b

θ

3

Write the ratios for sin θ, cos θ and tan θ for each triangle in question 1.

4

Write the ratios for sin θ, cos θ and tan θ for each triangle in question 2.

5

Name the trigonometric ratio represented by the following fractions. 26

a

θ

10 24

i

ii

iii

10 26 24 26 10 24

b

12

θ 20 16

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i ii

iii

12 20 16 12 16 20

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364

Preliminary Mathematics General

Development 6

Find the sine, cosine and tangent ratios in simplest form for each angle. a

angle θ ii angle φ i

b

11 φ



angle θ ii angle φ i

32

θ 40

60

φ

61 θ c

angle θ ii angle φ i

45

d

θ



51

7

φ

angle θ ii angle φ i

θ

r

24

24

p

φ q

Find the sine and cosine ratios in simplest form for angle A and B for each triangle. a b A 20 C A

34

16

30

C

8 

9

15 B

25

B

Draw a right-angled triangle for each of the following trigonometric ratios and i find the length of the third side. ii find the other two trigonometric ratios in simplest form. a

tan θ =

3 4

c

cosθ =

7 25

b

sin θ =

8 10

Draw the following two triangles using a protractor and a ruler.

3

2 60° a b c

60°

Measure the length of the hypotenuse, adjacent and opposite sides in each triangle. What is the value of the cosine ratio for 60° in both triangles? What is the value of the sine ratio for 60° in both triangles?

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365

12.5 Using the calculator in trigonometry In trigonometry an angle is usually measured in degrees, minutes and seconds. Make sure the calculator is set up to accept angles in degrees. It is essential in this course that the degree mode is selected. Degrees

Minutes

1 degree = 60 minutes 1° = 60′

1 minute = 60 seconds 1′ = 60′′

Finding a trigonometric ratio A calculator is used to find a trigonometric ratio of a given angle. It requires the sin , cos and tan keys. The trigonometric ratio key is pressed followed by the angle. The degrees, minutes and seconds °''' or DMS is then selected to enter minutes and seconds. Some calculators may require you to choose degrees, minutes and seconds from an options menu.

Example 9

Finding a trigonometric ratio

Find the value of the following, correct to two decimal places. 3 a sin 60° b 2 cos 40.5° c d sin 34°20′ tan 75° Solution 1

Press sin 60 = or exe

a

sin 60° = 0.866 025 403 8 = 0.87

2

Press 2 cos 40.5 = or exe

b

2 cos 40.5° = 1.520 811 31 = 1.52

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Preliminary Mathematics General

3

Press 3 ÷ tan 75 = or exe

c

4

Press sin 34 °''' 20 °''' = or exe

d

3 = 0.803 847 577 3 tan 75° = 0.80 sin 34°20′ = 0.564 006 558 1 = 0.56

Finding an angle from a trigonometric ratio A calculator is used to find a given angle from a trigonometric ratio. Check that the degree mode is selected. To find an angle use the sin−1 , cos−1 and tan−1 keys. To select these keys press the SHIFT or a 2nd function key. The degrees, minutes and seconds °''' or DMS is then selected to find the angle in minutes and seconds.

Example 10 a b

Finding an angle from a trigonometric ratio

Given sin θ = 0.6123, find the value of θ to the nearest degree. Given tan θ = 1.45, find the value of θ to the nearest minute. Solution 1

Press SHIFT sin−1 0.6123 = or exe .

a

sin θ = 0.6123 θ = 37.755 994 38 = 38°

2

Press SHIFT tan−1 1.45 = or exe Convert the answer to minutes by using the °''' or DMS .

b

tan θ = 1.45 θ = 55°24′ 27.76′′ = 55°24′

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367

Exercise 12E 1

2

What is the value of the following angles in minutes? a 1° b 7° c 20° d 1° e 0.5° f g 0.2° h 3

60° 5°

What is the value of the following angles in degrees? a 120 minutes b 480 minutes c 60 minutes e 30 minutes f 15 minutes g 45 minutes

d h

600 minutes 20 minutes

3

Find the value of the following trigonometric ratios, correct to two decimal places. a sin 20° b cos 43° c tan 65° d cos 72° e tan 13° f sin 82° g cos 15° h tan 48°

4

Find the value of the following trigonometric ratios, correct to two decimal places. a cos 63°30′ b sin 40°10′ c cos 52°45′ d cos 35°23′ e sin 22°56′ f tan 53°42′ g tan 68°2′ h cos 65°57′

5

Find the value of the following trigonometric ratios, correct to one decimal place. a 4 cos 30° b 3 tan 53° c 5 sin 74° d 6 sin 82° e 11 sin 21°30′ f 7 cos 32°40′ g 4 sin 25°12′ h 8 tan 39°24′

6

Given the following trigonometric ratios, find the value of θ to the nearest degree. a sin θ = 0.5673 b cos θ = 0.1623 c tan θ = 0.2782 1 5 3 sin θ = cosθ = tan θ = d e f 2 8 4

7

Given the following trigonometric ratios, find the value of θ to the nearest minute. a tan θ = 0.3891 b sin θ = 0.6456 c cos θ = 0.1432 3 1 1 sin θ = d e f tan θ = 1 cosθ = 5 3 4

8

Find the value of x. Answer correct to two decimal places. a x = 4 tan 27° b x = 7 sin 15°17′ c cos x = 0.5621

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Development 9

10

11

Find the value of the following trigonometric ratios, correct to two decimal places. a

5 tan 40°

b

e

4 cos 38°9′

f

1 sin 63° 5 tan 72°36′

c

12 cos 25°

d

3 sin 42°

g

6 sin 55°48′

h

7 cos 71°16′

Given the following trigonometric ratios, find the value of θ to the nearest degree. a

sin θ =

3 2

d

tan θ =

4 6



b

tan θ =

1



c

cosθ =

5 6

e

cosθ =

3 2

f

sin θ =

1

5

2

Given the following trigonometric ratios, find the value of θ to the nearest minute. a

cosθ =

2



b

sin θ =

3 4

c

tan θ =

5 12

d

sin θ =

7 7

e

tan θ =

2 7

f

cosθ =

3

7

11

12

Given the following trigonometric ratios, find the value of θ to the nearest degree. a tanθ = 1 : 6 b sinθ = 2 : 5 c cosθ = 3 : 8

13

Given that sin θ = 0.4 and angle θ is less than 90°, find the value of: a θ to the nearest degree. b cos θ, correct to one decimal place. c tan θ, correct to two decimal places.

14

Given that tan θ = 2.1 and angle θ is less than 90°, find the value of: a θ to the nearest minute. b sin θ, correct to three decimal places. c cos θ, correct to four decimal places.

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Chapter 12 — Similarity and right-angled triangles

369

12.6 Finding an unknown side

12.6

Trigonometric ratios are used to find an unknown side in a right-angled triangle, given at least one angle and one side. The method involves labelling the sides of the triangle and using the mnemonic SOH CAH TOA. The resulting equation is rearranged to make x the subject and the calculator used to find the unknown side. Finding an unknown side in a right-angled triangle 1 2 3 4 5

Name the sides of the triangle – h for hypotenuse, o for opposite and a for adjacent. Use the given side and unknown side x to determine the trigonometric ratio. The mnemonic SOH CAH TOA helps with this step. Rearrange the equation to make the unknown side x the subject. Use the calculator to find x. Remember to check the calculator is set up for degrees. Write the answer to the specified level of accuracy.

Example 11

Finding an unknown side

Find the length of the unknown side x in the triangle shown. Answer correct to two decimal places.

25 34°

Solution 1 2 3 4

5 6

Name the sides of the rightangled triangle. Determine the ratio (SOH). Substitute the known values. Multiply both sides of the equation by 25.

o h x sin 34° = 25 25 × sin 34° = x x = 25 × sin 34° = 13.979 822 59 sin θ =

Press 25 sin 34 = or exe . Write the answer correct to two decimal places.

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=13.98

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x

370

Preliminary Mathematics General

Finding an unknown side in the denominator It is possible that the unknown side (x) is the denominator of the trigonometric ratio. For example, in the triangle below, the unknown x is the hypotenuse of the triangle. This 2 results in the trigonometric ratio . x o h 2 sin 60° = x sin θ =

2

x 60°

To solve these types of equations multiply both sides by x. Then divide both sides by the trigonometric expression (sin 60°) to make x the subject.

Example 12

Finding an unknown side in the denominator

Find the length of the unknown side x in the triangle shown. Answer correct to two decimal places.

x 40° 12

Solution

5

Name the sides of the right-angled triangle. Determine the ratio (CAH). Substitute the known values. Multiply both sides of the equation by x. Divide both sides by cos 40°.

6

Press 12 ÷ cos 40 = or exe .

7

Write answer correct to two decimal places.

1 2 3 4

a h 12 cos 40° = x x × cos 40° = 12 12 x= cos 40° = 15.6664 887 47

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cosθ =

= 15.66

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Chapter 12 — Similarity and right-angled triangles

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Exercise 12F 1

Find the length of the unknown side x in each triangle, correct to two decimal places. a

b

15

31°

c

35

x x

24°

x

23 42°

d

e

9 43°

f

76 x

x

h

55

i

60 x

26.7° x

2

x

20°

60°

g

34

7 48.4° x

21.1°

Find the length of the unknown side x in each triangle, correct to two decimal places. a

b

47

c

67

58°45′ x

31 44°35′

x

26°20′

d

e

38°9′ 27

x

f

29 35°5′

x

x

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5 18°8′ x

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Preliminary Mathematics General

Development 3

Find the length of the unknown side x in each triangle, correct to two decimal places. a



34

b

48°



x

e

f

90

77

11

70°

x

42

x

44°

Find the length of the unknown side x in each triangle, correct to one decimal place. a



x

b

c

x

87 68°15′

25°6′ 29

d

16°50′



x

47

x

e

49°53′

28°7′

12

f

32°1′

51

x

x

5

x

6

47°

4 

c

33°

x

d



x

63°

47

Find the length of the unknown side x in each triangle, correct to three decimal places.

a

b



21.7°

12.5° x

c

80.9° 39

x

x

64

42

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Chapter 12 — Similarity and right-angled triangles

12.7 Finding an unknown angle

12.7

Trigonometric ratios are used to find an unknown angle in a right-angled triangle, given at least two sides. The method involves labelling the sides of the triangle and using the mnemonic SOH CAH TOA. The resulting equation is rearranged to make θ the subject and the calculator is used to find the unknown angle. Finding an unknown angle in a right-angled triangle 1 2

3 4 5

Name the sides of the triangle – h for hypotenuse, o for opposite and a for adjacent. Use the given sides and unknown angle θ to determine the trigonometric ratio. The mnemonic SOH CAH TOA helps with this step. Rearrange the equation to make the unknown angle θ the subject. Use the calculator to find θ. Remember to check the calculator is set up for degrees. Write the answer to the specified level of accuracy.

Example 13

Finding an unknown angle

Find the angle θ in the triangle shown. Answer correct to the nearest degree.

16

24 θ

Solution 1 2 3 4

Name the sides of the right-angled triangle. Determine the ratio (SOH). Substitute the known values. Make θ the subject of the equation.

o h 16 sin θ = 24 sin θ =

 16  θ = sin −1    24  5

Press SHIFT sin−1 (16 ÷ 24) = or exe

or 6

Press SHIFT sin−1 16 a bc 24 = or exe

7

Write answer correct to the nearest degree.

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= 41.810 314 9 θ = 42°

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Preliminary Mathematics General

Exercise 12G 1

Find the unknown angle θ in each triangle. Answer correct to the nearest degree. a

b

55 θ

d

c

12

θ

13

e

38

15

26

θ

40

θ

57

f

26

θ 19

16

33 θ

g

h

37 θ

2

θ 5

42

i

9

57 θ

40

Find the unknown angle θ in each triangle. Answer correct to the nearest degree. a

10.2

θ

b

c

2.4 θ

θ

42.3

3.0

7.9

50.6

d

e

9.8 8.3

f

3.3

98.3

θ θ

θ

4.5

67.5

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Chapter 12 — Similarity and right-angled triangles

Development 3

Find the unknown angle θ in each triangle. Answer correct to the nearest minute. a b c 8 22 θ

θ

19

5

16

θ 25

d



61

43

θ

4

e

66

θ

f

13 θ

7 47

Find the unknown angle θ in each triangle. Answer correct to the nearest degree. a b c 1 23

4 12

7 8

2 14 θ

θ

4 35

θ 3 34

5 

Find the unknown angle θ in each triangle. Answer correct to the nearest degree. a b c 3 29 17 θ 5

6

θ

θ 15

4

20

21

8

Find the angle θ and φ in each triangle. Answer correct to the nearest minute. a b c 26 8 30 θ

φ

10 24

θ

φ 34

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16

θ

10

6 φ

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Preliminary Mathematics General

12.8 Applications of right-angled triangles

12.8

Trigonometry is used to solve many practical problems. How high is that tree? What is the height of the mountain? Calculate the width of the river. When solving a trigonometric problem make sure you read the question carefully and draw a diagram. Label all the information given in the question on this diagram. Solving a trigonometric worded problem 1 2 3 4 5

Read the question and underline the key terms. Draw a diagram and label the information from the question. Use trigonometry to calculate a solution. Check that the answer is reasonable and units are correct. Write the answer in words and ensure the question has been answered.

Example 14

Application requiring the length of a side

An Australian flagpole casts a shadow 8.35 m long. The sun’s rays make an angle of 44° with the level ground. Find the height of the flagpole. Answer correct to two decimal places.

44° 8.35 m

Solution 1

Draw a diagram and label the required height as x.

x 44° 8.35 m

2 3 4 5 6 7

Name the sides of the right-angled triangle. Determine the ratio (TOA). Substitute the known values. Multiply both sides of the equation by 8.35. Press 8.35 tan 44 = or exe . Write the answer correct to two decimal places.

tan θ =

o a

x 8.35 8.35 × tan 44° = x x = 8.35 × tan 444° x = 8.06 tan 44° =

Height of the flagpole is 8.06 m. 8

Write the answer in words.

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Chapter 12 — Similarity and right-angled triangles

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Trigonometry has many applications, such as in building and construction. Any vertical parts of a structure make a right angle with horizontal parts. Sloping lines in the structure complete a right-angled triangle, and trigonometry can be used to calculate its other angles and side lengths.

Example 15

Application requiring an angle

The sloping roof of a shed uses sheets of Colorbond steel 4.5 m long on a shed 4 m wide. There is no overlap of the roof past the sides of the walls. Find the angle the roof makes with the horizontal. Answer correct to the nearest degree.

4.5 m

4m Solution 1

Draw a diagram and label the required angle as θ. 4.5 m θ 4m

2

Name the sides of the right-angled triangle.

3

Determine the ratio (CAH).

4

Substitute the known values.

5

Make θ the subject of the equation.

6

Press SHIFT cos−1 (4 ÷ 4.5) = or exe

7

Write the answer correct to the nearest degree.

8

Write the answer in words.

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a h 4 cosθ = 4.5 cosθ =

 4  θ = cos−1   4.5  = 27.266 044 45

θ = 27° The roof makes an angle of 27°.

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Preliminary Mathematics General

Exercise 12H 1

2

3

4

A balloon is tied to a string 25 m long. The other end of the string is secured by a peg to the surface of a level sports field. The wind blows so that the string forms a straight line making an angle of 37° with the ground. Find the height of the balloon above the ground. Answer correct to one decimal place. A pole is supported by a wire that runs from the top of the pole to a point on the level ground 5 m from the base of the pole. The wire makes an angle of 42° with the ground. Find the height of the pole, correct to two decimal places.

25 m 37°

42° 5m

Ann noticed a tree was directly opposite her on the far bank of Tree the river. After she walked 50 m along the side of the river, she found her line of sight to the tree made an angle of 39° with the river bank. Find the width of the river, to the nearest metre.

A ship at anchor requires 70 m of anchor chain. If the chain is inclined at 35° to the horizontal, find the depth of the water, correct to one decimal place.

39° 50 m Ann

Boat 70 m 35°

5

6

Vertical tent pole is supported by a rope of length 3.6 m tied to the top of the pole and to a peg on the ground. The pole is 2 m in height. Find the angle the rope makes to the horizontal. Answer correct to the nearest degree.

2m

A 3.5 m ladder has its foot 2.5 m out from the base of a wall. What angle does the ladder make with the ground? Answer correct to the nearest degree.

3.6 m

3.5 m

2.5 m

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Development 7

A plane maintains a flight path of 19° with the horizontal after it takes off. It travels for 4 km along the flight path. Find, correct to one decimal place: a the horizontal distance of the plane from its take-off point. b the height of the plane above ground level.

8

A wheelchair ramp is being provided to allow access to the first floor shops. The first floor is 3 m above the ground floor. The ramp requires an angle of 20° with the horizontal. How long will the ramp be, measured along its slope? Answer correct to two decimal places.

9

A shooter 80 m from a target and level with it, aims 2 m above the bullseye and hits it. What is the angle, to the nearest minute, that his rifle is inclined to the line of sight from his eye to the target?

10

A rope needs to be fixed with one end attached to the top of a 6 m vertical pole and the other end pegged at an angle of 65° with the level ground. Find the required length of rope. Answer correct to one decimal place.

11

Two ladders are the same distance up the wall. The shorter ladder is 5 m long and makes an angle of 50° with the ground. The longer ladder is 7 m long. Find: a the distance the ladders are up the wall, correct to two decimal places. b the angle the longer ladder makes with the ground, correct to the nearest degree.

12

7m

5m 50°

A pole is supported by a wire that runs from the top of the pole to a point on the level ground 7.2 m from the base of the pole. The height of the pole is 5.6 m. Find the angle, to the nearest degree, that the wire makes with the ground.

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Preliminary Mathematics General

12.9 Angles of elevation and depression The angle of elevation is the angle measured upwards from the horizontal. The angle of depression is the angle measured downwards from the horizontal. 12.9

Angle of elevation

Angle of depression Horizontal

θ Horizontal

Angle of elevation

θ

Angle of depression

The angle of elevation is equal to the angle of depression as they form alternate angles between two parallel lines. This information is useful to solve some problems. θ

Angle of depression θ

Example 16

Angle of elevation

Angle of elevation

A park ranger measured the top of a plume of volcanic ash to be at an angle of elevation of 41°. From her map she noted that the volcano was 7 km away. Calculate the height of the plume of volcanic ash. Answer correct to two decimal places.

Volcanic plume 41° 7 km

Solution 1

Draw a diagram and label the required height as x. x 41°

2

Name the sides of the right-angled triangle.

3

Determine the ratio (TOA).

4

Substitute the known values.

5

Multiply both sides of the equation by 7.

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7 km

o a x tan 41° = 7 7 × tan 41° = x x = 7 × tan 41° tan θ =

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Chapter 12 — Similarity and right-angled triangles

6 7 8

Press 7 tan 41 = or exe . Write the answer correct to two decimal places. Write the answer in words.

Example 17

381

x = 6.085 007 165 = 6.09 The height of the volcanic plume was 6.09 km.

Angle of depression

The top of a cliff is 85 m above sea level. Minh saw a tall ship. He estimated the angle of depression to be 17°. How far was the ship from the base of the cliff? Answer to the nearest metre.

Solution 1

Draw a diagram and label the required distance as x.

17° 85 m 17° x

2

Name the sides of the right-angled triangle.

3

Determine the ratio (TOA).

4

Substitute the known values.

5

Multiply both sides of the equation by x. Divide both sides by tan 17°. Press 85 ÷ tan 17 = or exe .

6 7

8 9

Write the answer correct to nearest metre. Write the answer in words.

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o a 85 tan17° = x x × tan17° = 85 85 x= tan17° = 278.0224726 tan θ =

x = 278 The ship is 278 metres from the base of the cliff.

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Preliminary Mathematics General

Exercise 12I 1

Luke walked 400 m away from the base of a tall building, on level ground. He measured the angle of elevation to the top of the building to be 62°. Find the height of the building. Answer correct to the nearest metre. 62° 400 m

2

3

4

5

The angle of depression from the top of a TV tower to a satellite dish near its base is 59°. The dish is 70 m from the centre of the tower’s base on flat land. Find the height of the tower. Answer correct to one decimal place.

59° x 70 m

When Sarah looked from the top of a cliff 50 m high, she noticed a boat at an angle of depression of 25°. How far was the boat from the base of the cliff? Answer correct to two decimal places.

The pilot of an aeroplane saw an airport at sea level at an angle of depression of 13°. His altimeter showed that the aeroplane was at a height of 4000 m. Find the horizontal distance of the aeroplane from the airport. Answer correct to the nearest metre.

25° 50 m

13° 4000 m

The angle of elevation to the top of a tree is 51° at a distance of 45 m from the point on level ground directly below the top of the tree. What is the height of the tree? Answer correct to one decimal place. 51° 45 m

6

A iron ore seam of length 120 m slopes down at an angle of depression from the horizontal of 38°. The mine engineer wishes to sink a vertical shaft, x, as shown. What is the depth of the required vertical shaft? Answer correct to the nearest metre.

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38° x

120 m

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7

A tourist viewing the Sydney Harbour from a building 130 m above the sea level observes a ferry that is 800 m from the base of the building. Find the angle of depression. Answer correct to the nearest degree.

θ 130 m

800 m 8

What would be the angle of elevation to the top of a radio transmitting tower 130 m tall and 300 m from the observer? Answer correct to the nearest degree.

130 m θ 300 m

9

Lachlan observes the top of a tree at a distance of 60 m from the base of the tree. The tree is 40 m high. What is the angle of elevation to the top of the tree? Answer correct to the nearest degree.

40 m θ 60 m

10

11

A town is 12 km from the base of a mountain. The town is also a distance of 12.011 km in a straight line to the mountain. What is the angle of depression from the top of a mountain to the town? Answer to the nearest degree correct to one decimal place.

θ

12.011 km 12 km

Find, to the nearest degree, the angle of elevation of a railway line that rises 7 m for every 150 m along the track.

7m 150 m

12

The distance from the base of the tree is 42 m. The tree is 28 m in height. What is the angle of elevation measured from ground level to the top of a tree? Answer correct to the nearest degree.

28 m 42 m

13

A helicopter is flying 850 m above sea level. It is also 1162 m in a straight line to a ship. What is the angle of depression from the helicopter to the ship? Answer correct to the nearest degree.

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850 m

1162 m

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Development 14

A rocket launching pad casts a shadow 35 m long when the angle of elevation of the sun is 52°. How high is the top of the launching pad? Answer correct to correct to nearest metre.

15

The angle of elevation to the top of a tree from a point A on the ground is 25°. The point A is 22 m from the base of the tree. Find the height of the tree. Answer correct to nearest metre.

16

A plane is 460 m directly above one end of a 1200 m runway. Find the angle of depression to the far end of the runway. Answer correct to the nearest minute.

17

A communication tower is located on the top of a hill. The angle of elevation to the top of the hill from an observer 2 km away from the base of the hill is 6°. The angle of elevation to the top of the tower from the observer is 8°. Find, to the nearest metre, the height of the: a hill. b hill and the tower. c tower.

18

Jack is on the top of a 65 m high cliff. He observes a man swimming out to sea at an angle of depression of 51°. Jack also sees a boat out to sea at an angle of depression of 30°. Find, to the nearest metre, the distance: a x of the man from the base of the cliff. b y of the boat from the base of the cliff. c from the man to the boat.

19

51°

30°

65 m

x y

The angle of elevation from a boat out to sea to the top of a 350 m cliff is 13°. After the boat travels directly towards the cliff, the angle of elevation from the boat is 19°. How far did the boat travel towards the cliff? Answer correct to the nearest metre.

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Chapter 12 — Similarity and right-angled triangles

Similar figures and scale factors

• • • •

Solving a worded problem using

1

similar figures

2 3 4 5 6

Trigonometric ratios

Finding an unknown side

sin θ =

o a o (SOH) cos θ = (CAH) tan θ = (TOA) h h a

1

4 1 2 3 4 Applications of right-angled

1

triangles

2 3 4

Angle of depression

Read the question and underline the key terms. Draw similar figures and label the information from the question. Use a pronumeral (x) to represent an unknown side. Write an equation using two fractions formed from matching sides. Solve the equation. Check that the answer is reasonable and units are correct. opposite adjacent opposite cos θ = tan θ = hypotenuse hypotenuse adjacent

3

Angle of elevation

Similar figures are the same shape but are different sizes. Corresponding (or matching) angles of similar figures are equal. Corresponding sides of similar figures are in the same ratio. Scale factor is the amount of enlargement or reduction.

sin θ =

2

Finding an unknown angle

Study guide 12

Name the sides of the triangle. Use the given side and unknown side x to determine the trigonometric ratio. Use SOH CAH TOA. Rearrange the equation to make the x the subject. Use the calculator to find x. Name the sides of the triangle. Use the given sides and unknown angle θ to determine the trigonometric ratio. Use SOH CAH TOA. Rearrange the equation to make θ the subject. Use the calculator to find θ. Read the question and underline the key terms. Draw a diagram and label the information. Use trigonometry to calculate a solution. Check that the answer is reasonable and units are correct. Angle of elevation θ Horizontal

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Horizontal θ Angle of depression

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385

Review

386

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

What is the scale factor for these triangles? 1 A B 3 3 C 4 D 6

6

2 3

9

2

A small tree 3 m high throws a shadow 1.5 m long. At the same time a large tree throws a shadow 24 m long. What is the height of the large tree in metres? A 12 B 25.5 C 32 D 48

3

What is the value of cos θ ? 9 12 A B 15 9 12 9 C D 15 12

4

5

6

What is the length of x? A 18 cos 58° B 18 sin 58° 18 18 C D sin 58° cos58°

θ

15

9

12

x 58° 18

What is the length of x? A 45 cos 34° B 45 sin 34° 45 45 C D cos 34° sin 34° How would angle θ be calculated?  5  4 A tan −1   B tan −1    4  5 C

 5 tan    4

D

x

34° 45

4 θ 5

 4 tan    5

7

What is the size of angle θ in question 6? (Answer correct to one decimal place.) A 38.6 B 39.0 C 51.0 D 51.3

8

What is the angle of elevation to the top of a tower 80 m tall and 100 m from the observer? Answer in degrees correct to one decimal place. A 51.3 B 51.4 C 38.6 D 38.7

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Review

Sample HSC – Short-answer questions 1

Find the length of the unknown sides for the following pairs of similar triangles. a



20

b

y

6 24

a b

4

12



c

6

m

6

9

x

16

d

6 g

12

14

h

21

4

28

n 8

2

The wall of a house casts a shadow 6 m long. At the same time a one-metre ruler casts a shadow 0.90 m long. What is the height of the building? Answer correct to the nearest metre. 1m 0.90 m

3

4

In the triangle shown, state the value of the: a hypotenuse. b opposite side. c adjacent side. What are these ratios in simplest form? a sin θ b cos θ c tan θ

18

6m

θ

30 24 24 26

θ

10

5

Find the value of the following trigonometric ratios, correct to two decimal places. a tan 68° b cos 13° c sin 23° d cos 82°

6

Given the following trigonometric ratios, find the value of θ to the nearest degree. 1 c tan θ = 0.2 a cos θ = 0.4829 b sin θ = 3

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388

7

Preliminary Mathematics General

Find the value of x, correct to two decimal places. a b x



x

63°

8

8

47°

24

c

x 65°15′ 36

Find the unknown angle θ in each triangle. Answer correct to the nearest degree. a b c 10 11 17 θ

8

θ 8

10 θ

9

Find the unknown angle θ in each triangle. Answer correct to the nearest minute. a b c 28 27 θ

θ

10

46 35

θ

33

10

A pole casts a shadow of 5.4 m long. The sun’s rays make an angle of 36° with the level ground. Find the height of the pole to the nearest metre.

Sun

36° 5.4 m 11

Susan looked from the top of a cliff 62 m high and noticed a ship at an angle of depression of 31°. How far was the ship from the base of the cliff ? Answer correct to one decimal place.

31° 62 m

Challenge questions 12

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C H A P T E R

13

Mathematics and communication Focus study — FSCo Mathematics and communication Read and interpret mobile phone plans and bills Calculate the cost of calls using the given time and duration Determine a suitable mobile phone plan using phone usage Construct and interpret tables and graphs of phone usage Use prefixes to describe the size of units of storage Convert units of file storage Calculate the time to download a file using the download speed Interpret statistics related to the effect of downloading files

13.1 Mobile phone plans Mobile phones that are used irresponsibly or with the wrong choice of mobile phone plan can result in unexpectedly large bills. Mobile phone bills are often a concern of people who make unnecessary calls on their mobile. Mobile phones provide immediate pleasure and can be somewhat addictive. Text messaging is usually cheaper than voice calls; however, premium text messaging services that use 1900 are billed at a higher rate. Extra costs are involved when mobile phones are used for internet access, 3D games, music, video downloads, email and photo messaging.

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The cost of making a mobile phone call is determined by the connection fee (or flagfall), call rate and the length of the call. The connection fee and call rate charged depends on the mobile phone network and mobile phone plan. In general the longer the phone call the greater the cost. There are thousands of mobile phone plans available from hundreds of providers. It is essential to do some research and choose the most appropriate plan. There are two broad categories: prepaid and postpaid. Mobile phone charge Call charge = Connection fee + Time used × Call rate

Prepaid plan A prepaid mobile phone requires the user to purchase credit in advance. This credit is used to pay for phone calls, text messages and data downloads. If users have no available credit then the mobile phone is blocked for use. Users are able to increase their credit at any time using their phone or at a retail store. A prepaid plan makes it easier to control spending by limiting debt. There are also fewer contractual arrangements. However, users often pay more for their calls and text messages.

Postpaid plan or fixed term contract Postpaid plan involves a contract that varies in length from one to two years. It involves a range of charges such as a connection fee, monthly access fees, call costs, disconnection fee and data charges. To compare different contracts you need to know your general phone usage. Be aware that some plans charge per second while others charge for blocks of 30 seconds or 60 seconds. Most contracts allow for calls up to a certain value as part of the access fee. This is referred to as free calls. When calls exceed that value they are charged at the specified rate.

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Chapter 13 — Mathematics and communication

Example 1

Calculating mobile phone charges

Plan A

Plan B

Monthly access fee

$19.00

$24.00

Free calls

$50.00

$20.00

Connection fee – Flagfall

$0.35

$0.25

Call rates (per 30 sec)

$0.47

$0.40

SMS

$0.25

$0.25

MMS

$0.50

$0.50

Messaging

391

The table above shows two mobile phone plans. Joel uses Plan A. Paige uses Plan B. a What is the call charge if Joel makes a 2-minute call? b What is the call charge if Paige makes a 5-minute call? c Mia makes 100 calls in a month with each call lasting 1 minute. What is the monthly cost on Plan B? Solution 1 2

3 4 5 6

7 8 9 10 11 12

Add the flagfall to the local rate. Local rate is charged for every 30 seconds. A 2-minute call is 120 seconds. Divide 120 by 30 to determine the cost. Evaluate. Write the answer in words. Add the flagfall to the local rate. Local rate is charge for every 30 seconds. A 5-minute call is 300 seconds. Divide 300 by 30 to determine the cost. Evaluate. Write the answer in words. Multiply the calculation for each call by 100. Evaluate. Subtract the free calls and add the monthly fee to the call charge. Write the answer in words.

a

Charge = 0.35 + (120 ÷ 30) × 0.47] = $2.23 Joel is charged $2.23.

b

Charge = 0.25 + (300 ÷ 30) × 0.40 = $4.25 Paige is charged $4.25.

c

Charge = 100 × [0.25 + (60 ÷ 30) × 0.40] = $105 Cost = 105 − 20 + 24 = $109 Cost of plan B is $109.

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Exercise 13A 1

A mobile phone fixed contract has a minimum monthly fee of $49, call connection fee of $0.37 and voice calls of $0.40 per 30-second block. What is the cost of these calls? a One-minute call b Three-minute call c Five-minute call d Ten-minute call e Twenty-minute call

2

The table below shows four mobile phone plans.

13A

Basic

b

c

d e

Super

Mega

Monthly access fee

$29

$49

$79

$99

Free calls

$120

$300

$600

$800

Connection fee – Flagfall

$0.30

$0.30

$0.30

$0.30

Call rates (per 30 sec)

$0.44

$0.40

$0.40

$0.35

SMS

$0.25

$0.25

$0.25

$0.25

MMS

$0.50

$0.50

$0.50

$0.50

Messaging a

Big

Jack uses the Basic plan. i What is his monthly access fee? ii What is the charge for a two-minute call? Chelsea uses the Super plan. i What value of free calls does she receive? ii What is the charge for a three-minute call? Adam uses the Mega plan. i What is the cost of an SMS message? ii What is the charge for a four-minute call? Lauren used the Big plan last month for SMS messages and was charged $20.00 in addition to her monthly access fee. How many SMS messages did she send? How many 30-second free calls are possible on the Mega plan?

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393

A part of a mobile phone bill is shown below. Telephone Bill User Name: Bill Sample Special Discounts From

To

Details

Quantity

15-Apr 15-Apr

14-May 14-May

Up to 50 free SMS 50 calls Talk Time included value ($10 Rollover to next month)

Amount 11.35CR 54.14CR $65.49CR

Equipment From

To

Details

Quantity

Rate

Amount

15-May 15-May

14-Jun 14-Jun

50 SMS for $10 Add on Talk Time Monthly Access

1 1

9.09 22.73

9.09 22.73 $31.82

Mobile Call Charges Date

Time

Origin

Destination Min:sec Rate

Amount

15-Apr 15-Apr 15-Apr 15-Apr 15-Apr 16-Apr 17-Apr 18-Apr 19-Apr 19-Apr

10:10am 11:34am 11:45am 12:02pm 12:15pm 10:10am 11:34am 11:45pm 08:02pm 10:10pm

Sydney Melbourne Brisbane Adelaide Brisbane Sydney Melbourne Brisbane Adelaide Sydney

Mobile Mobile Mobile Mobile Mobile Div-Mobile Mobile Mobile Mobile Info Access

0.45 0.63 0.50 3.77 0.05 5.46 5.46 0.00 0.00 2.16

a b c d e f g h 4

0:10:30 0:00:30 0:02:30 0:03:00 0:03:30 0:05:00 0:05:00 0:05:00 0:05:00 0:12:30

Peak Peak Peak Off Peak Peak Peak Peak Talk Time Talk Time Special

What is the total of all charges on this bill? How much was the mobile call charge on 16 April? How many free SMS messages were available on this plan? What is the monthly access fee? How much was the mobile call to Adelaide on 15 April? How many free calls are listed as having been made? What was the time of the call to Melbourne on 15 April? What was the length of the call to Sydney on 15 April?

A Prepaid phone has a call connection fee of $0.38 and voice calls of $0.48 per 30-second block. What is the cost of the following? a 30-second call b 1 minute and 30 seconds call c 3-minute call d 30-minute call.

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Development 5

The table below shows two mobile phone plans. Monthly access fee Free calls

Casual $39.00 $150

Frequent $69.00 $300

Connection fee – Flagfall

$0.32

$0.32

Call rates (per 30 sec)

$0.45

$0.38

Messaging

$0.22 $0.44

$0.20 $0.40

a b c d e f

g h

i

SMS MMS

John uses the Casual plan. i What is his monthly access fee? ii What is the charge for a call lasting 160 seconds? iii What is the charge for a call lasting 4.25 minutes? Georgina uses the Frequent plan. i What is her monthly fee? ii What is the charge for a call lasting 160 seconds? iii What is the charge for a call lasting 4.25 minutes? How many free 30-second calls do you receive on the Casual plan? How many free 30-second calls do you receive on the Frequent plan? How many more free 30-second calls do you receive on the Frequent plan compared to the Casual plan? How many 30-second calls are required to exceed the difference in the monthly access fee between the Casual and Frequent plans? (Use the Casual call rate and ignore free calls.) How many SMS messages are required to exceed the difference in the monthly access fee between the Casual and Frequent plans? (Use the Casual SMS rate.) Hannah is deciding on one of the above mobile phone plans. She will be using her mobile phone to make 300 calls (30 seconds) and 100 SMS messages each month. What plan should she choose? Jackson is deciding on one of the above mobile phone plans. He will be using his mobile phone to make 200 calls (30 seconds) and 500 SMS messages each month. What plan should he choose?

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6

A mobile phone plan has a monthly charge of $29 on a 24-month contract. The call rate is $0.98 per 60-second block plus there is a $0.38 flagfall. The contract includes $150 of free calls and 200 MB of data per month with excess data charged at $0.50 per MB. a What is the minimum cost of the contract? b What is the cost of making 200 calls in a month where the duration of each call is less than one minute? (Include monthly charge.) c What is the cost of a call lasting 4 minutes and 45 seconds? Assume the $150 free calls have been used. d The providers on the above plan decide to reduce the cost per MB to $0.125. Assuming the monthly charge remains at $29, what is the new amount of included data?

7 

The table below shows four mobile phone plans. Light

Normal

Heavy

Pro

$29

$49

$79

$99

Call rates (per minute)

$0.98

$0.95

$0.90

Unlimited

Connection fee

$0.38

$0.35

$0.35

Unlimited

SMS messages

$0.25

$0.25

Unlimited

Unlimited

Free calls

$100

$150

$300

Unlimited

500 MB

1 GB

3 GB

5 GB

$0.50/MB

$0.30/MB

$0.25/MB

$0.25/MB

Monthly access fee

Included data Excess data a b c

d

What is the charge for downloading 5 GB of data in the last month excluding the monthly access fee? i Light plan ii Normal plan iii Heavy plan iv Pro plan What is the total cost including monthly access fee of making 150 calls (60 seconds) in the last month? i Light plan ii Normal plan iii Heavy plan iv Pro plan Noah does not use his phone to make calls. However, his data usage for the past three months was 500 MB, 1 GB and 4 GB. Which mobile phone plan is more economical for Noah? Ava uses less than 200 MB of data each month. However, she made 100 calls, 150 calls and 70 calls in the past three months. Each call lasted for 1 minute. Which mobile phone plan is the most cost-effective for Ava?

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13.2 Phone usage tables and graphs A step-function or step graph consists of horizontal line segments or intervals. These line segments create a graph that looks like 13.2 staircase pattern. The end-points of the line segments consist of a closed circle or an open circle. The values on the vertical axis are read from the closed circle (shaded) and not the open circle. For example, in the graph below the cost of a 1-minute call is 20 cents and not 30 cents. A linear piecewise function is similar to the step-function. It consists of series of line segments which may not be horizontal. Each line segment is drawn separately and may not connect to the next line segment. Reading a step-function graph

A phone company charges 20 cents for the first minute and 10 cents for each additional minute rounded up to the next minute. The step-function graph is shown opposite. a What is the charge if Dylan makes a 2-minute call? b What is the charge if Sarah makes a 3.5-minute call?

Phone charges 50

Cost (cents)

Example 2

40 30 20 10 0

1

3 2 Time (min)

4

Solution 1 2

Read from the graph (when time = 2 min, cost = 30). Read from the graph (when time = 3.5 min, cost = 50).

a

Dylan is charged $0.30.

b

Sarah is charged $0.50.

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Chapter 13 — Mathematics and communication

Exercise 13B 1

The phone charges are described by the Phone charges step-function graph opposite. a How much is charged in the first 75 minute? 60 b What is the charge for each 45 additional minute? 30 c What is the charge for a 2-minute call? 15 d What is the charge for a 3.2-minute 0 call? 3 1 2 Time (min) e What is the charge for a 20-second call? f Copy the step-function graph and extend the graph to include the costs for a 5-minute call and a 6-minute call. Assume the same pattern of charges. Cost (cents)

13B

A mobile phone plan has a monthly Total monthly charge service fee plus call costs where the 50 rate changes after 40 calls as shown in the graph opposite. 40 a How much is the monthly service fee? 30 b How much does the company charge if you make 20 calls a 20 month? c How much does the company 10 charge if you make 40 calls a month? d How much does the company 0 20 60 80 100 40 charge if you make 60 calls a Monthly calls month? e How many calls are made if the total monthly charge is $40? f How many calls are made if the total monthly charge is $25? g How many calls are made if the total monthly charge is $15? h What is the gradient of the line segment between 0 and 40 calls? i What is call charge for making 0 to 40 calls? j What is the gradient of the line segment between 40 and 100 calls? k What is call charge for making 40 to 100 calls? l Copy the linear piecewise graph and extend the graph to include the call charges of $0.20 per call for 100 to 140 calls. Charge ($)

2

4

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Preliminary Mathematics General

Development 3

The call charges for a phone company are as follows: $0.15 for the first minute or part thereof, then $0.20 for the second minute or part thereof, then $0.25 for the third minute or part thereof and $0.30 additional for a call lasting between 3 and 4 minutes. a Draw a step-function graph to illustrate the call charges, with the Time (minutes) on the horizontal axis and Cost ($) on the vertical axis. b What is the call cost for 30 seconds? Use the step graph. c Max started a call at 11.20 a.m. and ended the call at 11.24 a.m. How much did he pay for call charges? d During the month, Hannah made 28 calls lasting 1 minute, 45 calls lasting 2 minutes and 35 calls lasting 3 minutes. What is her total call charge for the month? e Modify the step graph to include another line segment. The call charge is $0.20 additional for a call lasting between 4 and 5 minutes.

linear piecewise graph represents the total monthly call charge. Each line segment has a different call rate. a How much is the monthly service fee? Total monthly charge b The total monthly call charge for 30 calls 40 is $24. What is the gradient of the first 35 line segment? 30 c The total monthly call charge for 60 calls 25 is $30. What is the gradient of the second 20 line segment? 15 d The total monthly call charge for 80 calls 10 is $32. What is the gradient of the third 5 line segment? e What is the equation of the first line 0 60 80 20 40 segment? Monthly calls f What is the equation of the second line segment? g Copy the linear piecewise graph and extend the graph to include call charges of $0.08 per call for 80 to 100 calls. Charge ($)

4 The

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13.3 File storage Files are used to store data in a storage device. The data can be modified and transferred to other devices such as mobile phones. Electronic data is represented in digital form. It uses only two digits: 0 and 1. Two digits are easily represented electronically by circuits being off or on. Each on or off digit is called a bit (BInary digiT). A bit is the smallest unit of data stored in an electronic device. A byte is a group of 8 bits that represents a single unit. If the data is text then a byte would represent a character, such as a letter, a number, a punctuation mark or a space. The prefixes kilo, mega, giga and tera are then added and more commonly used to measure data. The lowercase ‘b’ is used to represent a bit while the uppercase ‘B’ is used to represent a byte. Unit

Symbol

Byte

B

Meaning

Exact value

One byte

Power of two 1

(20)

1024

(210)

Kilobyte

KB

Thousand bytes

Megabyte

MB

Million bytes

1 048 576

(220)

Gigabyte

GB

Billion bytes

1 073 741 824

(230)

Terabyte

TB

Trillion bytes

1 099 511 627 776

(240)

File storage units Convert file storage units by using the exact value. Use the calculator and the power of 2 to obtain the exact value.

Storage devices A storage device is any device that can store data and allow it to be retrieved when required. There are many different types of storage devices: • Hard disk has a storage capacity measured in the TB. • USB flash drive is available in range of different sizes such as 32 GB. • Mobile phones and media players will store different types of data and have different sizes. • Memory cards are used to increase the storage capacity of these devices. • A compact disc (CD) stores up to 800 MB.

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Preliminary Mathematics General

Converting file storage units

Example 3

Complete:

5 KB =

a

B

b

1.3 TB =

MB

Solution 1 2 3 4

Find the conversion factor. There are 210 or 1024 bytes in one kilobyte. To change KB to B, multiply by 210. Find the conversion factor. There are 220 or 1 048 576 megabytes in one terabyte. To change MB to TB multiply by 220.

b

5 KB = 5 × 210 Β = 5 × 1024 Β = 5120 B 1.3 TB = 1.3 × 220 MB = 1.3 × 1 048 576 ΜΒ = 1 363 148.8 ΜΒ

Converting file storage units

Example 4

Complete:

a

a

1536 KB =

MB

8 388 608 KB =

b

GB

Solution 1

2 3

4

Find the conversion factor. There are 210 or 1024 kilobytes in one megabyte. To change KB to MB, divide by 210. Find the conversion factor. There are 220 or 1 048 576 kilobytes in one megabyte. To change KB to GB divide by 220.

Example 5

a

1536 KB = 1536 ÷ 210 MB = 1536 ÷ 1024 ΜΒ = 1.5 ΜΒ

b

8 388 608 KB = 8 388 608 ÷ 220 GB = 8 388 608 ÷ 1 048 576 GB = 8 GB

Solving problems involving file storage

How many MP3 files of average size 3.2 MB can be stored on a 2 GB MP3 player? Solution 1 2 3

Convert to the same units. There are 210 or 1024 gigabytes in one megabyte. To change GB to MB multiply by 210 or 1024. Divide the file size into storage size of the MP3 player.

2 GB = 2 × 210 MB = 2048 MB

Files = 2048 ÷ 3.2 = 640

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Exercise 13C 1 13C

Complete the following. a

2 KB =

B

b

4 MB =

KB

c

7 TB =

KB

d

3 TB =

MB

e

9 TB =

GB

f

6 MB =

B

g

B = 8 TB

h

i

B = 4 GB

j

2.5 TB =

MB

l

5.3 TB =

MB

b

2 097 152 KB =

GB

d

3 145 728 KB =

GB

f

8 388 608 KB =

MB

k 2

4.8 KB =

B

GB = 5 TB

Complete the following. a

1024 B =

c

6 291 456 MB =

e

5 368 709 120 B =

g

KB TB GB

KB = 7168 B

h

TB = 3 221 225 472 MB

3

How many kilobytes are there in 2 gigabytes?

4

Madison has received a 50 KB file in JPEG format from a friend. How many files of this size could she store on a: a 1 GB USB flash drive? b 500 MB MP3 player? c 800 KB compact disc? d 100 GB external hard drive?

5

A USB flash drive has 2.5 GB of video data. What is the total file storage (in gigabytes) in the flash drive if the following files are added to the USB? Answer correct to one decimal place. a 330 MB b 1200 MB c 4000 MB

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Development 6

Complete the following. Answer correct to 4 decimal places when appropriate. a

3072 B =

c

2048 KB =

e

b

4194304 MB =

MB

d

3.2 GB =

MB = 2056 KB

f

KB

TB

TB

MB = 7168 KB

7

Tyler brought a video camera to the party and created 15 video files with average file size 250 MB. How much space (in MB) remains on his hard drive if it has the following storage capacity? a 200 GB b 50 TB c 1 TB d 10 000 MB

8

What is the difference in file storage between the following devices? a USB flash drive with a storage capacity of 4 GB and a compact disc with a storage capacity of 800 MB. Give your answer to the nearest MB. b Hard drive with a storage capacity of 2 TB and an external hard drive with a storage capacity of 300 GB. Give your answer to the nearest GB.

9

Order 200 MB, 0.002 TB, 2000 KB and 200 000 B in: a ascending order (smallest to largest) b descending order (largest to smallest).

10

Joel produces a 220 MB video file each day at work. a How many days will it take for the file storage to exceed a gigabyte? b How many days will it take for the file storage to exceed a terabyte?

11

A manufacturer claims that an 8 GB MP3 player holds 2000 songs. a What is the average file size (MB) of a typical song to the nearest whole number? b Do you think the manufacturer’s claims are correct? Give a reason for your answer.

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13.4 Digital downloads The speed of data transfer is measured by the number of bits per second (bps) or the bit rate. Common units of measurement for the transmission of data are kilobits (Kbps), megabits (Mbps), gigabits (Gbps) or terabits (Tbps). The prefix is used to convert the units of measurement. The meaning of the prefixes are kilo (1000), mega (1 000 000), giga (1 000 000 000) and tera (1 000 000 000 000). Unit

Symbol

Bit per second

Approximate value

bps

1

Kilobit per second

Kbps

1 000

Megabit per second

Mbps

1 000 000

Gigabit per second

Gbps

1 000 000 000

Terabit per second

Tbps

1 000 000 000 000

Converting data transfer rates 1 2 3

Learn the order of the prefixes. Use the prefixes to determine the conversion factor. An increase in the order of the prefix represents a conversion factor of 1000.

tera giga mega kilo unit

÷ 1000 ÷ 1000 ÷ 1000 ÷ 1000

Converting digital download times

Example 6

Complete

× 1000 × 1000 × 1000 × 1000

a

4 Mbps =

Kbps

b

2 000 000 bps =

a

4 Mbps = 4 × 1000 Kbps = 4000 Kbps

b

2 000 000 bps = 2 000 000 ÷ 1 000 000 Mbps = 2 Mbps

Mbps

Solution 1 2 3 4

Find the conversion factor. There are approximately 1000 Kbps in 1 Mbps. To change Mbps to Kbps, multiply by 1000. Find the conversion factor. There are 1 000 000 bps in 1 Mbps. To change bps to Mbps, divide by 1 000 000.

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Preliminary Mathematics General

Exercise 13D 1

Complete the following.

13D

a

6 Mbps =

Kbps

b

2 Tbps =

Kbps

c

3 Kbps =

bps

d

5 Tbps =

Gbps

e

9 Tbps =

Mbps

f

7 Mbps =

bps

g

Gps = 7 Tbps

h

i

bps = 2 Tbps

j

2.1 Kbps =

bps

l

4.8 Tbps =

Mbps

b

9000 bps =

Kbps

d

8 000 000 Mbps =

f

5 500 000 000 bps =

k 2

7.3 Tbps =

Mbps

bps = 8 Gbps

Complete the following. a

9 000 000 Kbps =

c

3000 Kbps =

e

4 780 000 Kbps =

Gbps

Gbps Mbps

g

Kbps = 6 700 000 bps

i

Tbps = 3 200 000 000 Mbps

h j

Tbps Gbps

Mbps = 4900 Kbps Kbps = 2400 bps

3

Search the internet to find an online data transfer rate converter to check your answers to questions 1 and 2.

4

How many megabits per second are in 7 terabits per second?

5

A file is 8 000 000 bits in size. How many seconds would it take to download the file at the following transfer rates? a 2 Mbps b 4 Kbps c 8 Tbps d 1 bps e 20 Mbps f 1000 Kbps

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Development 6

Order 700 000 bps, 700 Mbps, 0.007 Tbps and 7000 Kbps in: a ascending order (smallest to largest) of speed b descending order (largest to smallest) of speed.

7

A 306 byte file is being downloaded from the internet. a How many bits make up this file? Assume 1 byte is 8 bits. b How many seconds would it take to download this file at 3 bps? c How many seconds would it take to download this file at 6 Kbps? d How many seconds would it take to download this file at 2 Mbps?

8

A 413 696 byte music file is being downloaded from the internet. a How many bits make up this file? Assume 1 byte is 8 bits. b How many seconds would it take to download this file at 2 Kbps? c How many seconds would it take to download this file at 8 Mbps?

9

Hayley has taken a 696 320 byte picture on her camera. a How many bits make up this file? Assume 1 byte is 8 bits. b How many seconds would it take to download this file at 50 bps? c How many seconds would it take to download this file at 34 Kbps? d How many seconds would it take to download this file at 40 Mbps?

10

How many seconds would it take to upload a 2.4 MB file if the transfer rate is 4 Kbps? Answer correct to the nearest second. Assume 1 byte is 8 bits and 1 MB is 1024 KB.

11

Luke has a connection to the internet at a speed of 40 Mbps. What is the largest possible file size (MB) that can be downloaded in the following time? a 1 second b 1 minute c 1 hour

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13.5 Digital download statistics People are downloading digital media at an increasing rate. They use the internet to download files containing text, graphics, animation, music, video and software. Faster broadband has resulted in an increase in quantity and quality of music and video files that are downloaded from the internet. Downloading has also raised the issue of copyright. Digital download statistics Use summary statistics (mean, median, mode, range and interquartile range) to interpret the effect of downloading music and video files.

Example 7

Calculation of summary statistics

Tim collected data on the location where people have downloaded music files. He asked each person where they most preferred to download music. The result of his survey is shown below. Location

People

Home

105

Work

65

School

80

Internet cafe

75

a

What is the mode? Which location is the most popular for downloading music?

b

What percentage of the people preferred to download music from home? Answer correct to two decimal places. Solution 1 2 3 4

5 6

Mode is the score that occurs the most. Location with the mode is ‘home’. Find the total number of people surveyed by adding the people column. Divide the number of people downloading in the home by the total number of people. Multiply by 100 to express as a percentage. Write the answer in words.

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a b

Mode is downloading from home. Most popular location is home. Total = 105 + 65 + 80 + 75 = 325 Home =

105 × 100% 325

= 32.31% 32.31% preferred to download at home.

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407

Exercise 13E 1

The time (in minutes) spent sending SMS messages to friends by five teenagers was 82, 130, 57, 107 and 82. Find the following summary statistics. a Mean b Median c Mode d Range e Interquartile range f Sample standard deviation (to 3 decimal places) g Population standard deviation (to 3 decimal places) h What is the total number of minutes spent sending SMS messages to friends by these teenagers?

2

A 5.22 MB music file was downloaded from the internet on 15 consecutive days. The time taken (in seconds) to download the file was recorded: 24, 28, 25, 35, 21, 27, 29, 28, 23, 30, 28, 27, 33, 57 and 27. a What is the mean time (to 2 decimal places)? b What is the median time? c What is the mode? d Which is the better measure for the centre for the data? Explain your answer.

3

The number of video files downloaded by 30 students is recorded below.

13E

4 2 a b c d e

0

2

0

1

0

2

3

1

0

4

1

3

4

0

1 5 3 5 1 0 2 1 2 5 3 4 1 0 Construct a frequency table for this data. What was the most common number of video files that have been downloaded? Insert an fx column and calculate the mean. Insert a cumulative frequency column and calculate the median. What percentage of students have not downloaded any video files (to 1 decimal place)?

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Development 4

The sale of CDs from a traditional music store was 262, 288, 322, 276, 290, 301, 308, 318, 292 and 307. During the same period, sales from an online music store were 178, 195, 224, 190, 201, 228, 219, 216, 210 and 228. a What is the mean of sales from each store (to 1 decimal place). i  Traditional ii  Online b What is the sample standard deviation of sales from each store (to 1 decimal place)? i  Traditional ii  Online c Compare and contrast the mean and standard deviation from each store.

5

The grouped frequency table shows the number of people who have downloaded a video file. A sample of 200 people was taken in each age group. Age range

Class centre (x)

Frequency (f )

15–24

98

25–34

39

35–44

28

45–54

23

55–64

17

se your calculator to estimate the following summary statistics for the downloaders. U Answer correct to one decimal place. a Mean b Sample standard deviation c Population standard deviation

6 

The time taken to download a music file is shown in the table below.

File size (MB) 5 10 15 20 25 30   35   40   45 8 26 39 57 71 84 102 113 131 Time (sec) a Construct a column graph for this data. Use file size as the horizontal axis and time as the vertical axis. b What is the average increase in time taken to download the file when the file increases by 5 MB? Answer to the nearest second. c Estimate the time it would take to download a 50 MB file. d Describe the relationship between the file size and the time taken. © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Chapter 13 — Mathematics and communication

Study guide 13

Mobile phone plans

The cost of making a mobile phone call is determined by the connection fee (or flagfall), call rate and the length of the call. Call charge = Connection fee + Time used × Call rate Postpaid plan involves a contract that varies in length from one to two years. It involves a range of charges such as a connection fee, monthly access fees, call costs, disconnection fee and data charges.

Phone usage tables and

A step-function or step graph consists of horizontal line segments or intervals. The end points of the line segments consist of a closed circle or an open circle. The values on the vertical axis are read from the closed circle (shaded) and not the open circle.

graphs

File storage

Digital download

Convert file storage units by using the exact value. Use the calculator and the power of 2 to obtain the exact value. Unit

Symbol

Byte

B

1

(20)

1 024

(210)

One byte

KB

Thousand bytes

Megabyte

MB

Million bytes

1 048 576

(220)

Gigabyte

GB

Billion bytes

1 073 741 824

(230)

Terabyte

TB

Trillion bytes

1 099 511 627 776

(240)

Converting data transfer rates: 1 Learn the order of the prefixes. 2 Use the prefixes to determine the conversion factor. 3 An increase in the order of the prefix is a conversion factor of 1000.

Bit per second

statistics

Exact value

Kilobyte

Unit

Digital download

Meaning

Power of two

Symbol bps

Approximate value 1

Kilobit per second

Kbps

1 000

Megabit per second

Mbps

1 000 000

Gigabit per second

Gbps

1 000 000 000

Terabit per second

Tbps

1 000 000 000 000

Use summary statistics (mean, median, mode, range and interquartile range) to interpret the effect of downloading music and video files.

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Review

Chapter summary – Mathematics and communication

409

Review

410

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

A prepaid mobile phone plan charges a call connection fee of $0.32 and voice calls of $0.44 per 30-second block. What is the cost of making a ten-minute call? A $4.40 B $4.72 C $8.80 D $9.12

2

Alice has a mobile phone contract that charges a monthly access fee of $59, free calls $150, flagfall $0.25 and call rate of $0.45 per 30 seconds. What is the monthly charge if Alice made 200 calls of duration less than 30 seconds? A $59.00 B $81.50 C $96.50 D $209.00

3

How many kilobits per second are there in 5000 megabits per second? A 0.005 B 5 C 5 000 000 D 5 000 000 000

4

A file has 3 000 000 bits. How many seconds would it take to download the file at 6 Mbps? A 0.002 B 0.5 C 2 D 500

5

How many bytes in 6 MB? A 1024 B 6144 C 1 048 576 D 6 291 456

6

How many terabytes in 3072 GB? A 0.0029 B 3 C 3 145 728 D 3 221 225 472

7

How many data files of average size 7.2 MB can be stored on a 4 GB USB drive? A 568 B 4096 C 7372 D 582 542

8

The time taken (in seconds) to download 12 files was recorded: 48, 56, 49, 70, 34, 58, 62, 60, 41, 37, 45 and 39. What is the median? A 48.5 B 49 C 49.9 D 50

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1

What is the cost of a call of duration 3 minutes and 40 seconds given that there is a connection fee of 29 cents and the call rate is 45 cents per 30-second block?

2

The table below shows two mobile phone plans. $59 plan

$99 plan

Monthly access fee

$59

$99

Included allowance

$550

$900

Connection fee

$0.35

$0.32

Call rates (per minute)

$0.90

$0.88

Unlimited

Unlimited

Text

3

a

What is the charge for a call lasting 3 minutes and 20 seconds on the $59 plan?

b

What is the charge for a call lasting 3 minutes and 20 seconds on the $99 plan?

c

What is the maximum number of free calls (60 seconds) on the $59 plan?

d

What is the maximum number of free calls (60 seconds) on the $99 plan?

e

Determine the monthly charge for making 400 calls (60 seconds) on the $59 plan?

f

Determine the monthly charge for making 400 calls (60 seconds) on the $99 plan?

g

Determine the monthly charge for making 900 calls (60 seconds) on the $59 plan?

h

Determine the monthly charge for making 900 calls (60 seconds) on the $99 plan?

i

Isaac is deciding on one of the mobile phone plans show in the table above. He will be using his mobile phone to make 600 calls (60 seconds) and 500 SMS messages each month. What plan should he choose?

j

Tahlia is deciding on one of the above mobile phone plans. She will be using her mobile phone to make 300 calls (30 seconds) and 100 SMS messages each month. What plan should she choose?

A mobile phone plan has a monthly charge of $49 on a 24-month contract. The call rate is $0.90 per 60-second block with a $0.35 flagfall. The contract includes $450 of free calls, text and MMS. What is the monthly charge for making 400 voice calls (60 second)?

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Review

Sample HSC – Short-answer questions

Preliminary Mathematics General

Chapter summary Mathematics and communication 4 The phone charges–are described by the step-function graph below. Phone charges 1

Cost (dollars)

Review

412

0.8 0.6 0.4 0.2 1

        a b c d e f

5

6

4

How much is charged in the first minute? What is the charge for each additional minute? What is the charge for a 3-minute call? What is the charge for a 2-minute call? What is the charge for a 1.5-minute call? What is the charge for a 30-second call?

Complete the following. a

7 Kbps =

c

13 Mbps =

e

4 000 000 Mbps =

g

9 000 000 Kbps =

b

2 Tbps =

d

0.005 Tbps =

Tbps

f

1 230 000 Kbps =

Gbps

h

2 800 000 000 bps =

b

GB = 4.5 TB

bps Kbps

Gbps Kbps Mbps Gbps

Complete the following. a

7

3 2 Time (min)

6 TB =

MB

c

B = 3 GB

e

5 242 880 MB =

g

2 147 483 648 B =

TB GB

d

0.2 TB =

f

6144 MB =

h

24 576 KB =

GB GB MB

The time taken (in seconds) to download seven different YouTube videos was 121, 145, 168, 118, 140, 164 and 175. Calculate, correct to two decimal places, the: a mean b median c interquartile range. Challenge questions 13

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C H A P T E R

14

Mathematics and driving Focus study — FSDr Mathematics and driving Calculate the percentage decrease in the value of a vehicle Determine the cost of repayments and total amount repaid on a loan Describe the different types of motor vehicle insurance Calculate the cost of stamp duty on a vehicle Calculate the fuel consumption and running costs of a vehicle Determine the straight-line and declining balance depreciation Use the formula for distance, speed and time, and calculate stopping distance Calculate and interpret blood alcohol content Construct and interpret tables and graphs related to motor vehicles

14.1 Cost of purchase The cost of a purchasing a motor vehicle depends on many factors such as whether it is new or used, the make, the model, whether it is manual or 14.1 automatic, the number of kilometres travelled and the engine size. In addition, the list of installed optional equipment such as alloy wheels or cruise control has an effect on the purchase price. A motor vehicle is not an investment. It decreases in value immediately. In the first year of ownership, a new car can lose up to 20% of its value, and by the fifth year, your car will decrease in price by over 65%. The percentage decrease is determined by dividing the price decrease by the purchase price and multiplying the result by 100. 413 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

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Preliminary Mathematics General

Example 1

Calculating the percentage decrease

A new vehicle is bought for $25 000 and sold one year later for $19 000. Calculate the percentage decrease in the value of the new vehicle. Solution 1 2

3

Subtract $19 000 from $25 000 ($6000). Divide the price decrease ($6000) by the purchase price ($25 000). Express as a percentage (multiply by 100).

$6000 × 100 $25 000 = 24%

Percentage decrease =

Percentage increase is 24%.

Finance Many car dealers allow people to borrow money using the dealer’s finance arrangements. The purchaser pays a deposit and then makes a large number of repayments. The total cost in purchasing a motor vehicle using finance is greater than the sale price for cash. Buying on finance Total cost = Deposit + Total repayments Total repayments = Repayment × Number of repayments Interest paid = Total cost − Sale price

Example 2

Calculating the cost of repayments

A four-wheel drive is for sale at $45 000. Finance is available at $5000 deposit and monthly repayments of $1470 for 5 years. a b c

What is the total cost of the repayments? How much is the total cost using this finance package? What is the interest paid? Solution 1 2 3

Multiply the repayment by the number of repayments. Add the deposit to the total cost of the repayments. Subtract the sale price from the total cost.

a b c

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Total repayment = 1470 × 12 × 5 = $88 200 Total cost = 5000 + 88 200 = $93 200 Interest paid = 93 200 − 45 000 = $48 200

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Chapter 14 — Mathematics and driving

415

Exercise 14A 1

Calculate the percentage decrease in the price of a new vehicle after one year. a Purchase price is $25 500. Market value after one year is $21 420. b Purchase price is $36 800. Market value after one year is $27 600. c Purchase price is $54 250. Market value after one year is $48 825. d Purchase price is $23 826. Market value after one year is $20 900.

2

Calculate the price of the following cars after the trade-in. a Sale price is $35 500. Trade-in is worth $6000. b Sale price is $16 850. Trade-in is worth $2980. c Sale price is $24 120. Trade-in is worth $9460. d Sale price is $64 870. Trade-in is worth $11 820.

3

Calculate the amount of the deposit needed to purchase the following cars. a Sale price is $21 400. Deposit 25%. b Sale price is $19 240. Deposit 15%. c Sale price is $45 100. Deposit 35%. d Sale price is $65 200. Deposit 40%.

4

Calculate the total repayments to purchase the following cars. a Sale price is $14 800. Monthly repayments of $410 for 5 years. b Sale price is $19 240. Monthly repayments of $1120 for 2 years. c Sale price is $45 100. Weekly repayments of $360 for 3 years. d Sale price is $85 200. Weekly repayments of $610 for 4 years.

5

Charlotte has been offered terms to purchase a car. The price of the car is $24 560 or 50% deposit and repayments of $90 per week for 200 weeks.

14A

a b c

What is the amount of the deposit? Find the total cost of the repayments. What is the cost of purchasing the car on terms?

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Preliminary Mathematics General

Development 6

A utility vehicle is for sale at $42 000. Finance is available at $7500 deposit and monthly repayments of $1280 for 5 years. a What is the total cost of repayments? b How much will the car cost if you use the finance package? c What is the interest paid?

7

Jacob has seen a used car he would like to buy, priced at $13 400. He has saved $7000 towards the cost of the car. His parents have offered to lend him the balance to pay for it. Jacob agrees to pay $40 each week to repay his parents. a How much will Jacob need to borrow from his parents? b How long will it take Jacob to repay the loan from his parents?

8

A used car is for sale at $27 000. Finance is available at 10% deposit and monthly repayments of $630 for 4 years. a How much deposit is to be paid? b What is the total cost of repayments? c How much will the car cost if you use the finance package? d What is the interest paid?

9

Emily has two choices of finance packages for a new car. Package A: Deposit of $3000, $1400 per month over 5 years. Package B: No deposit, $1540 per month over 6 years. a Determine the total cost of package A. b Determine the total cost of package B. c How much will be saved by selecting the cheaper package?

10

A prestige car is for sale at $65 000. Finance from the car dealer is available at a deposit of 40% and weekly repayments of $530 for 4 years. A personal loan of $39 000 is available from the bank at 15% p.a. simple interest for 4 years. a How much deposit is required? b What is the interest paid using the finance from the car dealer? c What is the interest paid using the finance from the bank?

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Chapter 14 — Mathematics and driving

11

417

Registration of a motor vehicle involves the payment of a fee. These are not current fees. Registration fees Size of vehicle

Tare weight

Private use

Business use

Cars, station wagons and trucks

up to 975 kg

$218

$321

976 kg to 1154 kg

$239

$354

1155 kg to 1504 kg

$269

$404

1505 kg to 2504 kg

$383

$583

up to 254 kg

$52

$133

255 kg to 764 kg

$143

$200

765 kg to 975 kg

$218

$321

976 kg to 1154 kg

$239

$354

1155 kg to 1504 kg

$269

$404

1505 kg to 2499 kg

$383

$583

$101

$101

Trailers (including caravans)

Motor cycle a b c d

What is the cost of registering a car for private use whose weight is 1000 kg? What is the cost of registering a truck for business use whose weight is 1500 kg? What is the cost of registering a car for business use whose weight is 925 kg? What is the cost of registering a motor cycle for private use?

12 Bailey

is buying a used car for $12 000. He is required to pay a transfer fee of $26 and stamp duty of $360. Finance from the car dealer is available at a deposit of 20% and monthly repayments of $380 for 4 years. How much above the price is Bailey paying the car dealer?

13 Personal



loan calculators are used to determine the monthly repayments.

Use a personal loans calculator with a monthly gross salary of $5000 and monthly expense details of $2000 to determine the maximum amount to be borrowed. a Current variable rate of interest and a loan term of 4 years. b Current variable rate of interest and a loan term of 2 years. c Current fixed rate of interest and a loan term of 4 years. d Current fixed rate of interest and a loan term of 2 years.

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14.2 Insurance Insurance is a major cost in keeping a motor vehicle on the road. There are three main types of insurance: • Green slip or Compulsory Third Party insurance protects vehicle owners and drivers who are legally liable for personal injury to any other party in the event of a personal injury claim made against them by other road users. • Third Party Property insurance covers you for damage caused by your car to property owned by a third party in the event of an accident. • Comprehensive insurance covers you for damage to your own vehicle as well as damage your car may cause to another person’s vehicle or property. Insurance premium is the cost of taking out insurance cover. Many insurance companies offer an online calculator for your vehicle insurance premium. It requires information on the make/ model of car, your age/driving history, finance, modifications/accessories and location. The cost of insurance is also affected by: • No claim bonus is a discount on an insurance premium. This discount increases if no claim is made on the policy until it reaches the maximum discount level. • Excess is paid when a claim is made on the policy. The standard excess can be varied plus there are excesses for younger drivers. Example 3

Calculating the insurance premium

Elle has been quoted $960 for comprehensive car insurance. She has a no claim bonus of 40%. How much is Elle required to pay? Solution 1 2 3 4

No claim discount of 40% requires payment of 60%. Calculate 60% of $960. Evaluate. Write the answer in words.

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Premium = 60% of $960 = 0.60 × 960 = $576 Elle is required to pay $576.

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419

Exercise 14B 1

What is the cost of comprehensive car insurance for the following premiums? a Premium of $1080 with a no claim bonus of 60%. b Premium of $1690 with a no claim bonus of 30%. c Premium of $880 with a no claim bonus of 40%. d Premium of $1320 with a no claim bonus of 70%. e Premium of $2350 with a no claim bonus of 50%.

2

The graph below shows the percentage of claims for each age group.

14B

Percentage making a claim

Insurance claims 70 60 50 40 30 20 10 0 a b c d

3

10

20

30 40 Age in years

50

60

70

What is the percentage of claims for people 50 years old? What age group made the least number of claims? Calculate the gross percentage change in claims between the ages of 60 and 70. How do insurance companies cater for the large number of claims made by people 20 years old?

Dan is 20 years old and has received this quote for comprehensive insurance.

a b

Premium details

Excesses

Cost

12 month policy $678.00 30% No claim bonus

Standard

$500

Male under 21

$1200

Female under 21

$900

Calculate the cost of the insurance. What is Dan’s excess if he makes a claim?

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Development 4

Connor has been quoted an insurance premium of $980.60 from his insurance company. The company had given him a 20% no-claim bonus as he had not made a claim in the previous year. What would the insurance premium have been without his no-claim bonus?

5

The sector graph shows the road crash costs according to categories. The total insurance cost was $1.2 billion. a What is the insurance cost for minor injury? b What is the insurance cost for serious injury? c What is the insurance cost for a fatal accident? d What is the insurance cost for property damage?

6 

19.8

21.7

8.3 50.2

Fatal

Minor injury

Serious injury

Property damage

The premiums quoted below are for clients with a maximum no claim bonus. The car is owned outright by a mature age driver and driven for private use.



Mosman

Penrith

Mosman

Brand A

$20 100

$540

$605

$600

$760

Brand B

$38 890

$810

$899

$770

$1500

Brand C

$24 400

$615

$650

$615

$860

b c d e f g h j

Premium B

Model of car

a

i

Premium A

Agreed value

Penrith

Which suburb has the highest premium? Suggest a reason. Do expensive cars have higher premiums? What is the best quote for the Brand B? What is the best quote for the Brand A? Which model has the lowest premium? What is the average premium for the Brand B? What is the average premium for the Brand C? What is the average premium for Mosman? What is the average premium for Penrith? Premium A is being increased by 3%. What would be the new premium for a Brand A car at Mosman?

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14.3 Stamp duty Stamp duty is the tax you pay to the government when registering or transferring a motor vehicle. The amount of stamp duty payable is based on the price of the motor vehicle. For example, a new passenger car purchased for $40 000 would require a duty of $3 per $100 or $1200 (0.03 × 40 000). That is, for every $100 you paid for the vehicle, the stamp duty is $3 or a tax of 3%.

Stamp duty on vehicles 1 2 3

Round up the cost of the vehicle to the nearest $100 (per $100), $200 (per $200) etc. Express the stamp duty as a fraction or decimal. ($3 per $100 is 3/100 or 0.03.) Multiply the answer obtained in step 1 by the fraction or decimal obtained in step 2.

Example 4

Calculating stamp duty on vehicles

A used car is bought for $17 730. Calculate the stamp duty payable if the charge is $3 per $100 or part $100. Solution 2

Round $17 730 up to the nearest $100. Express the stamp duty as a fraction.

3

Multiply $17 800 by

4

Evaluate. Write the answer in words.

1

5

3 . 100

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Value of vehicle = $17 800 $3 per $100 is the fraction Stamp duty = 17 800 ×

3 . 100

3 100

= $534 Stamp duty payable is $534.

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Exercise 14C 1

The table below is used to calculate the stamp duty payable on a vehicle.

14C

Value of vehicle

Stamp duty payable

$0–$45 000

$3 per $100 or part $100

More than $45 000

$1350 plus $5 per $100 (or part $100) over $45 000

Calculate the stamp duty payable on the following vehicles. a $32 600 b $26 500 c $45 000 d $13 790 e $35 521 f $23 802 g $52 700 h $98 435 i $120 080 2

The table below is used to calculate the stamp duty payable on a vehicle. Value of vehicle

Stamp duty payable

$0–$60 000

$5 per $200 or part $200

More than $60 000

$1500 plus $7 per $200 (or part $200) over $60 000

Calculate the stamp duty payable on the following vehicles. a $13 200 b $29 790 c $45 410 d $73 800 e $61 670 f $88 605 g $57 326 h $79 190 i $91 456 3

The table below is used to calculate the stamp duty payable on a used vehicle. Value of vehicle Passenger

Non-passenger

All prices

$7 per $300 or part $300

$5 per $300 or part $300

Calculate the stamp duty payable on the following vehicles. a Passenger car $21 300. b Passenger car $69 500. c Non-passenger car $45 880. d Non-passenger car $36 614.

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Development 4

Stamp duty is calculated at 3% of the market value of a vehicle up to $45 000, plus 5% of the value of the vehicle over $45 000. Use the following graph to answer the questions below. 3500 Stamp duty ($)

3000 2500 2000 1500 1000 500 0

a b c d e f

5 

20 000 40 000 60 000 80 000 Market Value ($)

How much stamp duty is payable on a car whose market value is $20 000? How much stamp duty is payable on a car whose market value is $60 000? How much stamp duty is payable on a car whose market value is $45 000? How much stamp duty is payable on a car whose market value is $70 000? What is the market value if the stamp duty paid was $300? What is the market value if the stamp duty paid was $2300?

Construct a line graph to represent the following stamp duty charge. Stamp duty is calculated at 2.5% of the market value of a vehicle up to $60 000, plus 4% of the value of the vehicle over $60 000. Use your graph to answer the questions below. a How much stamp duty is payable on a car whose market value is $30 000? b How much stamp duty is payable on a car whose market value is $60 000? c How much stamp duty is payable on a car whose market value is $80 000? d What is the market value if the stamp duty paid was $500? e What is the market value if the stamp duty paid was $1000? f What is the market value if the stamp duty paid was $3000?

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14.4 Running costs (fuel) The running costs covered here are fuel costs, which depend on the price of fuel and the fuel consumption. A motor vehicle’s fuel consumption is the number of litres of fuel it uses to travel 100 kilometres. The fuel consumption is calculated by filling the motor vehicle with fuel and recording the kilometres travelled from the odometer. When the motor vehicle is again filled with fuel, record the reading from the odometer and how many litres of fill it takes to refill the tank. The distance travelled is the difference between the odometer readings. Fuel consumption Fuel consumption =

Amount of fuel (L) × 100 Distance travelled (km)

The cost of fuel for a journey can be calculated from the price of fuel ($/L) multiplied by the amount of fuel used (L). Fuel prices can be found in your local area from websites such as the one shown below.

Example 5

Calculating fuel consumption

A medium-sized car travelled 750 km using 60 L of petrol. What was the fuel consumption? Solution 1 2 3 4

Write the fuel consumption formula. Substitute 60 for the amount of fuel and 750 for the distance travelled. Evaluate. Write the answer in words.

Amount of fuel × 100 Distance travelled 60 × 100 = 750 = 8.0 L / 100 km

Fuel Consumption =

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Exercise 14D 1

Calculate the fuel consumption (litres per 100 km) for each of the following: a Abbey’s car uses 38.2 litres of petrol to travel 400 km. b A sports car travelled 900 km using 79.38 litres of petrol. c Joel’s sedan uses 30.36 litres of LPG to travel 600 km. d A small car uses 41.05 litres of petrol to travel 500 km. e Lucy’s car uses 139.8 litres of petrol to travel 1500 km. f Max’s motorbike uses 70 litres of LPG to travel 2000 km.

2

Chelsea has bought a used car whose fuel consumption is 10 litres of petrol per 100 kilometres. She is planning to travel around Australia. Calculate the number litres of petrol Chelsea’s car will use on the following distances: a A trip of 2716 km from Perth to Adelaide. b A trip of 732 km from Adelaide to Melbourne. c A trip of 658 km from Melbourne to Canberra. d A trip of 309 km from Canberra to Sydney. e A trip of 982 km from Sydney to Brisbane. f A trip of 3429 km from Brisbane to Darwin. g A trip of 4049 km from Darwin to Perth.

3

Riley’s car uses 11.26 litres of petrol per 100 km. a How many litres of petrol will his car use on a trip of 155 km from Sydney to Newcastle? b The petrol cost is $1.60 cents per litre. How much will the petrol cost for the trip?

4

Sienna filled her car with petrol. The odometer read 64 080 km at that time. When she next filled the petrol tank, the odometer read 64 605 km. The car took 42 L of petrol. a How far has the car travelled between fills? b What was the average fuel consumption in kilometres per litre?

14D

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Development 5

Stephanie travels 37 km to work and 37 km from work each day. a How many kilometres does she travel to and from work in a 6-day working week? b Stephanie drives an SUV with a fuel consumption of 8.38 L/100 km to and from work. How many litres of petrol does Stephanie use travelling to and from work in a week? Answer correct to one decimal place. c What is Stephanie’s petrol bill for work if petrol costs are $1.35 per litre?

6

A family car uses LPG at a rate of 15 L/100 km and the gas tank holds 72 litres. How far can it travel on a tank of LPG?

7

Grace drives a four-wheel drive whose petrol consumption is 15.2 L/100 km and the petrol tank is 95 litres. She is planning a trip from Sydney to Bourke via Dubbo. The distance from Sydney to Dubbo is 412 km and from Dubbo to Bourke is 360 km. Grace filled her petrol tank at Sydney. How many times will she need to fill her tank before arriving at Bourke? Give reasons for your answer.

8

The graph below shows a motor vehicle’s fuel consumption at various speeds. Petrol used to travel 200 km 50 Litres

40 30 20 10 0

a b c d e f

30

50

70 90 Speed (km/h)

110

How many litres of fuel were used at 70 km/h? How many litres of fuel were used at 110 km/h? What is the fuel consumption rate at 30 km/h? What is the fuel consumption rate at 90 km/h? What speed used fuel the most efficiently? How many litres of fuel were saved by travelling at 90 km/h instead of 110 km/h?

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9

Dylan owns a V8 car with a fuel consumption of 11 L/100 km in the city and 8 L/100 km in the country. Dylan travels 8000 km per year in the city and 10 000 km per year in the country. The average cost of petrol is $1.50 per litre in the city and 10 cents higher in the country. a Determine the cost of petrol to drive in the city for the year. b Determine the cost of petrol to drive in the country for the year. c What is the total cost of petrol for Dylan in one year? d What is the total cost of petrol for Dylan in one year if the average cost of petrol increased to $1.80 per litre in the city?

10

Holly is planning a trip from Sydney to Brisbane using a car with a fuel consumption of 13 litres/100 km. The distance from Sydney to Brisbane via the Pacific Highway is 998 km and via the New England Highway it is 1027 km. The cost of LPG is 79.2 cents per litre. a How much will the trip cost via the Pacific Highway? b How much will the trip cost via the New England Highway? c How much money is saved by travelling via the Pacific Highway?

11

Tyler buys a new car with a fuel consumption of 11.2 litres/100 km. Oscar buys the LPG version of Tyler’s new car, with a fuel consumption 15.4 litres/100 km. Both Tyler and Oscar average 300 km in a week in the same conditions. The average price of ULP is $1.40 cents/litre and LPG is $0.79 cents per litre. a How many litres of fuel are used by Tyler in a week? b How many litres of fuel are used by Oscar in a week? c Calculate each car’s yearly consumption of fuel. d What is Tyler’s yearly fuel bill? e What is Oscar’s yearly fuel bill? f Oscar paid an additional $1500 for the LPG version of the Ford Falcon. How many years will it take for the fuel savings to reach $1500 or the break-even point? Answer correct to the nearest whole number. g Research the current fuel prices of ULP and LPG. How long will it take for the fuel saving to exceed the initial costs?

12

Investigate the costs for two common cars on a family trip in your local area. Calculate the cost for the return trip in each case. You will need to determine the distance of the trip, fuel consumption for each car and the average price of fuel in the local area.

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14.5 Straight line depreciation Straight line depreciation occurs when the value of the item decreases by the same amount each period. For example, if you buy a car for $20 000 and it depreciates by $2000 each year, the value of the car after one year is $20 000 − $2000 or $18 000. After the second year the value of the car is $20 000 − $2000 − $2000 or $16 000. Straight line depreciation S = V0 − Dn S – Salvage value or current value of an item. Also referred to as the book value. V0 – Purchase price of the item. Value of the item when n = 0. D – Depreciated amount per time period. n – Number of time periods.

Example 6

Calculating the straight line depreciation

Molly pays $14 500 for a used car. It depreciates $1100 each year. How much will it be worth after three years? Solution 1 2 3

Write the straight line depreciation formula. Substitute V0 = 14 500, D = 1100 and n = 3 into the formula. Evaluate. Write the answer in words.

Example 7

S = V0 − Dn = 14 500 − 1100 × 3 = $11 200 The value of the car is $11 200.

Calculating the salvage value

A new car is purchased for $25 800. After 4 years its salvage value is $15 160. What is the annual amount of depreciation, if the amount of depreciation is constant? Solution 2

Write the straight-line depreciation formula. Substitute V0 = 25 800, S = 15 160 and n = 4 into the formula.

3

Evaluate.

15160 = 25800 − D × 4 25800 − 15160 D= 4 = $2660

4

Write the answer in words.

Annual depreciation is $2660.

1

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S = V0 − Dn

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Exercise 14E 1

Mia bought a used car for $15 200. She estimates that her car will depreciate in value by $3040 each year. a What is the loss in value (depreciation) during the first year? b What is the value of the car at the end of the first year? c What is the loss in value (depreciation) during the second year? d What is the value of the car at the end of the second year? e What is the loss in value (depreciation) during the third year? f What is the value of the car at the end of the third year?

2

Harrison pays $9500 for a motor bike. It depreciates $850 each year. What will be the value of the bike after: a three years? b five years? c seven years? d nine years?

3

Patrick buys a car for $55 500 and it is depreciated at a rate of 10% of its purchase price each year. What is the salvage value of the car after four years?

4

The graph shows the depreciation of a car over four years. a What is the initial value? b How much did the car depreciate each year? c What is the value of the car after 3 years? d When was the car worth $8000? e What is the value of the car after 1 3 years? 2 f What is the value of the car after 6 months?

16000 12000 Value ($)

14E

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8000 4000

0

1

2 Years

3

4

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Development 5

Ryan bought a commercial van three years ago. It has a salvage value of $36 000 and depreciated $4650 each year. How much did Ryan pay for the van?

6

Lucy bought a used car four years ago. It has a salvage value of $16 400 and depreciated $1250 each year. How much did Lucy pay for the used car?

7

Ethan has a car worth $9220. It depreciates by $420 each year. a When will the car be worth $5440? b When will the car be worth $3340?

8

A ute is purchased for $18 600. After two years it has depreciated to $14 800 using the straight line method of depreciation. a When will the ute be worth $3400? b When will the ute be worth $1500?

9

A truck is purchased new for $64 000. After 3 years its market value is $44 800. a What is the annual amount of depreciation, if the amount of depreciation is constant? b Determine the book value of the truck after 7 years.

10

Grace bought an SUV costing $38 000. It is expected that the SUV will have an effective life of 10 years and then be sold for $14 000. Assume the SUV depreciated by the same amount each year. What is the annual depreciation?

11

A utility van is purchased new for $24 000. After 3 years its book value is $15 000. What is the annual amount of depreciation, if the amount of depreciation is constant?

12

A caravan is bought for $82 000. It is expected to be used for 4 years and then sold for $50 000. Assume the caravan depreciates by the same amount each year. a How much does the caravan depreciate each year? b What is the total amount of depreciation for 4 years? c Copy and complete the following depreciation table for the first four years. Year

Current value

Depreciation

Depreciated value

1 2 3 4 d

Graph the value in dollars against the age in years.

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14.6 Declining balance depreciation Declining balance depreciation occurs when the value of the item decreases by a fixed percentage each time period. For example, if you buy a car for $20 000 and it depreciates by 10% each year then the value of the car after one year is $20 000 − $2000 or $18 000. After 14.6 the second year the value of the car is $20 000 − $2000 − $1800 or $16 200. Notice that the amount of depreciation has decreased in the second year. Declining balance depreciation S = V0(1 − r)n S – Salvage value or current value of an item. Also referred to as the book value. V0 – Purchase price of the item. Value of the item when n = 0. r – Rate of interest per time period expressed as a decimal. n – Number of time periods.

Example 8

Calculating the declining balance depreciation

Eva purchased a new car two years ago for $32 000. During the first year it had depreciated by 25% and during the second it had depreciated 20% of its value after the first year. What is the current value of the car?

Solution 1 2 3 4 5 6 7

Write the declining balance depreciation formula. Substitute V0 = 32 000, r = 0.25 and n = 1 into the formula. Evaluate. Write the declining balance depreciation formula. Substitute V0 = 24 000, r = 0.20 and n = 1 into the formula. Evaluate. Write the answer in words.

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First year S = V0(1 − r)n = 32 000 × (1 − 0.25)1 = $24 000 Second year S = V0(1 − r)n = 24 000 × (1 − 0.20)1 = $19 200 Current value is $19 200.

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Example 9

Calculating the salvage value

Angus buys a car that depreciates at the rate of 26% per annum. After five years the car has a salvage value of $17 420. How much did Angus pay for the car, to the nearest dollar? Solution 1 2 3 4 5 6

Write the declining balance depreciation formula. Substitute S = 17 420, r = 0.26 and n = 5 into the formula. Make V0 the subject of the equation. Evaluate. Express the answer correct to the nearest whole dollar. Write the answer in words.

Example 10

S = V0(1 − r)n 17 420 = V0 × (1 − 0.26)5 17 420 (1 − 0.26 )5 = $78 503.596 21

V0 =

= $78 504 Angus paid $78 504 for the car.

Calculating the percentage rate of depreciation

Madison bought a delivery van four years ago for $27 500. Using the declining balance method for depreciation, she estimates its present value to be $8107. What annual percentage rate of depreciation did she use? Answer to the nearest whole number. Solution

3

Write the declining balance depreciation formula. Substitute S = 8107, V0 = 27 500 and n = 4 into the formula. Make (1 − r)4 the subject of the equation.

4

Take the fourth root of both sides.

5

Rearrange to make r the subject. Evaluate. Express the answer correct to the nearest whole number. Write the answer in words.

1 2

6 7 8

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S = V0(1 − r)n 8107 = 27 500 × (1 − r)4 (1 −r )4 =

1− r =

8107 27500

4

8107 27500

8107 27500 = 0.263 145 28 = 26%

r = 1− 4

Rate of depreciation is 26%.

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Exercise 14F 1

A motor vehicle is bought for $22 000. It depreciates at 16% per annum and is expected to be used for 5 years. What is the salvage value of the motor vehicle after the following time periods? Answer to the nearest cent. a one year b two years c three years

2

Emma purchased a used car for $6560 two years ago. Use the declining balance method to determine the salvage value of the used car if the depreciation rate is 15% per annum. Answer to the nearest dollar.

3

Bailey purchased a motor cycle for $17 500. It depreciates at 28% per year. Answer to the nearest dollar. a What is the book value of the motor cycle after three years? b How much has the motor cycle depreciated over the three years?

4

A new car is bought for $52 000. It depreciates at 22% per annum and is expected to be used for 4 years. How much has the car depreciated over the 4 years? Answer to the nearest dollar.

5

Chloe purchased a car for $19 900. It depreciates at 24% per year. Answer to the nearest dollar. a What is the salvage value of the car after five years? b How much has the car depreciated over the five years?

6

The depreciation of a used car over four years is shown in the graph opposite. a What is the initial value of the used car? b How much did the used car depreciate during the first year? c When is the value of the used car $2000? d When is the value of the used car $1500? e What is the value of the used car after 4 years? f What is the value of the used car 1 after 1 years? 2

4000 3000 Value ($)

14F

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2000 1000

0

1

2 Years

3

4

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Preliminary Mathematics General

Development 7

A hatchback vehicle was purchased for $16 980 three years ago. By using the declining balance method of depreciation, the current value of the vehicle is $9614. What is the annual percentage rate of depreciation, correct to two decimal places?

8

A new car is valued at $35 000. After one year using the declining balance method, it is valued at $25 500. a Determine the annual percentage rate of depreciation. Answer correct to 3 decimal places. b What is the value of the new car after three years? Answer correct to the nearest dollar.

9

Philip and Amy spent $200 000 on a luxury car 7 years ago. Its current value is $104 350. Using the declining balance method, find the percentage depreciation rate over this period. Answer correct to one decimal place.

10

Jessica invested $18 820 to buy a new car for her business. How many years would it take for this car to depreciate to $4520? Assume declining balance method of depreciation with a rate of depreciation of 30%. (Answer to the nearest year.)

11

A motor vehicle is bought for $32 000. It depreciates at 16% per annum and is expected to be used for 5 years. a How much does the motor vehicle depreciate in the first year? b Copy and complete the following depreciation table for the first five years. Answer to the nearest dollar. Year

Current value

Depreciation

Depreciated value

1 2 3 4 5 c

Graph the value in dollars against the age in years.

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14.7 Safety Distance, speed and time Speed is a rate that compares the distance travelled to the time taken. 14.7 The speed of a car is measured in kilometres per hour (km/h). The speedometer in a car measures the instantaneous speed of a car. They are not totally accurate but have a tolerance of 5%. GPS devices are capable of showing speed readings based on the distance travelled per one-Hertz interval. Most cars also have an odometer to indicate the distance travelled by a vehicle. Distance, speed and time D D or T = or D = S × T T S D – Distance S – Speed T – Time S=

Example 11 a b

Road sign on the right is used to remember the formulas. Hide the required quantity to determine the formula.

D S

T

Finding the distance, speed and time

Find the distance travelled by a car whose average speed is 65 km/h if the journey lasts 5 hours. (Answer correct to the nearest kilometre.) How long will it take a vehicle to travel 150 km at a speed of 60 km/h? Solution 1 2 3 4 5

6

Write the formula. Substitute 65 for S and 5 for T into the formula. Evaluate and express answer correct to the nearest kilometre. Write the formula. Substitute 150 for D and 60 for S into the formula. Evaluate and express answer correct to the nearest hour.

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a

b

D=S×T = 65 × 5 = 325 km D T= S 150 = 60 = 2.5 h

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Preliminary Mathematics General

Stopping distance The stopping distance is the distance a vehicle travels from the time a driver sees an event occurring to the time the vehicle is brought to a stop. It is calculated by adding the reaction distance and the braking distance. Reaction distance (or thinking distance) is the distance travelled by the vehicle when a driver decides to brake to when the driver first commences braking. The reaction time averages 0.75 second for a fit and alert driver. The braking distance is affected by the road surface (wet, slippery, uneven or unsealed), slope of the road (uphill or downhill), weight of the vehicle and condition of the brakes. Stopping distance Stopping distance = Reaction distance + Braking distance 5Vt V 2 d= + (formula is an approximation using average conditions) 18 170 d – Stopping distance in metres. V – Velocity or speed of the motor vehicle in km/h. t – Time reaction in seconds.

Example 12

Calculating the stopping distance

Tahlia was driving at a speed of 45 km/h and reaction time of 0.75 seconds. Calculate the stopping distance using the formula 5Vt V 2 d= + . 18 170 Answer correct to the nearest whole metre.

Solution 1 2

3 4 5

Write the stopping distance formula. Substitute V = 45 and t = 0.75 into the formula. Evaluate. Express the answer correct to one decimal place. Write the answer in words.

5Vt V 2 + 18 170 5 × 45 × 0.75 452 = + 18 170 = 21.28676471 ≈ 21 m

d=

Stopping distance is 21 m.

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Exercise 14G 1

Find the average speed (in km/h) of a vehicle which travels: a 180 km in 2 hours b 485 km in 5 hours 1 c 360 km in 4.5 hours d 21 km in hour 4 1 e 240 km in 2 hours f 16 km in 20 minutes 2

2

Find the distance travelled by a car whose average speed is 56 km/h if the journey lasts (answer correct to the nearest kilometre): a 3 hours b 7 hours 1 c 2.6 hours d 1 hours 4 1 3 e 3 hours f 2 hours 2 4

3

How long will it take a vehicle to travel (answer correct to the nearest hour): a 160 km at a speed of 80 km/h b 150 km at a speed of 60 km/h c 120 km at a speed of 48 km/h d 225 km at a speed of 45 km/h e 240 km at a speed of 40 km/h f 556 km at a speed of 69.5 km/h

4

The Melbourne Formula 1 track is 5.303 km in length. The track record is 1 minute and 24 seconds. What is the average speed (km/h) for the lap record? Answer correct to two decimal places.

5

Caitlin lives in Wollongong and travels to Sydney daily. The car trip requires her to travel at different speeds. Most often she travels 30 kilometres at 60 km/h and 40 kilometres at 100 km/h. a What is the total distance of the trip? b How long (in hours) does the trip take? c What is her average speed (in km/h) when travelling to Sydney? (Answer correct to two decimal places.)

6

Thomas drives his car to work 3 days a week. The length of the trip is 48 km. The trip took 43 minutes on Monday, 50 minutes on Tuesday and 42 minutes on Wednesday. a Calculate the average time taken to travel to work. b What is the average speed (in km/h) for the three trips?

14G

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7

The graph opposite shows the reaction distance and the braking distance.

0

5

Metres 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

50 km/h

Reaction

55 km/h

Braking

60 km/h



Travelling at 60 km/h: 65 km/h a what is the reaction 70 km/h distance? b what is the braking distance? c what is the stopping distance?

8

What is the stopping distance for each of the following? a Reaction-time distance of 25 metres and braking distance of 22 metres. b Reaction-time distance of 19 metres and braking distance of 30 metres.

9

Michael is driving with a reaction time of 0.75 seconds. Calculate the stopping distance 5Vt V 2 (to the nearest metre) using the formula d = for each of the following speeds. + 18 170 a 30 km/h b 50 km/h c 70 km/h d 90 km/h e 110 km/h f 130 km/h

10

Sarah was driving her car at 40 km/h through a school zone (reaction time is 0.50 seconds). A school student ran onto the road 12 metres in front of her. a Do you think Sarah was able to stop without running over the child? Give a reason for your answer. b What would have happen if Sarah was driving her car at 60 km/h? Explain your answer.

11

Oliver uses the freeway to travel to work. His reaction time is 0.60 seconds. Oliver usually drives at the speed limit of 110 km/h. 5Vt V 2 d = + ? a What is the stopping distance on the freeway using the formula 18 170 b Determine a safe distance between cars on the freeway that are travelling at 110 km/h. Give a reason for your answer.

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Development 12

Find the average speed (in km/h) of a vehicle that travels (answer correct to the nearest whole number): a 500 km in 6 hours and 10 minutes b 64 km in 1 hour and 30 seconds c 36 000 m in 45 minutes d 320 m in 10 seconds

13

Find the distance travelled by a car whose average speed is 68 km/h if the journey lasts (answer correct to the nearest kilometre): a 30 minutes b 2 minutes c 1 hour and 20 minutes d 4 hours 10 seconds

14

How long will it take a vehicle to travel (answer correct to the nearest minute): a 450 km at a speed of 82 km/h b 50 km at a speed of 60 km/h c 250 km at a speed of 49 km/h d 580 000 m at a speed of 62 km/h e 24 000 m at a speed of 72 km/h f 100 km at a speed of 1 km/h

15

The land speed record is 20.4 km/min. a Express this speed in km/h. b How far does this vehicle travel in 5 minutes? c How far does this vehicle travel in 1 second? d How long would it take for this vehicle to travel from Sydney to Brisbane (982 km)? Answer to the nearest minute.

16

The Bathurst 1000 motor race has a lap record of 2 min and 12.339 seconds. The length of the lap is 6.213 km. a What is the average speed (to nearest km/h) for the lap record? b How long is the race if the winning car travels the 161 laps at the average speed for the lap record? Answer to the nearest minute.

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17

‘If you double your speed you need to double your reaction distance.’ 5Vt to complete the table. Assume reaction time of 0.75 seconds. a Use d = 18 Speed (km/h)

Reaction distance (m)

  10   20   40   80 b 18

Do you agree with the above statement? Give a reason.

‘If you double your speed you need to quadruple your braking distance.’ V2 to complete the table. a Use d = 170 Speed (km/h)

Braking distance (m)

  10   20   40   80 b

Do you agree with the above statement? Give a reason.

19

Joshua is driving with a speed of 30 km/h. 5Vt V 2 + with t as the subject. a Write the formula d = 18 170 b Find the value of t when d = 10 metres. Answer correct to one decimal place. c Find the value of t when d = 20 metres. Answer correct to one decimal place. d Find the value of t when d = 30 metres. Answer correct to one decimal place.

20

Liam is driving at a speed of 60 km/h. 5Vt a Use the formula d = to complete the table below. 18 Reaction time (sec)

Reaction distance (m)

0.50 1.00 1.50 b

What effect does increasing the reaction time have on the stopping distance? Use the calculations in the above table to reach your conclusion.

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14.8 Blood alcohol content Blood alcohol content (BAC) is a measure of the amount of alcohol in your blood. The measurement is the number of grams of alcohol in 100 millilitres of blood. For example, a BAC 0.05 means 0.05 g or 50 mg of alcohol in every 100 mL of blood. BAC is influenced by the number of standard drinks consumed in a given amount of time and a person’s mass. Other factors that affect BAC include gender, fitness, health and liver function. Blood alcohol content (BAC) BACMale = BAC N – H – M –

(10 N − 7.5 H ) (10 N − 7.5 H ) or BACFemale = 6.8 M 5.5 M

– Blood alcohol content. Number of standard drinks consumed. Hours drinking. Mass in kilograms.

Example 13

Calculating the BAC

Madison is 82 kg and has consumed 7 standard drinks in the past two hours. She was stopped by police for a random breath test. What would be Madison’s BAC? Answer correct to 3 decimal places.

Solution 1

Write the formula.

2

Substitute the 7 for N, 2 for H and 82 for M into the formula. Evaluate. Express the answer correct to 3 decimal places. Write the answer in words.

3 4 5

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(10 N − 7.5 H ) 5.5 M (10 × 7 − 7.5 × 2 ) = (5.5 × 822 ) = 0.121 951 219 5 ≈ 0.122

BACFemale =

Madison’s BAC is 0.122.

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NSW has three blood alcohol limits: zero, 0.02 and 0.05. Zero or 0.02 BAC laws apply in Australia for people under 25 who have held a licence for less than three years, including learner and probationary drivers. This means you cannot drink at all and then drive, as you will be over the limit and likely to lose your licence. The BAC is measured with a breathalyser or by analysing a sample of blood. Hours to wait before driving BAC 0.015 BAC − Blood alcohol content. Number of hours =

Using BAC tables

Example 14

The table below shows BAC and body weight (kg). Body weight (kg) Drinks

45

55

65

75

85

1

0.008

0.007

0.006

0.005

0.004

2

0.041

0.033

0.028

0.025

0.022

3

0.074

0.060

0.051

0.044

0.039

4

0.106

0.087

0.074

0.064

0.056

5

0.139

0.114

0.096

0.083

0.074

6

0.172

0.140

0.119

0.103

0.091

Terry weighs 65 kg and consumes four standard drinks in an hour. Calculate the number of hours to wait before driving. (Answer to the nearest hour.) Solution 1 2 3 4 5

Write the formula. Substitute the 0.074 for BAC into the formula. Evaluate. Write answer correct to nearest hour. Write the answer in words.

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BAC 0.015 0.074 = 0.015 = 4.933 33 ≈ 5

Number of hours =

Terry waits 5 hours to drive.

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Exercise 14H 1

Calculate the BAC for the following females. Answer correct to two decimal places. a Sarah is 48 kg and has consumed 4 standard drinks in the past 2 hours. b Sienna is 59 kg and has consumed 3 standard drinks in the past hour. c Alyssa is 81 kg and has consumed 6 standard drinks in the past 2 hours. d Kayla is 65 kg and has consumed 8 standard drinks in the past 6 hours. e Tahlia is 71 kg and has consumed 13 standard drinks in the past 3 hours. f Mia is 55 kg and has consumed 9 standard drinks in the past 5 hours.

2

Calculate the BAC for the following males. Answer correct to two decimal places. a Dylan is 53 kg and has consumed 3 standard drinks in the past 3 hours. b Riley is 64 kg and has consumed 5 standard drinks in the past hour. c Thomas is 98 kg and has consumed 2 standard drinks in the past 2 hours. d Zachary is 47 kg and has consumed 10 standard drinks in the past 5 hours. e Charlie is 85 kg and has consumed 12 standard drinks in the past 4 hours. f Jacob is 104 kg and has consumed 7 standard drinks in the past 6 hours.

3

James and Olivia are twins and both weigh 73 kg. At a party they consume 6 standard drinks in two hours.

14H

a b c 4

What is James’s BAC? Answer correct to 2 decimal places. What is Olivia’s BAC? Answer correct to 2 decimal places. How long does James need to wait before he drives home?

Calculate the number of hours to wait before driving. Answer to the nearest minute. a BAC of 0.056. b BAC of 0.123. c BAC of 0.087. d BAC of 0.153. e BAC of 0.092. f BAC of 0.172.

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5

The table below shows the BAC after one hour. Body weight (kg)



Drinks

45

55

65

75

85

95

105

115

1

0.008

0.007

0.006

0.005

0.004

0.004

0.004

0.003

2

0.041

0.033

0.028

0.025

0.022

0.019

0.018

0.016

3

0.074

0.060

0.051

0.044

0.039

0.035

0.032

0.029

4

0.106

0.087

0.074

0.064

0.056

0.050

0.046

0.042

5

0.139

0.114

0.096

0.083

0.074

0.066

0.060

0.054

6

0.172

0.140

0.119

0.103

0.091

0.081

0.074

0.067

Calculate the number of hours to wait before driving. (Answer to the nearest minute.) a Joshua weighs 85 kg and consumes 5 standard drinks in an hour. b Mitchell weighs 115 kg and consumes 3 standard drinks in an hour. c Harrison weighs 45 kg and consumes 6 standard drinks in an hour. d Cooper weighs 65 kg and consumes 2 standard drinks in an hour. e Zachary weighs 95 kg and consumes 4 standard drinks in an hour. f Angus weighs 75 kg and consumes 1 standard drink in an hour. 6

Use the above table to construct three separate column graphs. Make the number of drinks the horizontal axis and the BAC the vertical axis. a Body weight of 45 kg. b Body weight of 115 kg.

7

The formula for calculating ‘standard drinks’ is S = V × A × 0.789 where S is the number of standard drinks, V is the volume of drink in litres and A is the percentage of alcohol. How many standard drinks are in each of the following drinks? Answer correct to one decimal place. a 345 mL bottle of full strength beer at 5.2% alcohol. b 750 mL bottle of champagne at 13.5% alcohol. c 150 mL glass of white wine at 12.5% alcohol. d Mixed drink with a 30 mL of brandy at 38% alcohol. e 360 mL can of light beer at 2.1% alcohol.

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Development 8

a b

9 

b

11

(10 N − 7.5 H ) if: 6.8 M BACMale = 0.066, M = 60 and N = 5. (Answer correct to the nearest minute.) BACMale = 0.050, M = 79 and N = 7. (Answer correct to the nearest minute.)

Find the value of H in the formula BACMale = a

10

BAC if: 0.015 Number of hours to wait before driving is 5. (Answer correct to 3 decimal places.) Number of hours to wait before driving is 3. (Answer correct to 3 decimal places.)

Find the value of BAC in the formula Number of hours =

Find the value of N in the formula BACFemale =

(10 N − 7.5 H ) if: 5.5 M

a

BACFemale = 0.066, M = 48 and H = 2. (Answer correct to one decimal place.)

b

BACFemale = 0.120, M = 57 and H = 4. (Answer correct to one decimal place.)

The graph below relates the lifetime risk of death to the number of standard drinks consumed per day.

Lifetime risk per 100

10 9

Men

8

Women

7 6 5 4 3 2 1 0

a b

1

2

3

4 5 6 7 Australian standard drinks per day

8

9

10

What is lifetime risk for a female and a male who consumes 7 drinks per day? Why is the effect of alcohol greater on a female than on a male?

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Preliminary Mathematics General

14.9 Driving statistics Motor vehicle tables and graphs A distance-time graph describes a journey involving different events. Each event is a line segment on the distance-time graph and represents travelling at a constant speed. The steeper 14.9 the line segment the faster the object is travelling. If the distance-time graph has a horizontal line then the object is not moving or is at rest. Distance-time graphs Line graph with time on the horizontal axis and distance on the vertical axis. Vertical Rise Distance = = Speed (velocity). 1 Gradient of the line = Horizontal Run Time 2 Horizontal line indicates that the object is stationary or not moving.

Example 15

Reading a distance-time graph

Distance (km)

The distance-time graph describes a car trip taken last Sunday. a How long was the rest stop? 140 b How far did the car travel from 120 its starting point? c What was the total distance 100 travelled? 80 d Determine the average speed 60 during the first hour of the trip. 40 20 0

1

2 3 Time (h)

4

Solution 1 2 3 4

Car is at rest when it is not travelling (horizontal line). Largest value for distance. (140 km). The car has travelled on a trip of 140 km and returned. Average speed is distance travelled divided by the time taken.

a

Time for rest stop is 1 hour.

b

Distance is 140 km.

c

Total distance = 140 × 2 = 280 km

d

S=

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D 60 = = 60 km/h T 1

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Accident statistics Accidents that result in death, injury and damage have always happened. Governments collect, present and interpret data on road incidents to try to reduce the problem. There are many factors that may cause a road accident such as poor driving, speeding, alcohol, fatigue, bad road design or lack of vehicle maintenance. Driving statistics Use summary statistics (mean, median, mode, range and interquartile range) to measure the centre and spread of the data.

Example 16

Calculation of summary statistics

The table below shows the number of road accidents involving fatigue in the last four months of the year. Killed

Injured

September

112

1488

October

197

4365

November

89

2019

December

134

3487

Find the median, mean and sample standard deviation of the accidents involving a death. Solution 1 2 3 4 5 6 7

Write the scores in increasing order. The median is the average of 112 and 134. Enter the statistics mode of the calculator. Clear the contents of the memory. Enter the data into the calculator. Select the x key to view the mean. Select the σ n−1 key to view the sample standard deviation.

89, 112, 134, 197 Median is 123.

Mean is 133. Standard deviation is σ n−1 = 46.5.

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Preliminary Mathematics General

Exercise 14I The distance-time graph describes Ella’s car trip. a How long was the rest stop? b How far did the car travel from its starting point? c How long was the trip? d Determine the average speed during the third hour of the trip.

80

Distance (km)

1 14I

60 40 20

0

2

2

3 Time (hr)

4

5

The table below shows the running costs as cents per km for five motor vehicles.

a b c d e

3

1

Brand A

Brand B

Brand C

Brand D

Fuel

5.06

6.90

9.69

8.99

Tyres

1.03

1.18

0.88

1.28

Service

2.51

3.73

3.02

3.88

What is the cost of service for the Brand D vehicle? Which of the above cars has the best fuel economy? Harry has driven his Brand C vehicle 7580 kilometres this year. What is the fuel cost of the Brand C vehicle for the year? Calculate the difference in service costs between Brand A and Brand B if both cars are driven 15 000 km in a year. What is the difference in tyre costs between Brand D and Brand A when they are both driven 100 000 km?

The table below shows the petrol used at different speeds for the same distance. Litres a b c

50 km/h

70 km/h

90 km/h

110 km/h

34

38

43

49

How much petrol would you save by travelling at 50 km/h instead of 70 km/h? How much petrol would you save by travelling at 70 km/h instead of 110 km/h? What is the difference in cost by travelling at 50 km/h instead of 90 km/h? Assume the petrol costs are $1.45 cents per litre.

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4

The table opposite shows the number of road accidents involving speed that caused an injury in the first five months of the year. Find the following summary statistics. a Mean Month Injured b Median January 2814 c Mode d Range February 1652 e Interquartile range March 1786 f Sample standard deviation April 1589 g Population standard deviation May 2182 h What percentage of the road accidents occurred in January?

5

The speed (in km/h) of some motor vehicles travelling through an intersection was 42, 36, 36, 44, 30, 34, 38, 36 and 39. a What is the mean, correct to the nearest whole number? b What is the mode? c Find the median. d What is the sample standard deviation, correct to two decimal places?

6

The frequency table below shows the number of motor bikes passing through a checkpoint each hour for the past 24 hours. Motor bikes (x)

a b c

Frequency (f )

11

4

12

7

13

6

14

5

Frequency × Score ( fx)

How many motor bikes passed through the checkpoint? Find the mean of this data. Answer correct to two decimal places. What is the median of this data?

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Development 7     8

Construct a distance-time graph using the following data: Event 1: Started from home and travelled at a speed of 30 km/h for 2 hours. Event 2: Stopped for 1 hour to do some shopping. Event 3: Travelled 90 km in 2 hours to reach the destination. Event 4: Returned home in 3 hours. A local community were concerned about the number of accidents at an intersection. The number of accidents at an intersection in the past 13 days is recorded below. 0

6

1

0

3

0

2

0

3

0

1

0

a b

9 

0

Find the mean, median and mode of this data. Which is the better measure for the centre for the data? Explain your answer.

The grouped frequency table shows the age of the driver involved in a fatal road accident during the past year. Class

a b c d e

Class centre (x)

Freq. ( f )

20–29

85

30–39

72

40–49

71

50–59

55

60–69

36

f×x

Copy and complete the above grouped frequency table. How many road accidents occurred in the past year? Find the mean of this data to the nearest whole number. What percentage of road accidents had a driver younger than 30? Answer correct to two decimal places. What percentage of road accidents had a driver older than 49? Answer correct to two decimal places.

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Chapter 14 — Mathematics and driving

Study guide 14

Cost of purchase

Total cost = Deposit + Total repayments Total repayments = Repayment × Number of repayments Interest paid = Total cost − Sale price

Insurance

The cost of insurance is affected by make/model of car, your age/ driving history, finance, modifications/accessories and location. No-claim bonus and excess amount are major factors.

Stamp duty

1 2 3

Round the cost of the vehicle up to the nearest $100 (as required). Express the stamp duty as a fraction or decimal. Multiply the answer in step 1 by the answer in step 2. Amount of fuel (L) × 100 Distance travelled (km)

Running costs

Fuel consumption =

Straight-line

S = V0 - Dn

S  –  Salvage value or current value. V0  –  Purchase price of the item. D  –  Depreciated amount per time period. n  –  Number of time periods.

S = V0 (1 − r)n

S  –  Salvage value or current value. V0  –  Purchase price of the item. r  – Rate of interest per time period (decimal). n  –  Number of time periods.

depreciation

Declining balance depreciation

Safety

S=

D D or T = or D = S × T T S

D  –  Distance S  –  Speed T  –  Time

Stopping distance = Reaction-time distance + Braking distance Blood alcohol content

BACMale = BACFemale

(10 N − 7.5 H ) or 6.8 M (10 N − 7.5 H ) = 5.5 M

Number of hours = Driving statistics

BAC 0.015

BAC  –  Blood alcohol content. N  –  Number of standard drinks. H  –  Hours drinking. M  –  Mass in kilograms. BAC  –  Blood alcohol content.

Use summary statistics (mean, median, mode, range and interquartile range) to interpret the effect of downloading music and video files.

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Review

Chapter summary – Mathematics and driving

451

Review

452

Preliminary Mathematics General

Sample HSC – Objective-response questions 1

A motor bike is for sale at $13 000. Finance is available at $3000 deposit and monthly repayments of $520 for 4 years. What is the interest paid? A $14 960 B $17 960 C $24 960 D $27 960

2

Jake has been quoted $1280 for comprehensive car insurance. He has a no claim bonus of 60%. How much is Jake required to pay? A $512 B $768 C $1220 D $1280

3

A new car is bought for $28 810. Calculate the stamp duty payable if the charge is $3 per $100 or part $100. A $840 B $864 C $867 D $870

4

Mia’s car uses 8.25 litres per 100 km. How many litres of petrol will her car use on a trip of 1150 km from Broken Hill to Sydney? A 94.875 L B 139.73 L C 1397.3 L D 9487.5 L

5

Mitchell purchased a used car for $7500 and it depreciated by $700 each year. What is its depreciated value after three years? A $4700 B $5400 C $6100 D $6800

6

A car depreciates in value from $39 000 to $12 250 in four years under the declining balance method. What is the annual rate of depreciation, to the nearest whole number? A 17% B 18% C 25% D 26%

7

How long will it take a vehicle to travel 342 km at a speed of 70 km/h? A 0.20 h B 2.394 h C 4.89 h D 272 h

8

Layla is 61 kg and has consumed 5 standard drinks in the past four hours.



What is Layla’s blood alcohol content using the formula BACFemale = A 0.007 B 0.048 C 0.059 D 0.060

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(10 N − 7.5 H ) ? 5.5 M

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1

Michael buys a car for $18 000. After one year the market value of the car is $15 000. What is the percentage decrease in the price? Answer correct to one decimal place.

2

A new car is for sale at $39 000. Finance is available at 20% deposit and monthly repayments of $900 for 5 years. a How much will the car cost if you use the finance package? b What is the interest paid?

3

Lucy is 18 years old and has received this quote for comprehensive insurance. Premium details

Excesses

Cost

12 month policy $850.00 10% No claim bonus

Standard

$600

a b

Male under 21

additional $1400

Female under 21

additional $1000

Calculate the cost of the insurance if Lucy is eligible for a no-claim bonus. What is Lucy’s excess if she makes a claim?

4

Logan has bought a used car for $12 460. Calculate the stamp duty payable if the charge is $4 per $100 or part $100.

5

Thomas travels 51 km to work and 51 km from work each day. a How many kilometres does he travel to and from work in a 5-day working week? b Thomas drives a car with a fuel consumption of 7.5 L/100 km to and from work. How many litres of petrol does Thomas use travelling to and from work? c What is Thomas’s petrol bill for work if petrol costs are $1.52 per litre?

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Review

Sample HSC – Short-answer questions

Review

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Preliminary Mathematics General

Sample HSC – Short-answer questions 6

A caravan is purchased for $12 986. After 3 years it has a salvage value of $6020. a What is the annual amount of depreciation, if the amount of depreciation is constant? b Determine the book value of the caravan after 5 years.

7

Alexis purchased a car for $19 900. It depreciates at 24% per year under the declining balance method. a What is the salvage value of the car after five years? Answer to the nearest dollar. b How much has the car depreciated over the five years?

8

What is the stopping distance if the reaction-time distance is 15 m and the braking distance is 21 m?

9

A car is travelling at 70 km/h and the reaction time of the driver is 0.50 seconds. 5Vt V 2 + a Find the stopping distance of this car using the formula d = . 18 170 b How much further would it take to stop this car if the reaction time of the driver was 2 seconds?

10

Levi weighs 74 kg and consumes five standard drinks in an hour. a

Calculate Levi’s blood alcohol content? (Answer correct to 2 significant figures.) (10 N − 7.5 H ) ).   ( BACMale = 6.8 M

What is the number of hours to wait before driving? (Answer correct to 1 decimal place.) BAC ) .        ( Numbers of hours = 0.015 b

11

The percentage of killed drivers with a blood alcohol concentration of 0.05/100 mL for the past ten years is listed below. Drivers (%) a b c

41

32

25

23

23

31

30

23

30

14

What are the mean, median and mode for the above data? Which is the better measure for the centre for the data? Explain your answer. What are the range, interquartile range and population standard deviation for the data? Challenge questions 14

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HSC Practice Paper 2 Section I Attempt Questions 1–15 (15 marks) Allow about 20 minutes for this section 1

Stella sells shoes for a retail store and receives wages of $1875 per month plus 4% commission on all her sales. What were her sales in a month if she received a total pay of $1953? (A) $878.00 (B) $1946.88 (C) $1950.00 (D) $2190.12

2

The number of significant figures in the number 0.00206 is (A) 2 (B) 3 (C) 4 (D) 5

3

A car mechanic charges a total of $165 to repair a motor vehicle. The Goods and Services Tax (GST) of 10% was included in this total. Which of the following statements is correct? (A) 90% of $165 was the price of the repair before the GST was added. (B) The total repair price included $16.50 GST. (C) The price before adding the GST is $165 ÷ 1.10. (D) The GST cannot be determined without knowing the original repair cost.

4

For the scores 20, 22, 22, 14, 19, 22, 20, 21, 24 and 12 consider the following statements. I The median is greater than the mean. II The mode is greater than the median. (A) Both statement I and II are true. (B) Both statement I and II are false. (C) Statement I is false and statement II is true. (D) Statement I is true and statement II is false. 455

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Preliminary General Mathematics

5

Mason has planted red and white rose bushes in the ratio 2:3. How many white rose bushes are there if he planted a total of 30 rose bushes? (A) 6 (B) 12 (C) 18 (D) 20

6

A traverse survey of an area of land was taken. The field book entry is shown opposite and all measurements are in metres. What is the approximate length of AB? (A) 30 m (B) 69 m (C) 75 m (D) 81 m

D 150 C 70 110 75 0 A

30 B

7

Which of the following expressions would give the height (h), of the tree in the diagram? (A) 42 × tan 34° 42 (B) ta 34° tan h (C) 42 × cos 34° 34° 42 42 m (D) cos 34°

8

What is the value of 2x - x2 if x = -3? (A) -15 (B) -3 (C) 3 (D) 15

9

A wage sheet of a small business shows one employee’s details. Employee Terry Brown

Rate per hour $20.00

Normal hours 30

Overtime (× ×2) x

Wage $840

Terry worked some overtime at double-time rate, which is missing from the wage sheet. Using the information on the wage sheet, how many hours of overtime did Terry work? (A) 4 (B) 5 (C) 6 (D) 8

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HSC Practice Paper 2

457

10

Joshua invests $1600 for 3 years at 8% p.a. compounding quarterly. How much compound interest will he receive? (A) $415.54 (B) $429.19 (C) $2015.54 (D) $2029.19

11

What type of data is information collected on the type of motor vehicle? (A) Discrete (B) Continuous (C) Categorical (D) Stratified

12

A menu at the local club is shown opposite. How many different 3 course meals can be ordered? (A) 1 (B) 7 (C) 8 (D) 12

13

Find the value of x given a = 32, y = 2 and the formula a = 2xy2. (A) 3 (B) 4 (C) 5 (D) 6

14

A used car was bought for $(x + 10) and sold for $(x - 20), Which of the following statements is true? (A) There was a profit of $(x - 30). (B) There was a loss of $30. (C) There was a profit of $(x + 10). (D) There was a loss of $(x - 30).

15

Tennis balls are sold in a box of 5 yellow balls or a box of x white balls. A tennis coach needs 400 balls and purchases 20 boxes of yellow balls and a certain number of white balls. Which of the following expressions describes the number of boxes of white balls purchased? (A) (B) (C) (D)

Entrees Soup Salad

Mains Beef Chicken Vegetarian

Desserts Apple pie Ice-cream

400 − x 5 300x 300 + x 300 x

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458

Preliminary General Mathematics

Section II Attempt Questions 16-18 (45 marks) Allow about 70 minutes for this section All necessary working should be shown in every question Question 16 (15 marks) (a)

A communications company pays overtime at a rate of time-and-a-half for the first 4 hours overtime and double time thereafter. Natalie is employed as a personal assistant. During a normal week she works 35 hours at $27.80 an hour. (i) How much did Natalie earn in a week when she worked 42 hours? (ii)

(b)

Marks

Natalie receives annual leave loading of 17 12 % of 4 weeks basic pay. 1. What is the value of Natalie’s leave loading? 2. Calculate the total amount Natalie is paid for her 4 weeks annual leave. Taxable income

Tax payable

0-$6000

Nil

$6001-$35 000

Nil + 15 cents for each $1 over $6000

$35 001-$80 000

$4350 + 30 cents for each $1 over $35 000

$80 001-$180 000

$17 850 + 38 cents for each $1 over $80 000

$180 001 and over

$55 850 + 45 cents for each $1 over $180 000

Daniel earns a gross income of $63 500 during the financial year. He has allowable deductions of $4500. (i) What is Daniel’s taxable income? (ii) Calculate the tax payable on Daniel’s taxable income. (iii) Daniel must pay 1.5% of his taxable income for the Medicare Levy. Calculate how much Daniel pays in Medicare Levy. (iv) What is Daniel’s total tax payable including the Levy? (v) Daniel has paid $12 255 in tax during the financial year. Determine whether Daniel receives a refund of tax, or whether he is required to pay more tax, and determine this amount. (c)

2 2

Simplify: (i) 1 - 2 (5a + 3) (ii) 45x6y5 ÷ 9xy5 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1 2 1 1 2

2 2 Cambridge University Press

459

HSC Practice Paper 2

Question 17 (15 marks) (a)

Marks

The diagram below shows a small and a large vertical pole joined by length of wire at the top of each pole. A h C 2m

44°

B

10 m

The smaller pole is 2 metres tall and is 10 metres from the taller pole, which is of height h metres. The angle of elevation of the top of the tall pole from the top of the small pole is 44°. (i) Use trigonometry to find the distance AB in the diagram. Answer correct to one decimal place. (ii) Find the height of the tall pole, correct to one decimal place. (iii) Find the length of the wire AC that joins the two poles. Answer correct to one decimal place. (b)

(c)

(d)

(e)

Blake plays cards with a normal deck and draws a card from the deck. (i) What are the chances of drawing a 3 or a black card? (ii) What are the chances of not drawing an ace? A paddock is surveyed using an offset survey. The field book entry is shown opposite. (i) Draw a sketch of the paddock. (ii) Find the area of the paddock.

D 140 C 60 90 50 0 A

2 1 2

2 1

45 B

1 1

Zoe invests $24 000 in an account earning 3% p.a. compounding monthly for 5 years. (i)

Calculate the total amount of interest earned.

2

(ii)

Calculate the annual percentage rate of simple interest that would produce the same amount of interest. (Answer correct to 2 decimal places.)

1

Solve the equation 2 x − 5 = 1 3

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2

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460

Preliminary General Mathematics

Question 18 (15 marks) (a)

The following box-and-whisker plot shows the weight (kg) of 400 people.

50

(b)

60

80

90

100

What is the range of weights?

1

(ii)

Determine the median weight.

1

(iii)

Determine the interquartile range of weights.

1

(iv)

How many people weighed less than 83 kg?

2

Henry’s room measures 5850 mm by 4950 mm and needs carpeting. (i) The cost of the carpet is $90 per m2, and a tradesperson charges $40 per m2 to lay the carpet. What is the cost to have the floor covered with carpet? (ii) Henry’s room has a ceiling height of 2800 mm. He is considering buying heater A, B or C as shown in the table below.

2

Rooms up to 70 m3 Rooms up to 80 m3 Rooms up to 90 m3

Determine the most suitable heater and give a reason for your answer.

2

An above ground swimming pool is the shape of a cylinder. It has a radius of 3 metres and contains water to a uniform depth of 0.9 metres. (i) What is the volume of water in the pool, in cubic metres (to 2 decimal places)?

2

(ii)

(d)

70

(i)

Heater A Heater B Heater C

(c)

Marks

How many litres of water are in the pool? (1 m3 = 1000 L). Answer correct to the nearest litre.

1

The two figures below are similar. x

5 cm Not to scale

12 cm

8 cm

(i)

What is the scale factor, in simplest form?

1

(ii)

What is the length of the unknown side in the above trapezium?

2

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HSC formulae and data sheet Financial Mathematics Simple interest I = Prn P is initial amount r is interest rate per period, expressed as a decimal n is number of periods

V0 is initial value of asset r is depreciation rate per period, expressed as a decimal n is number of periods

Data Analysis Mean of a sample x=

Compound interest A P r n

A = P(1 + r)n is final amount is initial amount is interest rate per period, expressed as a decimal is number of compounding periods

Present value and future value PV =

FV , FV = PV(1 + r)n (1 + r )n

r is interest rate per period, as expressed as a decimal n is number of compounding periods

Straight-line method of depreciation S = V0 - Dn S is salvage value of asset after n periods V0 is initial value of asset D is amount of depreciation per period n is number of periods

Declining-balance method of depreciation S = V0(1 - r)n S is salvage value of asset after n periods

sum of scores number of scores

z-score For any score x, x − x− z= s x is mean s is standard deviation

Outlier(s) score(s) less than QL - 1.5 × IQR or score(s) more than QU + 1.5 × IQR QL is lower quartile QU is upper quartile IQR is interquartile range

Least-squares line of best fit y = gradient × x + y-intercept standard deviation of y scores gradient = r × standard deviation of x scores y-intercept = y - (gradient × x) r is correlation coefficient x is mean of x score y is mean of y scores

Normal distribution • • •

approximately 68% of scores have z-scores between -1 and 1 approximately 95% of scores have z-scores between -2 and 2 approximately 99.7% of scores have z-scores between -3 and 3

461

Spherical Geometry Circumference of a circle C = 2pr or C = pd r is radius d is diameter

Arc length of a circle l=

θ 2π r 360

r is radius θ is number of degrees in central angle

Radius of Earth (taken as) 6400 km

Time differences For calculation of time differences using longitude: 15° = 1 hour time difference

Area Circle A = pr2 r is radius

Sector A=

θ πr2 360

r is radius θ is number of degrees in central angle

Annulus A = p(R2 - r2) R is radius of outer circle r is radius of inner circle

462

Preliminary Mathematics General

Trapezium

Volume

h A = (a + b) 2 h is perpendicular height a and b are the lengths of the parallel sides

h V ≈ ( AL + 4 Am + AR ) 3 h distance between successive measurements AL is area of left end AM is area of middle AR is area of right end

Area of land and catchment areas unit conversion: 1 ha = 10 000 m2

Trigonometric Ratios

Surface Area Sphere

Closed cylinder A = 2pr2 + 2prh r is radius h is perpendicular height

Volume



sinθ =

opposite side hypotenuse

cosθ =

adjacent side hypotenuse

tanθ =

opposite side adjacent side

Prism or cylinder

Sine rule

V = Ah r is radius h is perpendicular height

In ∆ABC a b c = = sin A sin B sin C

Pyramid or cone

Cosine rule

1 Ah 3 A is area of the base h is perpendicular height

a2 + b2 − c 2 or cos C = 2 ab

V=

Approximation Using Simpson’s Rule Area h ( d f + 4 dm + dl ) 3 h distance between successive measurements df is first measurement dm is middle measurement dl is last measurement A≈

(10 N − 7.5 H ) or 6.8 M

(10 N − 7.5 H ) 5.5 M N is number of standard drinks consumed H is number of hours of drinking M is person’s mass in kilograms BAC Female =

D = ST , S = average speed =

1 byte = 8 bits 1 kilobyte = 210 bytes = 1024 bytes 1 megabyte = 220 bytes = 1024 kilobytes 1 gigabyte = 230 bytes = 1024 megabytes 1 terabyte = 240 bytes = 1024 gigabytes

total distance travelled total time taken

Probability of an Event The probability of an event where outcomes are equally likely is given by: P(event) =

number of favourable outcomes total number of outcomes

Straight Lines Gradient m=

Units of Memory and File Size

D D ,T= T S

 reaction-time   braking  stopping distance =  +   distance    distance 

In ∆ABC c2 = a2 + b2 - 2ab cos C

Volume and capacity unit conversion: 1 m3 = 1000 L

BACMale =

Distance, Speed and Time

θ

A = 4pr3 r is radius

Blood Alcohol Content Estimates

vertical change in position horizontal change in position

Gradient–intercept form y = mx + b m is gradient b is y-intercept

Glossary A Absolute error The difference between the actual value and the measured value indicated by an instrument. Adjacent side A side in a right-angled triangle next to the reference angle but not the hypotenuse. Allowable deduction Deductions allowed by the Australian Taxation Office such as work-related, self-education, travel, car or clothing expenses. Analysing data A process that interprets data and transforms it into information. Angle of depression The angle between the horizontal and the direction below the horizontal. Angle of elevation The angle between the horizontal and the direction above the horizontal. Annual leave loading A payment calculated as a fixed percentage of the normal pay over a fixed number of weeks. Annual leave loading is usually at the rate of 17½%. Appreciation An increase in value of an item over time. It is often expressed as the rate of appreciation. Area The amount of surface enclosed by the boundaries of a shape.

B Bar chart A graph that displays categorical data using horizontal bars. Bias When events are not equally likely.

Bonus An extra payment or gift earned as reward for achieving a goal. Boxplot See Box-and-whisker plot. Box-and-whisker plot A graph using five-number summary: lower extreme, lower quartile, median, upper quartile and upper extreme. Budget A plan used to manage money by listing a person’s income and expenditure.

C Capacity The amount of liquid within a solid figure. Casual rate An amount paid for each hour of casual work. Categorical data Data that is divided into categories, such as hair colour. It uses words not numbers. Census Collecting data from the whole population. Chance The likelihood of an event occurring. Coefficient A number in front of a particular letter in an algebraic expression. For example, the term 3y has a coefficient of 3. Collecting data A process that involves deciding what to collect, locating it and collecting it. Commission A payment for services, mostly as a percentage of the value of the goods sold. Complementary event The outcomes that are not members of the event. Compound interest Interest calculated from the initial amount borrowed or principal plus any

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464

Preliminary Mathematics General

interest that has been earned. It calculates the interest on the interest. Compounding period The length of time between interest payments in a compound interest investment. Concentration A measure of how much of a given substance is mixed with another substance. Cone A solid figure, with a circular base, that tapers to a point (apex). Continuous data Numerical data obtained when quantities are measured rather than counted. Conversion graph A graph used to change one quantity from one unit to another unit. Cosine ratio The ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cumulative frequency The frequency of the score plus the frequency of all the scores less than that score. It is the progressive total of the frequencies. Cumulative frequency histogram A histogram with equal intervals of the scores on the horizontal axis and the cumulative frequencies associated with these intervals shown by vertical rectangles. Cumulative frequency polygon A line graph constructed by joining the top right hand corner of the rectangles in a cumulative frequency histogram. Cylinder A prism with a circular base.

D Data Raw scores. Information before it is organised. Decile A band of 10% of the scores in a data set. Deduction A regular amount of money subtracted from a person’s wage or salary. Degree A unit for measuring angles. Dependent variable A variable that depends on the number substituted for the independent variable. Discrete data Data obtained when a quantity is counted. It can only take exact numerical values. Displaying data A process that involves the presentation of the data and information. Distributive law A rule for expanding grouping symbols by multiplying each term inside the

grouping symbol by the number or term outside the grouping symbol. Divided bar graph A graph that shows the relationship or proportions of parts to a whole. It consists of bars or rectangles drawn to scale. Dividend A payment given as an amount per share or as a percentage of the issued price. Dividend yield The annual dividend divided by the share’s market price and expressed as a percentage. Dot plot A graph that consists of a number line with each data point marked by a dot. When several data points have the same value, the points are stacked on top of each other. Double time A penalty rate that pays the employee twice the normal hourly rate.

E Elevation A scale drawing of a building from one side. Enlargement A similar figure drawn larger than the original figure. Equally likely outcomes Outcomes of an event that have the same chance of occurring. Equation A mathematical statement that says that two things are equal. Evaluate An instruction to work out the exact value of an expression. Expand An instruction to remove the grouping symbols. Experimental probability See Relative frequency. Expression A mathematical statement written in numbers and symbols.

F Face value The initial price when a company firstly lists on the stock or securities exchange (ASX). Factorise An instruction to break up an expression into a product of its factors. Field diagram A diagram used to calculate the area of irregularly shaped blocks of land. Five-number summary A summary of a data set consisting of the lower extreme, lower quartile, median, upper quartile and upper extreme.

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Glossary

Flat interest See Simple interest. Formula A mathematical relationship between two or more variables. Fortnight Two weeks or 14 days. Frequency The number of times a certain event occurs. Frequency histogram A histogram with equal intervals of the scores on the horizontal axis and the frequencies associated with these intervals shown by vertical rectangles. Frequency polygon A line graph constructed by joining the midpoints at the tops of the rectangles of a frequency histogram. Frequency table A table that lists the outcomes and how often (frequency) each outcome occurs. Fundamental counting principle A law that states if we have ‘p’ outcomes for first event and ‘q’ outcomes for the second event, then the total number of outcomes for both events is p × q. Future value The amount an investment will grow under compound interest.

G General form A linear equation written in the form ax + bx + c = 0. Goods and Services Tax (GST) A tax added to the purchase price of each item. The GST rate in Australia is 10% of the purchase price of the item except for basic food items and some medical expenses. Gradient The steepness or slope of the line. It is calculated by dividing the vertical rise by the horizontal run. Gradient–intercept formula A linear equation written in the form y = mx + b. Gross income The total amount of money earned from all sources. It includes interest, profits from shares or any payment received throughout the year. Gross pay The total of an employee’s pay including allowances, overtime pay, commissions and bonuses. Grouped data A data organised into small groups rather than as individual scores.

465

Grouping symbol Symbols used to indicate the order of operations such as parentheses ( ) and brackets [ ].

H Histogram A graph using columns to represent frequency or cumulative frequency. See Frequency histogram and Cumulative frequency histogram. Hypotenuse A side in a right-angled triangle opposite the right angle. It is the longest side.

I Income tax Tax paid on the income received. Independent variable A variable that does not depend on another variable for its value. Inflation A rise in the price of goods and services or Consumer Price Index (CPI). It is often expressed as annual percentage. Intercept The position where the line cuts the axes. Interest The amount paid for borrowing money or the amount earned for lending money Interest rate The rate at which interest is charged or paid. It is usually expressed as a percentage. Interquartile range The difference between the third quartile and first quartile.

L Like term Terms with exactly the same pronumerals, such as 2a and 5a. Limit of reading The smallest unit on measuring instrument. Line of best fit A straight line used to approximately model the linear relationship between two variables. Linear equation An equation whose variables are raised to the power of one. Linear function A function when graphed on a number plane is a straight line. Linear modelling A mathematical description of a practical situation using a linear function. Lower extreme Lowest score in a data set. Lower quartile The lowest 25% of the scores in the data set.

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466

Preliminary Mathematics General

M

P

Market value Current price of a share. Mean A measure of the centre. It is calculated by summing all the scores and dividing by the number of scores. Measurement Determining the size of a quantity. Median The middle score or value. To find the median, list all the scores in increasing order and select the middle one. Medicare levy An additional charge to support Australia’s universal health care system. Mode The score that occurs the most. It is the score with the highest frequency. Multistage event Two or more events such as tossing a coin and rolling a die.

Pay As You Go (PAYG) Tax deducted from a person’s wage or salary throughout the year. Per annum Per year. Percentage change The increase or decrease in the quantity as a percentage of the original amount of the quantity. Percentage error The maximum error in a measurement as a percentage of the measurement given. Percentile A band of 1% of the scores in a data set. Pie chart See Sector graph. Piecework A fixed payment for work completed. Population The entire data set. Population standard deviation A calculation for the standard deviation that uses all the data or the entire population. (σn) Prefix The first part of a word. In measurement it is used to indicate the size of a quantity. Present value The current value of an investment Principal The initial amount of money borrowed. Prism A solid shape that has the same cross-section for its entire height. Probability Probability is the chance of something happening. The probability of the event is calculated by dividing the number of favourable outcomes by the total number of outcomes. Pronumeral A letter or symbol used to represent a number. Pyramid A solid shape with a plane shape as its base and triangular sides meeting at an apex. Pythagoras’ theorem The square of the hypotenuse is equal to the sum of the squares of the other two sides. h2 = a2 + b2

N Net pay The amount remaining after deductions have been subtracted from the gross pay. Number pattern A sequence of numbers formed using a rule. Each number in the pattern is called a term.

O Offset survey A survey involving the measurement of distances along a suitable diagonal or traverse. The perpendicular distances from the traverse to the vertices of the shape are called the offsets. Ogive See Cumulative frequency polygon. Opposite side A side in a right-angled triangle opposite the reference angle. Organising data A process that arranges, represents and formats data. It is carried out after the data is collected. Outcome A possible result in a probability experiment. Outlier Data values that appear to stand out from the main body of a data set. Overtime Extra payments when a person works beyond the normal working day.

Q Quantitative data Numerical data. It is data that has been measured. Quartile A band of 25% of the scores in a data set. See Upper quartile and Lower quartile.

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Glossary

Questionnaire A series of questions to gather specific information.

R Random sample A sample that occurs when members of the population have an equal chance of being selected. Range The difference between the highest and lowest scores. It is a simple way of measuring the spread of the data. Rate A comparison of different quantities in a definite order. Rate of interest See Interest rate. Ratio A comparison of like quantities in a definite order. Reduction A similar figure drawn smaller than the original figure. Relative error A measurement calculated by dividing the limit of reading (absolute error) by the actual measurement. Relative frequency The frequency of the event divided by the total number of frequencies. It estimates the chances of something happening or the probability of an event. Retainer A fixed payment usually paid to a person receiving a commission. Royalty A payment for the use of intellectual property such as book or song. It is calculated as a percentage of the revenue or profit received from its use.

S Salary A payment for a year’s work which is divided into equal monthly, fortnightly or weekly payments. Sample A part of the population. Sample space The set of all possible outcomes. Sample standard deviation A calculation for the standard deviation when the data set is a sample (σn - 1). Scale factor The enlargement or reduction of a shape by a scale factor.

467

Scientific notation A number between 1 and 10 multiplied by a power of ten. It is used to write very large or very small numbers more conveniently. Sector graph A graph that represents data as sectors of a circle (‘slices’ of a ‘pie’). It shows the relationship or proportions of parts to a whole. Share A part ownership in a company. Significant figures A statement to specify the accuracy of a number. It is often used to round a number. Similar figure Figures that have exactly the same shape but they are different sizes. Simple interest A fixed percentage of the amount invested or borrowed. It is calculated on the original amount. Simultaneous equations Two or more equations whose values are common to all the equations. It is the point of intersection of the equations. Sine ratio The ratio of the opposite side to the hypotenuse in a right-angled triangle. Standard deviation A measure of the spread of data about the mean. It is an average of the squared deviations of each score from the mean. Standard notation See Scientific notation. Statistical inquiry A process of gathering statistics that involves six steps: posing questions, collecting data, organising data, displaying data, analysing data and writing a report. Stem-and-leaf plot A method of displaying data where the first part of a number is written in the stem and the second part of the number is written in the leaves. Stock See Share. Strata A group within a population that reflects the characteristics of the entire population. Stratified sample A sample using categories or strata of a population. Members from each category are randomly selected. For example, one student is selected from each year 7, 8, 9, 10, 11 and 12. Substitution It involves replacing the pronumeral in an algebraic expression with one or more numbers. Summary statistic A number such as the mode, mean or median that describes the data.

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Preliminary Mathematics General

Surface area The sum of the area of each surface of the solid. Systematic list An orderly method of determining all the possible outcomes. Systematic sample A sample that divides the population into a structured sample size. For example, sorting the names of people in alphabetical order and selecting every fifth person.

T Tangent ratio The ratio of the opposite side to the adjacent side in a right-angled triangle. Taxable income The gross income minus any allowable deductions. Time-and-a-half A penalty rate that pays the employee one-and-a-half times the normal hourly rate. Traverse survey See Offset survey. Tree diagram A technique used to list the outcomes in a probability experiment. It shows each event as a branch of the tree. Trigonometry A branch of mathematics involving the measurement of triangles.

Upper extreme Highest score in the data set. Upper quartile The highest 25% of the scores in the data set.

V Value Added Tax (VAT) A tax added to the purchase price of each item. VAT is used in many countries with the rate ranging from 2% to 25%.
 Variable A symbol used to represent a number or group of numbers. Volume The amount of space occupied by a threedimensional object.

W Wage A payment for work that is calculated on an hourly basis.

X x-intercept The point at which the graph cuts the x-axis.

Y

U Unitary method A method used in ratio and percentages involving the calculation of a unit.

y-intercept The point at which the graph cuts the y-axis.

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Answers Chapter 1 24

6 7 8 9 10 11

pay are incorrect as not every month has 4 weeks. 18.75 hrs

Exercise 1A 1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 21 22 23

a c

$1782 b $3563 $7721 a $1108, $2217 b $1509, $3019 c $2073, $4146 d $851, $1702 $1656 a $30 160 b $39 520 c $39 216 d $125 736 a $3000 b $78 000 $41 860 $81 120 Stephanie $49 348 Tahlia $45 852 Stephanie by $3496 Laura $32 110 Ebony $29 508 Laura by $2602 Tran $98 696 Jake $99 960 Difference $1264 a $892.50 b $943.50 c $1020.00 d $1071.00 $1130.50 a $444.00 b $351.50 c $858.40 d $511.71 a $15 787.20 b $31 720.00 c $39 062.40 d $24 731.20 a 40.50 hrs b $911.25 42 hrs 9 hrs a $193.60 b $968.00 c $1936.00 d $50 336.00 $332.10 Alyssa $320.00 Connor Alyssa by $12.10 $635 481.60 $83 790.00 Computer application $1640.85 Weekly pay $1777.58 Calculations for weekly

Exercise 1B 1

a c a c

b d b d

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

$108.00 $379.20 $227.94 $292.68 $1560.00 $135.30 $1023.40 a 45.5 hrs $9.80 $16.00 $1000.68 $1237.50 a $257.50 $820.88 $1761 $3589.60 $2212 $3720 $451.05 $680.00

19

a $448.00 b $268.80 c 37 %

20 21

5.89 hrs a $453.60 c $22.68

2

$237.00 $262.98 $114.30 $370.74

Exercise 1D 1

b

$1041.60

$1287.50

1 2

b d

4 hrs $567.00

Exercise 1C 1

2 3 4 5

a c e a c

$1092.00 $1862.00 $1176.00 $1421.00 $784.00 $3564.95 $2326.50 a $2901.60 c $2065.80 e $2214.50

2 3 4

b

$222 $5810.95 $542 Computer application a $1424.00 b $35.60 a $1400.00 b $980.00 c $73 528.00 d Pay increases by $728 however holiday loading was $980.

5 6 7

8 9 10 11 12

a c a c

$352.80 b $1368.80 $9120.00 b $5710.00 d $1620.00 a $431.00 b c $1640.00 $960.00 a $512.00 b a $1000, $3250 b $1000, $4500 c $1000, $5750 d $1000, $7000 $50 600.00 a $3400.00 b c $5200.00 2% $460.00 a $11 600.00 b c $2260.00 d

$669.60 $8400.00 $11 814.00 $642.00

$500.00

$4700.00

$41 600.00 $1220.00

Exercise 1E b d

$897.75 $966.00

b d

$791.00 $1074.85

b d f

$2896.20 $2943.90 $2055.00

1

2 3 4 5 6

a b c

$2250 $3870 $2880 $5140 $500 $4473.60 a $13 096.32 c $15 712.35 a $12 105.60

b

$36 864.00

b

$9240.40

469 © The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Cambridge University Press

Answers

470

7 8 9 10 11 12 13

Preliminary Mathematics General

a c a c a c a

$8762.00 b $28 580.76 $58.20 b $107.60 d $142.80 b $154.20 d $1005.00 b $358.20 a 11 240 copies b c $37 859.13 $832.00

$18 312.84

5 6

$97.00 $165.80 $156.40 $103.20 $595.00

Computer application a $526 b $658 total c $77 total Income

Expenses

Job $1896 Mortgage

$526

Groceries

$360

Entertain

$120

Medical

$18

Car

$160

Electricity

$20

Telephone

$14

Rates

$43

Balance

$635

$42 065.70

Exercise 1F 1 2 3 4 5 6 7 8

9

10 11 12 13

a c a c a a c

$511.00 $493.00 $407.40 $962.73 $15 892 $1689.95 $4003.61 $79 485 a $464.75 a $9.60 c $16 140.80 a $1434.00 c $129.06 e $891.82 a $1120.00 c $100.80 e $756.20 a $1976.00 c $1769.71 a $989.40 a $2420.10 c 24.7% a $2403.00 c 17%

b

$720.00

b

$1419.60

b b

$32 858.00 $3850.70

b b

$343.75 $310.40

b d

$258.12 $542.18

b d

$168.00 $363.80

b

$177.84

b b d b

33.0% $1210.05 $2400.00 $1259.40

Exercise 1G 1 2 3 4

a c

$1536 $18 720 $13 473 a $61 460.37 c $1690.60

b d

$2550 $23 400

b

$59 769.77

a

Income

Expenses

Job

$74

Sport

$24

Allow

$30

Movies

$22

School

$16

Food

$20

Balance

$22

$104 b

$22.00

$104

3

d

$1896

4

5

i k a c e g i k a c e g i a c e g i

$1896

k

Review

6

Section I 1 4 7

2 5 8

C C D

$1855.20 $641.25 $5371.25 a $839.16 a $4725.00 $4440 a $31 461.46 $1056.00 $5.35 a $1833.85 $512 $2781.12 $1526 $27103.50

b

B D D

3 B 6 A

a c e

7

Section II 1 2 3 4 5 6 7 8 9 10 11 12 13 14

a

$38.65

b b

$5634.36 $235 275.00

b

$26 742.24

8 9 10 11

b

25.8%

Chapter 2 Exercise 2A 1

2

a c e a c e g

4r, r –a, 5a 2m, 9m 7 7y h 5d –5d

b d f b d f h

5x, 2x xy, 4xy, 3yx cd, dc 20 20p 2x –6 –6y 22t

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

16 16f 7ab 7c + 4 17r + 8 5a + 3b 3de 6xy 9a – 3b 7x2 – 3x + 4 t2 + t 2e2 + e w2 + 3w + 5 –3r2 + r – 7 2a 3 5m 4 3d 11 13s 3 3y 4 7g 6 Yes No Yes

j l b d f h j l b d f h b d f h j l b d f

10gh 4xyz 8 –7 8f 3x + 4y 4 2dd – 5h 9a – b 5ab – 2b 4g + 9h –2a2 + 5a 2m2 + 4m 2dd2 – 4d –v2 + v + 2 x 5 2x 7 4y 15 5f 8 5e 6 r 10 No No No

4x

6x

4x + y

x + 7y 7

3x + 7y 7

x + 8y 8

2x – y

4x – y

2x

x + 2y 2

3x + 2y 2

x + 3y

y − 6x P = 2l + 2b P = 7x + 4y 4 7w a 12 17 x c 21 13h e 24 11u g 30 11w i 12 13 x k 15

b d f h j l

a 20 2z 15 7r 24 13e 20 17a 20 7d 5

Exercise 2B 1

a c e g i

24g 56d 100x2 42e2 −45f 45 2 45f

b d f h

35m 7a2 30s2 −16w2

Cambridge University Press

471

Answers

3 4

4m2n2 6x3 15s2t4 2 4u w3 a 2y mn c 3 e m2

i a c e g i

6

d

7 yz 13 r 3s

h b d f h

11w2 3 5j 1 4s

10p3

5p2q

6p2q

12p3q

6p2q2

2pq2

4p2q2

2pq3

32 y 2 3 10w 2 9 1

a

e

b d f

15a 2 c 3e

g i

h

8

4

2 t

12t

6t

3

4r

2r

r t

A=

11

x 2y a c e

12 d 2 5 15 10w 2 63 2m2 45

4

5

7

8 9

15 x 3 2

15h2 k 2 4 a2b2 2 3m 2 4

12 b

u2v

d

18m2n

f

1 6 y2

Exercise 2C 1

3

10 11

9

10

2

6

5p2

c

12

4a

4 y2 3

7

8

c –10p2qr f 48q4 i 20z4

b

f

x 8 5p 5z –9h2 3b x 7 1 3m 2

g 5

b 45y2z e 8r5 h 5d3e3

a d g

9x – 6. Did not multiply −2 by 3.

3a + 6 b 2d + 2 7b − 14 d 6x + 8 10x − 14 f 36b + 4 20 + 8t h 6 − 12w 15 + 45d j 40e − 16d 20a + 45d l 14h + 56g −4x − 12 b −3y − 15 −b − 8 d −7k + 14 −6w + 6 f −2x + 26 −8 − 4q h −15 + 20r −56 + 14s y2 + y b v2 + 4v 2 n + 10n d 2x2 − 3x 3e2 + 5e f 6d2 − 2d 7ez + 3fz h 2ab − 3ac cd + 4ce 6g + 2 b 8s + 14 y − 27 d x+8 8z − 2 f −4q + 35 6x + 1 b 25y − 16 9b − 4 d 9r + 2 5n − 2 f 4q − 7 7x − 3 b 5y + 8 8a + 22 d 13c − 9 10s + 34 f 9h + 21 10x h 2z + 59 7c − 41 j 21g + 3 13u − 18 l −7d + 10 18x − 12 − 2x − 6 = 16x − 18. Expansion of 2nd parenthesis. a 2x2 − 3x b 2b2 + 4b 2 c 2y + 5y d g2 2 e 7v − 3v f b2 − 3b g 3u2 + 5u h 3n2 − 25n i 5d2 + 26d j −6e2 + 65e 2 k 7k − 15k l −10t2 + 19t 2 2 a b − 5ab a 2x3 + 3x2 − 2x − 2 b a3 + 2a2 − 4a − 12 c 5y3 + 2y2 − 3y − 21 d −3b3 − b2 + 7b e z3 − 2z2 − z f −2e3 − 7e2 + 7e g 5x h 3a3 + 3a i v j a3 + a2b − ab − b2 k x4 + x2y − x2 − 3xy l y3 + 4y2z − yz2 − y −6n3 + 3n2r a c e g i k a c e g i a c e g i a c e a c e a c e g i k

3 4

5 6

7

8

9 10

11

Exercise 2E 1

2

Exercise 2D

3 1

2

a c e g a c e g

3 10 7 4 y+4 x−1 2v + 10 2d − 3

b d f h b d f h

4 5 2 3 a+4 h−3 3s − 9 2+x

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

2(x − 3). Error not taking out the factor of 2 from 6. a 5(a + 4) b 3(x + 6) c 8(p − 7) d 7(z + 3) e 4(8 + d) f 9(3 − t) g 2(2n + 5) h 4(4c + 9) i 5(3f + 4) j 8(3 − 2x) k 4(7g − 2) l 3(3w − 7) Expanding a −2(a + 6) b −3(q + 5) c −5(m + 4) d −9(s − 10) e −4(4 + y) f −8(4 + h) a x+4 b 6−y c 2 + 3e2 d 6 + 7k3 e 5r − 7 f 3x + 1 g q2 + 5 h 1 + 2w i 3n2 + 4 j 2t4 + 3 a y(y − 5) b 5m(2m + 1) c 4x(x − 4) d v(3 + v) e 4g(2 + g2) f 4d2 (1 + 3d2) g 2a (b + 2) h 5xy(y + 2) i 3b2(2c − 5) j ef (d − 2e) k xy(5x + 3y) l r2t2(r − 1) 4x(3 + 2x). Did not take out the largest common factor. a 4(x + 3y + 2) b 5(2x − y + 3z) c 2(a + 2b − 3c) d 3(3y2 + 2y − 4) e 2(4 − 2r + 3r2) f 3(1 + 2h2 + 3h) g 5 (2v − 3v2 + 5) h 4(2m − mn + 3) i 3(3c + 5cd + 6d) a a(2b + c + g) b x(4 + y + z) c g(7 − h + 14g) d d(d − 3 + e) e b(b + c −5) f k(1 − 4kh + 2h) g xy(z + 5y − 2x) h ab(2c + 5abc2 − 3) i mn(1 − 2m + n)

a d g a d a d

b e h b e b e

a d

19 25 48 25 –25 –2 1 3 −2 4 3 –2

g

–4

h –6

g 4

19 −96 2 10 4 6 –2

h 6.61 b 5 e 0

c f i c f c f

7

24 3 0 7 97 1 24 i −2 25 c 9 f 13 1 i −6 2

Cambridge University Press

Answers

2

Answers

472

5

6

7 8 9 10 11 12 13 14

Preliminary Mathematics General

1 2

a

1

b 4

d

2

e 3

f

a

485

b −153

c

d

11

e 25

f

15 16 48 10 49

150.8 cm 64 3 3.24 8 0.008 9.20 6

8

2

3

a c e g i k a c e g i k a c e g i k

4

5

6

7

a c e g i k a c e g i k a c e g i k a

y=9 a=5 m = −5 h=3 q = −11 m=5 a = 10 d = 23 z=5 v=7 j = −1 f = 15 x=3 v=4 h = −5 1 w=6 2 1 e = −1 7 5 d = −10 7 y=8 w = 24 a = −35 d = −108 x = 20 m = 20 x=2 q=2 v=4 g = −2 p = −4 n=3 y=4 a=6 m = −1 q=3 e=4 d=4 1 m= 6 4

b=2

f

d =−2 1 5 1 q=5 2 8 x= 9

1 8

h=4

h j l

9 10 b d f h j l b d f h j l b d f h j l b d f h j l b d f h j l b d f h j l b

x = 15 c = −11 d = −9 r = −5 x=3 g = −8 k = 10 s=1 k = −2 x=4 h = 14 c = −5 w=9 t = −6 a = −10 2 c = −7 3 1 k=2 8 2 e= 4 3 d = 56 f = −15 g = 18 s = 33 y = −12 w = −81 y=6 d = 10 m=3 x = −1 v = 16 b = 16 c=1 y=3 d = −5 r = −1 h=3 w = −6 2 c= 3 3

11 12 13 14

15 16 17

a a c e g i a b

a c e g i

2

3

4

a c e g i a c e g i a c

1 2 11 m= 5 15 x=8

s=

2 5

2 5 4 h=8 7 k = 100 v = 48 a=4 k = 24 n = 24 e = 18 d=4 y=9 y=3 v = 35 3 h = −2 4 3 a=2 7 x=6

y = 14

d

a =1

f

v=

h

8

b d f h

x=8 s = 18 b = −45 r=9

b

y=−4

d

6

29 50 1 r=6 4

d = −28 d=8 z = 10 q = 12

d

x = 10

f h

m=6 c=7

b b d f

g

x=6

h

i

y = 60

j

k

w = −40

l

a c

y=3 y=3

b d

e

d=6

f

4320 x = 18 c = 28 m = 126 1 a=7 2 r = 12 2 x=2 5 x=1 x=8 2 x=3 5 3 a=5 4

7 8 9 10 11 12

13 14 15 16

1 3

e=1

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g

y = 14

i

x=5

3 7

h

4 5

Exercise 2H 1 2 3 4 5

13 18

b d f h

t = −11

i

x=3 x=6 n = 16

b

b

a

e g

Exercise 2G 1

5

c

2 x = . Error in 2nd line. 7 2+x=7 b x=5 x=5 b y=2 v = −15 d b=3 d=5 f z=2 a=9 h w=2 e=1 6 × p = 93 p = $15.50

b b b

x = −1 a=3

i

6 7

6 y=3 4x − 3 = 11 + 6x −2x = 14 x = −7 a 10 + 3x = 19 a 2(x 2( + 4) = 20 a 4 + 2n = 36

f h

a a c e

h = 14 c=3 2 u=2 7 b = 77 2 x=5 3 w = −17 a = −1 3 e= 10 C = 36 × T x=6 n = 10 s = 60

e g

2 5

d

7x = 2

Exercise 2F 1

y = 11 2 3 e x=7 4 1 g z=3 2 1 i e=2 2 1 k y = 12 3 7x + 6 = 8 c

c 8

17 18

a a a a a

196 c 300 36 c 1024 15 c 3 26 c 24 1.0 1 a 14 b 17 c 19 2 2 7 a 7 b 6 c –36 3 11 a $1060 b $1165 350 km a 2.6 × 10−13 b 3.9 × 10−7 a 4ºC b 43ºC (v − u) a u = v − at b a= t (v − u) c t= a C a r= 2π b i 0.48 cm ii 1.10 mm a m = Bh2 b i 73 ii 110 cm a v = 13.6 b v = 10.2 a R = 1.42 b R = 2.19 c R = 2.88 a s = 42 b s = 592.5 a $2700.00 b $19 712.00 c $3720 32 81 20 36 7.2

b b b b b

Cambridge University Press

Answers

a

3V πr2

20

a

3

 3V   4π 

b

h = 4.30

b

r = 0.8 2

Review Section I 1 4 7

2 5 8

B D D

3 D 6 C 9 B

A A C

3

Section II 1 2 3 4

5

6

7

7st + 1 b 8r − 6s 35 x 2 a b 2 c a b 3 19m c 24 a 7x − 7 b c 35x − 7 d e 7d + 49e f g −9b + 2h h i y2 + 2yz 2 a 7(b + 5) b c −3(v − 5) d e 2(2x − 7y 7 ) f g 3(7 − 5x) h 4(6b − 4c + 1) i 3(4s − 5v + 3) a d = 11 b c r=3 d e x=5 f a

d2 − d w4 16 y 3

4

y

10 + 10r 12a − 3b −16v + 4s 6w − 12 2(v − 7) 3(2 + 3) 3(2y 3(4x + 7y 7 )

h=5 t=4 n=4

1 2

h

v=−2

a c

r = 72 11 m= 20 x = −11 n=3

b d

e

y=2

f

12

a c a a c a

13

a

u = 17 b x=6 d = 15 b 1.24 b 3.49 4.5 × 1019 b C − 2000 t= 20 3

r = −1 b=2 2 v=3 13 z = 32

g i 8

9 10 11

b

h=8 1.72 1.8 ×

50

7

8 9 10 11 12 13 14

15

16 17 18

1

Exercise 3A a

6

b

2000 d 89 000 570 f 600 000 9.4 h 60 81 j 0.49 22 l 5.1 3000 b 45 000 76 000 d 8 100 000 4 000 000 f 520 6.8 h 9.3 45 j 0.3 2.3 l 60 2000 b 12 000 9 000 000 d 7 800 000 50 000 f 300 000 000 6.1 h 0.4 0.21 j 0.08 0.079 l 8 150 b 120 480 d 2400 108 f 600 12 h 800 4 j 45 6.5 l 13 Centimetre b Metre Millimetre d Kilometre Centimetre f Metre Tonne b Gram Kilogram d Kilogram Tonne f Gram Minute b Minute Year d Week Second f Hour 19.25 L b 50 2.2 kg 2 495 000 m 3068 kg 12 km a 4L b 40 L a 0.005 km, 5000 cm, 500 m, 5 000 000 mm b 5 000 000 mm, 500 m, 5000 cm, 0.005 km a 1 000 000 b 1 000 000 c 100 d 0.1 e 20 f 0.005 g 39 000 h 310 000 000 i 4 700 000 j 0.0743 k 65 l 0.4 a 0.08 b 8 a 1.5 km b 50 a 17 h b 32%

Exercise 3B

Chapter 3

1

1019

5

c e g i k a c e g i k a c e g i k a c e g i k a c e a c e a c e a

7800

a b d

A = 10, B = 44, C = 72, D = 89 1 mm c 0.5 mm A Lower limit = 9.5 A Upper limit = 10.5 B Lower limit = 43.5 B Upper limit = 44.5

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

e

f

2

a b d

e f 3 4 5 6 7

a c a c a c a c a c d

C Lower limit = 71.5 C Upper limit = 72.5 D Lower limit = 88.5 D Upper limit = 89.5 A ±0.050 B ±0.011 C ±0.007 D ±0.006 A − 5.0% B – 1.1% C − 0.7% D − 0.6% A − 1.82 kg B − 4.24 kg 0.02 kg c 0.01 kg A Lower limit = 1.81 kg A Upper limit = 1.83 kg B Lower limit = 4.23 kg B Upper limit = 4.25 kg A ±0.005 B ±0.002 A −0.5% B −0.2% 0.40 kg b 0.008 1% 16 g b 0.068 6.81% 0.29 kg b 0.097 9.667% 0.15 m b 0.006 0.6% 0.3 m b 0.2 m Cooper − 0.0053 Filip − 0.0036 Cooper − 0.533% Filip − 0.358%

Exercise 3C 1

2

3 4 5

6

7.6 × 103 b 5.9 × 105 d 3.5 × 104 f 7.71 × 107 h 9.54 × 1010 5.6 × 10−4 b 8.12 × 10−7 d 5.8 × 10−5 f 2.6 × 10–1 h 1.67 × 10–10 5.0 × 10−6 s a 4.10 × 108 b a 112 000 b c 5200 d e 240 f g 3 900 000 h i 64 000 a 0.00035 b c 0.000000163 d e 0.049 f g 0.0000000412 a c e g i a c e g i

1.7 × 109 6.8 × 106 3.1 × 108 5.23 × 1011 6.87 × 10−5 4.3 × 10−3 3.12 × 10−6 9.2 × 10−2 4 × 108 534 000 000 86 780 000 7 800 000 000 28 0.0000079 0.00581 0.98

Cambridge University Press

Answers

19

473

Answers

474

Preliminary Mathematics General

h i 7 8 9 10

11

12 13 14 15 16 17 18 19 20 21 22 23

0.0000633 0.000000003 5.81 × 10−6 kg a 1.475 × 1010 c 2.982 × 10−6 a 3.25 × 107 c 1.5 × 101 a 1 600 000 c 789 000 e 778 000 g 821 100 i 49 000 a 0.004 c 0.00159 e 0.00003 g 0.0081 i 0.00042 0.000016 m 270 000 000 cg 4240 g a 2 880 000 4 × 108 a 1.5 × 108 a 262 000 13 920 3 × 108 m/s 2 × 106 2.76 × 1013 9.4 × 108 km

b d b

2.961 × 108 1.86 × 10–8 1.5 × 1010

b d f h

3 678 000 3 000 000 3 194 700 7 000

b d f h

0.1918 0.111222 0.019833 0.0927

b b b

0.00004802 1.2 × 2.43

10−5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

a d g j

5:1 7:15 1:2:1 5:2

1

2

3

4 5 6 7 8 9 10 11

1 b e h k

1:4 2:3 1:2:4 14:9

c f i l

3:2 7:2 3:1 3:4

2

6h a c

a c e g a c e a c e g h i a c a a a a b a a b a

$12.50 b $175 d $2.90 f 12 mL h 5 km/L b $128/m d 6 mg/g f 654 m/s b 880 mm/h d 400 m/s f 6090 mg/mL 4 800 000 mL/kL 12 600 000 mg/kg $337.50 b $78 000 0.00175 L/L b 14 mL b $183 b $11/$1000 $10.80/$1000 $13.50 b 600 km 6.5 L/100 km 891 km/h b

$196 $195 36 km 20 mL 1.5 m/s 112 L/min 14 g/L 200 cm/s 920 m/min 0.0575 km/s

160:80 b 144:96 40:200 d 140:100 150 g 5026 a $120 000 b $100 000 c $20 000 150 g of flour and 100g of sugar a $2.56 b $25.60 c $35.84 d $15.36 a $14.20 b $56.80 c $85.20 d $142.00 5 L or 1.67 L b 3 L a 3 a $218 750 b $156 250 c $125 000 a 3:2 b $900 c $1740 1 3 or 3.125 kg 8 $0.75 million $45 $6.95 29.16 mm

3

4 5 6 7

8 9 10

a c e g a c e g a c e a c a c a a c e

$4.80 $9.90 $536 $6.75 $22 $112 $3 $80 $24 $135 $37.20 $49 014.40 $131736.80 $185.25 $546 $15 810 $96 $20 $993.60 25% a $501.60 c $510 a 10%

1 2 3 4 5 6 7 8 9

32 h 0.175% 20.16 L 11 000 MJ

10

11

370 km

171 kL

Exercise 3F

Exercise 3D 1

Section II

Exercise 3E

12 13 14

b d f h b d f h b d f b d b d b b d f

$36 $22 $70.56 $0.12 $156 $822 $11.52 $0.70 $19.80 $6.60 $15 $100 551.60 $68 878.80 $234 $68.25 $13 438.50 $323.40 $1263.50 $84

b d b

$336 $249.60 $198

5

3 D 6 A 9 D

6

15

2 5 8

B D D

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1.2 m 0.000625 3.6 × 10−3 2.0 × 106 2:3 2:3:6 7:11 3:2 $73.60 $12.88 $0.46

$1408 $4.66 7.98

Exercise 4A 1

2

3

4

Section I D D C B

400 000

Chapter 4

Review 1 4 7 10

6.246 tonnes 0.05 b 3 000 000 0.1 m b 0.299% 4 800 000 b 190 5.08 × 104 b 3.81 × 108 1.512 × 1010 b 3.6 × 1012 2.3 × 106 a 5:1 b c 4:1 d e 4:3 f g 3:1 h a $18.40 b c $5.52 d e $0.92 f a $0.015/g b 240 m/min c 20 mm/min d 4.8 kg/mg e 14 000 000 mL/kg f 360 c/mg 640 km a $1473 b a $7.22 b c $13.49 d $110.16 a c a c a c a c a

a c e a c e

inquiry b primary organising d readable information f graphs False b True True d False True The six steps of a statistical inquiry: posing questions, collecting data, organising data, displaying data, analysing data and writing a report. Many people believe that information is more important than the natural resources as a source of social and economic power. Primary sources – interviewing people, conducting questionnaires or observing a system in operation. Secondary sources – data collected or created by someone else such as information gathered from newspapers, books and the internet. Frequency tables are used to organise ungrouped and grouped data. Cambridge University Press

475

Answers

9

10 11 12

2

3

4 5 6 7 8 9 10 11 12

a c e g i k m a c e g i k a c e g i k m o a a

Categorical b Quantitative Quantitative d Categorical Quantitative f Quantitative Categorical h Quantitative Categorical j Quantitative Quantitative l Categorical Quantitative Discrete b Discrete Continuous d Discrete Continuous f Discrete Discrete h Continuous Continuous j Discrete Continuous Continuous b Continuous Categorical d Categorical Categorical f Discrete Continuous h Discrete Categorical j Discrete Categorical l Continuous Continuous n Discrete Continuous p Categorical Quantitative b Continuous Quantitative b Discrete Categorical and nominal Categorical and ordinal Quantitative and discrete Categorical and nominal Quantitative and discrete Quantitative and continuous a Quantitative and discrete b Categorical and ordinal

Exercise 4C 1

a c

Sample Census

3 4 5 6 7 8 9

10

14

Census Sample

f h j b d f h j

Sample Census Census Random Stratified Random Random Random

173 6% a a a

124 200 440

b b b

8 22 15

c c c

116 28 18

36 a b

440 i 11

1

ii 15 iv 16

23 b 65% School population does not represent the views of the entire population. 56 students a 13 females b 12 males a No opportunity to give a reason or state what part of the policy is good. b Biased as it states that the boss is lazy. This may not be the interviewee’s opinion. a Inaccurate. Survey is biased as people at the festival have a liking for country music. b Systematic survey of people across the country.

a

f

g

3

a b c a b c a b c

The questionnaire has seven questions with a category for the interviewee to choose. Categorical 7 d 0 How do you rate the decorations? Excellent, good, average, disappointing How far have you travelled to Carter’s Place? 0–5 km, 5–10 km, 10–15 km, more than 15 km Where would you improve the menu? Entree, main, dessert Investigation Investigation Investigation Investigation Investigation Investigation Investigation Investigation Investigation

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Biased question. It assumes the interviewee likes the local member.

Review Section I 1 4 7 10

2 5 8

D C B C

A C C

3 C 6 A 9 D

Section II 2

a c

b c e

2

5

1

Exercise 4D

4 b d

Sample Census Sample Systematic Stratified Systematic Systematic Stratified

iii 13

11 12 13

Exercise 4B 1

2

e g i a c e g i

3

4 5

6 7 8

9 10

a c a c e g i a c e g

False b False False d True Categorical b Categorical Quantitative d Quantitative Quantitative f Categorical Quantitative h Quantitative Quantitative Discrete b Continuous Continuous d Discrete Discrete f Discrete Continuous h Discrete Categorical and ordinal a Systematic b Stratified c Random d Systematic e Random f Stratified g Random h Systematic i Stratified a 100 b 33 c 67 a 21 b 19 a Stratified b Systematic c Random Stratified sample using year groups as the strata Biased question. It assumes the interviewee likes the swimming centre.

Chapter 5 Exercise 5A 1

a c

b

Cost (c) c

Weight (w)

Cost of chocolates

60 50 Cost ($)

8

Information is often displayed using graphs such as dot plots, sector graphs, histograms, line graphs, stem-and-leaf plots and box-and-whisker plots. Graphs make it easy to interpret data by making instant comparisons and revealing trends. They help people to make quick and accurate decisions. a Personal data would allow retailers to increase their sales by targeting their advertising campaign. b Opinion. c Opinion. d Opinion. Investigation. a Opinion. b Opinion. Questions may be asked to mislead the interviewee and result in inaccurate data.

Answers

7

40 30 20 10 w 1

d

2 3 4 Weight (kg)

5

$18 Cambridge University Press

a c

b

Cost (c)

d

Time (t)

0 −33 −22 −1 −11 1 2 3 −2 −3

1.2

Cost ($)

1 0.8 6

0.6

2

3

4

0.2

c

0

2.5

5

7.5

10

Science experiment

6

2

10

1 m

5.7 1, 2, 3, 4, 5 0, 4, 8, 12, 16 −2, 1, 4, 7, 10 3, 2, 1, 0, −1

0

1

2

3

4

v

30

25

20

15

10

11

12

13

14

J

13 12 11 10 9

1 2 3 4 Emily’s age (years ( )

v

0

10

20

30

40

NZD

0

12

24

36

48

Conversion rate NZD 50 40

20

Computer value

10

30

AU AUD

x

10 20 30 40

20 15

Exercise 5B

10

1

5 t

y

1

3 2 1 0 −33 −22 −1 −11 1 2 3 −2 −3

x

8

c e a

10 8 6 4 2

2 3 4 Time (years ( ) d

$3000 3 years

c

$2000

3 4

d

0

10

20

30

40

C

3

23

43

63

83

C

Taxi hire

a a

4 b −1 −12 d 4 2 b −5 1 − 2 2 b −1 Gradient of 2, y-intercept of 2

10 8 6 4

80

a

2

a c a

y

b

b

0 −33 −22 −1 −12 1 2 3

E

AUD

25

0 −33 −22 −1 −11 1 2 3 −2 −3

Age relationship

30 b

3 2 1

c

b

t

y

b

2 3 4 Weight (kg)

6

Value ($100)

5

7

c a

Cost ($)

4

10

a

w

2

b a b c d a

J

8

4

9 12 15 Mass

4

6 4

6

3

8

8

3

2

10

4.5 t

Cost of apples

c

Cost ($)

d a

b

6

1

14

1

3 4 5 Time (min)

0

a

0

2

E

b

w t

3

p

0.4

1

a

3 2 1

Mobile phone costs

c

9

q

Jack’s age (years ( )

2

Preliminary Mathematics General

Time

Answers

476

2

60

x

40

1

2

3

4

20 d 10 20 30 40 Distance (km)

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Cambridge University Press

Answers

Gradient of −2, y-intercept of −1

2 Gradient of − , y-intercept 3 of −3

d

y

0

1

2

x

−2 −4

c

2 2

x

4

6

c

8

1 Gradient of , y-intercept of 0 3

f

x

1

y

0 −1 −1 3 6 9 12

2 6

a

Gradient of 1, y-intercept of 3 y 6 5 4 3 2 1

3

t

0

1

2

3

4

d

0

150

300

450

600 4

b

x

y 3 2 1

f g h a c a c

y=

a b

Gradient is 2, y-intercept is 3

400 300

y

2 3 4 Time (hrs) d

150

0

n

0

1

2

3

4

c

0

16

32

48

64

3 2 1 0 −1 −1 1 2 3 4

x

0 −2−1 −1 1 2 −2

200

t

1 , y-intercept of 1 2

y = −3x + 12

d

y

500

1 c a

1 x −1 2

7 6 5 4 3 2 1

x

7

0

Gradient is 4, y-intercept is 2 Gradient is 3, y-intercept is −7 Gradient is 5, y-intercept is 0.4 Gradient is 1.5, y-intercept is −2 1 Gradient is , y-intercept is 3 2 Gradient is −3, y-intercept is 5 Gradient is 1, y-intercept is 0 Gradient is 5, y-intercept is 2 y = 3x + 2 b y = −2x + 10 y = −4x − 1 d y = 0.5x + 1 y=x+1 b y = −2x − 1

100

0 −2−1 −1 1 2 −2

d

16

600

Gradient of −1, y-intercept of 1

Gradient of

Train travel

d

Distance (km)

0 −1 −1 1 2 3

x

0 −2−1 −2 1 2 −4

x

a b c d e

4 2

4 3 2 1

2 3 4 Weight (kg)

Exercise 5C

Gradient of 2, y-intercept of 0

y

c

10 1

0 −1 −2 1 2 3 −4 −6

x

b

30 20

8 6 4 2

4

a

40

y

6

5

50

Gradient of 4, y-intercept of −3

e

Gradient of 1, y-intercept of −1 y

d

60 x

0 −1 −1 1 2 3 −2 −3 −4 −5 −6

Cost ($)

−1

70

1

2

Steak

y

y

4

−2

b

Answers

b

477

x

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

5

a b

y = 3x y 6 5 4 3 2 1 0 −1 −1 1 2 −2 −3

x

Cambridge University Press

Answers

478

6

Preliminary Mathematics General

a b

p = 21h p

100 c h 2

4

6

d = 50t d

y

7

5

x

4

y

2

1

4

a–f

6

8

1

x

0 −2 −1 −1 1 2 3 −2 −3 −4 −5

t 2

x 1

−1

0

1

−1

0 −1 −1 1 2 3 4 5

−2 f

e

y

a–f

y b

a

3 2 1 0 −2−1 −1 1 2 −2 −3 −4 f e

0 −1 −2 1 2 g

x

0 −4−2 −2 2 −4 −6 h

d

4 3 2 1

4

a b c d

2 boxes to break even. Profit of $10 Loss of $5 Initial costs are $10

5

a b c d a

10 packs to break even. Loss of $50 Profit of $100 Initial costs are $100

x

y 2

y

0 −2−1 −1 1 2

The point of intersection is (1, −3)

8 6 4 2

c

The graphs for equations a, b and c intersect on the y-axis at 1 (y-intercept of 1). The graphs for equations d, e and f intersect on the y-axis at −2 (y-intercept of −2)

a

x

y

f

x

0 −3 −2−1 −1 1 2 3 −2 −3 −4 −5 −6

d

The graphs for equations a, b and c are parallel with a positive gradient. The graphs for equations d, e and f are parallel with a negative gradient.

4

6 5 4 3 2 1

5 4 3 2 1

x

2

3

The point of intersection is (2, 4)

y

e

1

2

3

yc b a 2

10

(2, 0) (0, 1)

6

d

100

−2

b d

(−1, 2) (–2, 0)

3

200

g

y

0 −1 −1 1 2 3 4

300

9

a c

2

3 2 1

8

400

g

1

8

50

8

x

0 −1 −1 1 2 3 4 −2 −3 −4

150

a b

Exercise 5D

1

200

7

y

b

x 6

3 2 1

y

0 −2 −1 −1

2 1 0 −2−1 −1 1 2 −2

b

3 2 1 0 −2 −1 −1

1 2

x

x

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1 2

x

Simultaneous solution is 2 1 x = − and y = 3 3

x

y

i

y

y 4 3 2 1 0 −1 −1

1 2 3

x

Simultaneous solution is x = 2 and y = 3 Cambridge University Press

479

Answers

b

y 1

3000

x

y

x

2

a 3

Dollars ($)

100 Income

60

4

Costs

40

5

20 2

3

4 5 Books

Initial costs are $24. Break even point is 3 books. Profit of $24.

a b c

a = 100 + b a + b = 1500

6 7

e a c a

y

Cost ($) a = 100 + b

750 a + b = 1500

8

b d a

$15 $15

250 a 250 500 750100012501500

9

a

1500 L 10 minutes

Data (d)

250 CNY 70 AUD

50 40 30 20 10

$3.50 $3.00 $90 000

d 1

2

$2 500 300 000 $87.5 billion

d a

2

3

4 5 Data

6

2.5

AUD

0

10

20

30

40

JPY

0

9

18

27

36

b

$25 C = 0.5n + 15

JPY

AUD converted to JPY

40

Parking fee

20

20

15

10 AUD 10 20 30 40

5

1250

d

b

Cost (c)

10

1500

500

a c

30

b

1000

1

c Internet access plan charges

500 L b 2100 L d 1900 L 400 CNY b 20 AUD d Gradient is 5 $1.50 b $1.50 d $40 000 b 48 months v = −2.5t + 120 $115 000 f $125 billion b 1 d A= N 4 $250 billion $15 b 30 calls d

c

6

b c d

a c e a c e a c a c d e a

3 C 6 B

D C D

Section II

Cost ($)

1

80

Break even point is 40 items.

2 5 8

A C C

Exercise 5E

Simultaneous solution is x = −1 and y = −1

8

1 4 7

10 20 30 40 50 60 Items

0 −2−1 −1 1 2 −2 −3

1

Section I

Costs

500

c

16 GBP c 25 AUD Gradient is 0.4 and the vertical intercept is 0 GBP = 0.4 × AUD or y = 0.4x

Review

1500 1000

3 2 1

7

e

2000

Simultaneous solution is x = −4 and y = −19 d

Income

2500

Dollars ($)

0 −6 −4−2 −4 2 −8 −12 −16 −20 −24

b d

Amy’s wage is $800 and Nghi’s wage is $700. i C = 200 + 40n ii I = 45n

GBP

3 6 9 Time (hr)

3

x c

$10

4

a c a

3 2

b

1 −2 y 2 1 0 −2−1 −1 1 2 −2

Australian dollars to British pounds

x

Gradient is 1 and y-intercept is 1.

25

y

b

20

5 4 3 2 1

15 10 5 AUD 10 20 30 40 50

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

0 −1 −1 1 2 3

x

Gradient is −2 and y-intercept is 5. Cambridge University Press

Answers

c

Preliminary Mathematics General

c

x

7

8

8

n 1

2

15 16

9 seconds

a b

n I

5

f

6

7

2000

1

2

3

4

5

I

0

30

60

90

120

150

0

1000 n 1 b

Simple interest on $600 at 7% p.a.

c

250

d

7

2

3

4

5

6

7

1

$105 f $180 g 12 years Investigation graphics calculator Simple interest on $1000 at 4% p.a.

70

140

3 210

4 280

5

6

Simple interest on $50 000 at 7% p.a.

3000 2000 1000 n 2

300

4

6

8

10 12

b

250 200

Simple interest on $50 000 at 7.5% p.a.

I

150

4000

100

3000

50 n 1

2

3

4

5

2000 1000

6

$240 5

n

a

2

c

I Simple interest on $5000 9%

2500

2

4

4000

e

I

3

6% earns $500, 9% earns $750 and 12% earns $1000 6% earns $3000, 9% earns $4500 and 12% earns $6000 6% takes 4 months, 9% takes about 2.7 months and 12% takes 2 months.

I

n 1

2

a

50

2000

7%

1500

5%

1000

0

6%

3000

150

I = 70n

9%

4000

0

3 4

12%

6000 5000

$420

100

Exercise 6B 1

4

200

$5850 b $910 c $37 000 $1134 e $15 225 $600 b $1089 c $5020 $10 005 e $11 518.75 1.2% b 0.4% 2.4% d 3.6% $8 b $144 $1440 d $900 $55 200 $69.75 a $22 000 b $122 000 $24 $5040 a $7680 b $7560 c $7200 Computer application $71.75 $40 000 7.5% 1 3 years 3 a $51 000 b $2000

5 6 7 8 9 10 11 12 13 14

3

$175 3 years I = 30n

I

a d a d a c a c

4

2

n

Exercise 6A

3

e g a b

2

Chapter 6

2

I Simple interest on $100 000

100

c,d

6

a

200

6

2 4 Time (seconds)

1

6

300

4

t

c

d

400

Speed of a rocket

v

5% earns $625, 7% earns $875 and 9% earns $1125 5% earns $1250, 7% earns $1750 and 9% earns $2250 5% takes 4 years, 7% takes about 2.9 years and 9% takes 2.3 or fewer years.

c

500

Gradient is 3 and y-intercept is −2. 1 a y = 2x − 1 b y = − x 2 a 2 roses b $10 loss c $20 profit d Initial costs are $20 a $18 000 b $24 000 c v = −0.5t + 30 d $27 000 e 60 months f $500 a, b

6

Simple interest on $1000 at 7% p.a.

I

0 −2−1 −1 1 2 −2 −3

5

b

c,d

y 3 2 1

Speed (km/h)

Answers

480

4

6

8

10 12

Simple interest on $50 000 at 8.1% p.a.

I 4000 3000 2000

500

1000 n 1

2

3

4

5

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

n 2

4

6

8

10 12

Cambridge University Press

Answers

1 2 3 4 5 6 7 8 9 10 11 12 13

a c a c

$899.89 b $11 578.20 $15 577.17 d $27 621. 40 $10 063.79 b $102 028.69 $1307.84 d $23 509.61 $27 209.78 $12 107.45 $2536.50 $6433.75 a $20 766.90 b $7266.90 $39 604.68 $53 608.98 $49 662.32 Computer application $2340.76 Investment 2 by $80

0

n I

2

1400

70

1300

65

1200

60

1100

55

1000

50 2

4

b c d 6% 4%

1

2

1000 n b c

2000

d

n e f g a b

2

3

4

5

about $2290 about $2425 about 1.5 years A = 800 × (1.07)n 0

1

2

3

6

4

a

4% is about $1150, 6% is $1230 and 8% is about $1310 4% is about $1225, 6% is $1350 and 8% is about $1475 4% takes about 10 years, 6% takes about 7 years and 8% takes 5 years. Compound interest investments I (thousands) 170

5

800 856 916 980 1049 1122

160

c

150

I

140

Compound interest at 7% p.a.

10 15 20

3

4

5

6

12%

n

900

12 24 36 48

800

b n 1 e f g

2

3

4

about $950 about $1275 about 3.5 years

5

6

7 c

a c e a c e a c e a c e g a c e a c

$1200 $59 520 $22 400 $10 510 $2333 $153 956 $1330 $610 $5750 $5205 $71 760 $67 500 $101 340 $174 250 $243 600 $6339 $13660 $571 755

1 2

100

1000

6% is about $103 000, 9% is about $105 000 and 12% is about $106 000 6% is about $127 000, 9% is about $143 000 and 12% is about $161 000

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

6

8

10 12

Investment B is about $56 500 Investment C is about $56 300 About 1 year

6%

110

1100

4

Exercise 6F

120

1200

B A

9%

130

1300

C

Exercise 6E

1250 Compound interest at 4% p.a.

D

2

6

1500

2250

I

5

1750

2500

n

4

Compound interest investments I 8%

5

2

3

About $1500 a

c

1

Compound interest investments

n

n 1

2000 2080 2163 2250 2340

I

a

I (thousands)

2000

3

6% takes about 36 months, 9% takes about 24 months and 12% takes 18 months

1500

5

1

7

1600

A = 2000 × (1.04)n

a b

Compound interest at 7% p.a.

I

Exercise 6D 1

d

Investigation graphics calculator

3

4

b d

$38 880 $6720

b d

$86 400 $580 000

b d f b d f

$3120 $692 $10 080 $15 450 $220 600 $105 100

b d f b d

$113 208 $81 225 $395 580 $46 296 $561 507

$37.35 $3000 million $1000 million $1500 million $500 million $600 million 54% $35.00 b $11.40 d $8.10 f $30 b Bank B d $5 f $35.75 h Bank B j

a b c d e f a c e a c e g i

$2.80 $5.60 $25.00 $35 Bank D $15 $38.75 Bank D

Cambridge University Press

Answers

3 4

Exercise 6C

481

Answers

482

Preliminary Mathematics General

c,d

Exercise 6G 1 2 3 4 5 6 7

8 9 10 11 12

I

$88 373.39 $535 092.25 a $5151.43 c $1764.48 $2360.28 $1900 a $4.14 c $5.22 a $340 342 c $393 824 e $1 531 538 a $130 c $20 a 7.32% a $1215.32 a $29 386.56 c 9 years 5%

2 3

4 5

6 7 8 9 10 11 12

a c e

$2988 $5574 $10 351.88 $18 622.50 a 2.49% c 4.86% e 6.30% 6.67% a $124.80 c $1282.95 e $8805.32 $0.12 a $1297.05 a $14 155 c $965 $120 a $7.30 c 22.62% $0.64 $252.40

$7543.25

200 100 n

b d b d f b d b b b

$3.83 $6.03 $923 785 $704 994 $1 162 024 $50 V = 20n + 50 8.59% $515.32 $9386.56

b d f

$852.40 $3393 $219 330

b d f

7.38% 5.59% 2.50%

b d f

$310.11 $377.09 $794.80

b b

36.43% $900

b

$1.90

2 5 8

3 6

A D C

$300 $64 286 a $980 a I = 50n

b

6 7 8 9 10 11 12 13 14

d e f 17

B B

Section II 1 2 3 4

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Section I B D D

1

2

3

4

5

e a b

$275 $135 061 $151 497 $39 905 $52 071 a $122 360 c $637 600 a $38.15 a $710 517 a $3.02 $758.90 4.96% $273.50

e

1.08 × 109 $11.20 i $30 loss ii $10 profit iii 4 b $4000 3V 3 c i ii 0.78 m 4π d $1152 i Qualitative categorical data ii Quantitative discrete data iii Quantitative continuous data

e f a

Chapter 7 b d b b b

$586 950 $81 088 $8.60 $230 517 $4.95

Exercise 7A 1

a

HSC Practice Paper 1

Review 1 4 7

18

300 b

Exercise 6H 1

Simple interest on $1000 at 5% p.a.

C B C A D B A C B C D C C D B a

$367.20 ii 23.5% b 8 ii 16 c 780 kg i $4502.63 ii i 10x − 2 ii $25 000 a i –3 ii iii y = −3x + 2

b

n

0

1

2

3

4

5

I

0

50

100

150

200

250

i iii i iii

b

i

c

i iii

d

$2940

2

x = 15

ii

b a

$112.80 −12 $1002.63 288x7y6 2 1 x=− 2 26.1%

$1786.47 ii $1000.30 The sample has not been picked at random. The five students selected are not in school uniform and may not be representative of the school. It is possible that these students do not like the school uniform.

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

b d

Score

Tally

Freq.

20

|||

3

21

|||| ||

7

22

|||| |

6

23

||||

4

24

||

2 c

21

22

Number of calls

Tally

Freq.

0

|||| ||

7

1

|||| |||| ||

12

2

|||| |||

8

3

||||

4

4

||

2

5

|

1 c

1 27

2

3

Score

Tally

Freq.

0

|||| |

6

1

|||| ||

7

2

|||| |

6

3

||||

5

4

|||

3

5

||

2

6

|

1

Cambridge University Press

483

Answers

5

6

a

b e

7

a

b d

Score

Tally

Freq

7

|||

8 9

8

a

Score

Tally

Freq

3

1

||||

4

|||| ||

7

2

||||

4

||||

4

3

|||| |

11

4

||

2

5

|||| |||

8

6

|||| |

6

7

||||

4

b c d

8 23 − 27 36

3

Class

Class Centre

Tally

Freq

170–174

172

||

2

175–179

177

||

2

180–184

182

|||| |||| ||

12

10

|||

3

11

||

2

12

|

1

Score

Tally

Freq

8

|||

3

185–189

187

|||| |||

8

91

||||

4

9

||||

5

190–194

192

|||| |

6

92

||||

5

10

0

93

|||

3

11

0

94

||||

4

12

95

|||

3

96

|||

3

97

||||

5

98

|||

3

b c

Score

Tally

Freq

1

|||| |||| |

11

2

|||| ||||

9

3

||||

5

|

1

Exercise 7B 1

a

4

|||| |||| |

11

Class

5

|||| |

6

5–19

12

10

6

|||| ||||

9

20–34

27

8

35–49

42

6

50–64

57

4

Total

28

c

5 − 19

Score

Tally |||| |

6

33

||||

5

34

|||| ||||

9

35

|||| |||

8

36

|||| |||

8

37

|||| |

6

38

|||| ||||

9

51 34 and 38 s

c e

2

b d a

Freq

32

20 29.41%

a b c

3 letters Small amount of data indicates that the conclusions are not reliable. However a word length of 3 is significantly more common than any other word lengths.

Class Centre

51 c 31 d 1 and 4 No. The outcomes for a fair die are equally likely; however, this is a small sample. Many more rolls of the die are required to produce outcomes that are closer to the theoretical frequencies.

4

10 28

Freq

5

a

b

0–9, 10–19, 20–29, 30–39 and 40–49 4.5, 14.5, 24.5, 34.5 and 44.5

Class

Class Centre

Freq

0–9

4.5

3

10–19

14.5

11

20–29

24.5

29

30–39

34.5

46

40–49

44.5

1

130–139, 140–149, 150–159, 160–169, 170–179 and 180–189 134.5, 144.5, 154.5, 164.5, 174.5 and 184.5

c

Class

Class Centre

Freq

130–139

134.5

2

140–149

144.5

3

150–159

154.5

9

Class

Class Centre

Tally

Freq

3–7

5

||||

4

160–169

164.5

16

8–12

10

|||| |

6

170–179

174.5

8

13–17

15

||

2

180–189

184.5

2

18–22

20

|||| ||

7

23–27

25

|

1

28–32

30

|||

3

33–37

35

|||| |||

8

38–42

40

||||

5

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

6

a b

0–9, 10–19, 20–29, 30–39, 40–49, 50–59 and 60–69 4.5, 14.5, 24.5, 34.5, 44.5, 54.5 and 64.5

Cambridge University Press

Answers

4

484

Preliminary Mathematics General

Class Centre

Freq

0–9

4.5

7

10–19

14.5

11

20–29

24.5

6

30–39

34.5

3

40–49

44.5

1

50–59

54.5

1

60–69

64.5

1

4

Score

60%

1

2

b d a

3

Freq

Cum Freq

||||

4

4

5

||||

4

8

6

||||

5

13

7

||

2

15

8

||||

5

20

9

||||

5

25

10

||||

5

30

d f h

30 5 20

i

a

b d a

Tally

Score

Freq

Cum Freq

4

4

4

5

6

10

6

7

17

7

10

27

8

5

32 c e

5 10

5 6

Freq

Cum Freq

20

4

21

7 j 2 k l 15 m 26.6% Graphics calculator investigation a

Tally

Freq

Cum Freq

61

|||| |

6

6

62

|||| |||| |||

13

19

63

|||| |||| ||||

14

33

64

|||| ||||

9

42

65

||

2

44

66

|

1

45

c

45

d

4

e

33

f

3

7

g

4.44%

h

22

10

17

23

12

29

Exercise 7D

24

6

35

1

25

5

40 c

3 11

2

Tally

Freq

Cum Freq

8

||

2

2

9

||||

5

7

10

|||| |||| |

11

18

11

|||| |

6

24

12

||

2

26

13

||||

4

30

6 19 45 93.33%

8

9 10

11

a c e a c e g a c a c e a c e

b d f b d f h b d b d f b d f

3 41 101 4 5 21 11 10 10.5 146 110 185 27 6 30.5

21 201 99 5.5 27 9 10 6.5 4 210 122 63 33 15 37

Exercise 7E 1

a 10 8 6 4 2 3

4

5

6 7 Score

8

3

4

5

6 7 Score

8

b 10 8 6 4 2

35

Score

5 22 6, 8, 9 and 10 1 15 73.33%

Score

22 32

Number of calls

7

11 10

4

c e g

Exercise 7C

c e

30 5

Frequency

d

Class

b d a

Frequency

Answers

c

3 4

5

6

a c e g a c e a c a c e a c e a b

17 b 21 12 d 9 10 f 11 12 h 16 21 b 32 34 d 10 28 f 11 5 b 8 5 d 8 7 b 12 16 d 6.5 21 f 9 10 b 5.5 3 d 27 11 f 10 Range = 17; IQR = 12 Range = 10; IQR = 5.5

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

2

a

Score

Tally

Freq

1

|||| |

6

2

|||| ||

7

3

|||| |

6

4

||||

5

5

|||

3

6

||

2

7

|

1

Cambridge University Press

9

Answers

c

7

Score

Freq

Clm. Freq

6

60

2

2

5

61

3

5

4

62

3

8

63

4

12

64

2

14

65

5

19

66

3

22

67

8

30

7

6 5

Frequency

Frequency

5 a

4 3 2

3 2

1

1 1

2

3

4 5 Score

6

7

3

4

c

7

8

4 a

7

Tally

Freq

b

17

||||

4

30

3

18

||

2

2

19

|||

3

20

|||

3

21

|||

3

22

||

2

23

|||

3

24

||||

4

5 4

Clm. freq

Score

6 Frequency

5 6 Score

1 2

3

4 5 Score

6

7

3 a

Tally

Freq

4

|||| ||

7

5

|||| |

6

6

|||

3

7

|||

3

8

|

1

b 5 Frequency

Score

10

60 61 62 63 64 65 66 67 Score

c 30 Cum freq

1

20

20

10

4 3 2

59 60 61 62 63 64 65 66 67 Score

1

b

17 18 19 20 21 22 23 24 Score

7 c

5

6

5

4

Frequency

Frequency

6

3 2

d 12 days e 22 days

4 3 2

a c e g i k

10 50 0 0 50 500

b d f h j

90 50 50 150 50

1

1 3

4

5 6 Score

7

8

17 18 19 20 21 22 23 24 Score

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Cambridge University Press

Answers

b

485

486

Preliminary Mathematics General

d

Clm. Freq

30 25

10

10

10

20

90

100

30

50

150

40

50

200

50

0

200

60

50

250

70

0

250

80

150

400

90

50

450

100

50

500

a

25

15 10

e g a

22 f Q1 = 21, Q3 = 23 2 Suggested classes: 38–42, 43–47, 48–52, 53–57 and 58–62

b

Freq

Clm. Freq

1

38–42

40

5

5

2

43–47

45

6

11

20

7

7

48–52

50

8

19

21

6

13

53–57

55

7

26

22

6

19

58–62

60

4

30

23

4

23

c d

24

3

26 28

26

2

30

7

c

3 4

6 5

5

4

5

2

4

1

b d

30 60

240 225

0

5 10 15 20 25

0

5 10 15 20 25 30

0

2

4

6

8 10 12

a c a c d a c e f

27 b 42 22 5m b 8m 18 m at East Park East Park with IQR of 10.5 123 b 180 154 d 141.5 161.5

g

Data is symmetrical with a median of 154 and IQR of 20.

a c e f

30 73.5 77.5

3 2

35 40 45 50 55 60 65 Class centre

1

120 130 140 150 160 170 180

e 19 20 21 22 23 24 25 26 27 Score

6

8

c

7 6 Frequency

7 6 Frequency

a c a

b

7

6

10

See above

8

3

35 40 45 50 55 60 65 Class centre 50 h Q1 = 45, Q3 = 55

Exercise 7F

Class Centre

Clm. Freq

2

g i

Class

Freq

25

15

5

19 20 21 22 23 24 25 26 27 Score

8

20

10

5

Score

b

Frequency

30

20

Frequency

7

Freq

f

Frequency

Score

Cum freq

Answers

l

5 4

Grace

4

Holly 30 40 50 60 70 80 90 Assessment result

2

2

1

1 20 21 22 23 24 25 26 Score

94 67

5

3

3

b d

g 40 45 50 55 60 Class centre

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Grace was more consistent with a higher median than Holly. Grace had the highest and lowest score. Cambridge University Press

487

Answers

a

Sou Am th eric a

g

Asia

Andrew

9

8

9

b a

T N b

c

18% F

S

G c

20

68%

2 3 4

a c a c

b d b d

1

a

2

b d f a

72° a d

b e

108° 90°

c

36° 54°

5

72°

A

8

i iii

20 km/h ii 15 km/h 10 km/h iv 15 km/h Day 5 c Days 2, 6 and 8 25 km/h e 10 km/h 10 km/h

Feb Mar Apr

Sep

15

11 pm

Dogs

May Aug

7 am

10

Cats

b d

0

a

11 am

Mon

day

day Tues We dne sda y y

rsda

b d e

c

48

75

c

See above See below 30

2

See above

Wat er

15

Juice

a c a

b d

16 9 and 16

13 332

8 6

20 11 pm

See above c See above Sales have steadily increased over the 12-month period. The exception was the month of May.

10 3 am

25

a

Jun

Jul

Exercise 7I 1

3 pm

Thu

y

ida

Fr

7 pm

day

Sun

Saturday

7

See above

6 Jan 5 4 3 2 1 0

Oct

5

b

c

See above

Nov

3 am

B

20

No pets

6

b a

Dec

25

Other

5

6

Exercise 7H

240° 90° 270° 324°

3

4

C

5

180° 60° 144° 54°

2

7

Exercise 7G 1

1

12 10 8 6 4 2 0

USA

60 64 68 72 76 80 84 Golf scores

Lachlan has a lower spread of scores. He is more consistent than Andrew.

a

Europe

Lachlan

h

4

P aci fic

8

a c e

4

7 am

2

10 5

Soft drink

1

0

Milk

3 7 pm

b a

2

3

4 c

3

5

6

5

6

5

11 am

10

b J

M

S

8

W

3 pm

3

f a b c

See above i 30° ii 20° iii 15° iv 15° May and August 15° d July

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

6 4 2 1

2

3

4

Cambridge University Press

Answers

Graphics calculator investigation. 63 b 74 77 d 74 71 f 77

Canada

7 8

b a

b

5

6

7

c

5

2

9777664 5 58 733 6 1456889 75221 7 123889

b

Ages for admission is younger for males than females. The median for males is 63 for males and 68 for females. The admission for males has a greater spread than the females. 34 b 24 28 d 15 Lower extreme = 6; lower quartile = 15; median = 24; upper quartile = 29; upper extreme = 34

a

b c

0 1 2 3 4

4 79 0124566778 00113347 016

51

|||

3

3

52

||||

5

8

53

|||| ||||

9

17

20

54

|||| ||||

9

26

10

55

|||| |||

8

34

56

|||| |||

8

42

57

||||

5

47

58

||||

4

51

b e

c

51

d

26

Apr

8

51 52 53 54 55 56 57 58 Score

Class centre

Freq

4–8

6

5

9–13

11

6

14–18

16

8

19–23

21

4

Total

23

a b c

See above 26 14 30 Lower extreme = 9; lower quartile = 14; median = 25; upper quartile = 30; upper extreme = 35 8 20.5 24

Chapter 8 Exercise 8A

40 30

1

20

g a c a c e f

54 197.5 189.5 41 67.5 66.5

Q1 = 53, Q3 = 56 244 216.5 72 57.5

h b d b d

Nikolas

2

3

4 5 6

Samuel

Class

c a b c d

4

1

a

Mar

5

Section II 1

May

6

2

Feb

0

7

4

30

Jun

8

3

Jan

40

7

51 52 53 54 55 56 57 58 Score

3 A 6 C

D A D

50

See above

9

3 2 5 8

B D A

b

Cum Freq

10

Section I

Six sectors each with an angle of 60o

Freq

50

16 Lower extreme = 4; lower quartile = 23; median = 27.5; upper quartile = 33; upper extreme = 46

a

Tally

f

Review 1 4 7

6

12

Score

Australian tourists spent less nights away from home in the past year than New Zealand tourists. The median for the Australian tourist is 7 and the New Zealand tourist is 22. New Zealand tourist has a greater spread than the Australian tourist.

a

a c e

b a

5

876554433 0 56 75411 1 24479 Aus NZ 1 2 2233689 3 2

Frequency

4

Preliminary Mathematics General

Cum freq

Answers

488

40 50 60 70 80 Jelly bean guesses 5

Rent 2 3 Food

Petrol

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

10.0 cm b 13.0 cm 26.0 mm d 22.4 mm 4.9 cm f 83.1 cm 9.00 cm b 14.70 cm 11.53 mm d 21.17 mm 4.21 cm f 10.42 cm y = 55 b a = 55 x = 23 d d = 25 b = 62 f m = 14 9.0 m a x = 126.00 b y = 5.74, x = 8.31 a x = 3.6, y =1.7 b x = 21.1

Exercise 8B 1

Save

a c e a c e a c e

4

a b c

21.6 cm 41.2 m 59.0 m 49.2 m a 40.0 m b 12.4 cm c 26.5 mm 43.69 cm

Cambridge University Press

Answers

6

7 8

9

10 11

a b c a c e a c a b c a c e

18.8 m 12.6 cm 88.0 mm 25.1 cm 213.6 mm 69.1 m 25.7 m 41.1 m 25.00 m 10.71 mm 33.56 cm 36m 47 cm 24 m 28 cm a 8.49 cm

b d f b

b d f b

119.4 m 157.1 mm 18.8 cm 10.8 m

16 m 22 m 23 m 28.97 cm

Exercise 8C 1

2

3 4 5 6

7 8 9 10 11 12 13 14 15 16 17

149.5 m2 19.5 mm2 100.8 m2 37.2 cm2 198 m2 14 m2 42 cm2 11 046 m2 154 cm2 a 5.8 km2 c 15.0 mm2 e 75 mm2 7 m2 49 m2 27 tiles a 1380 cm2 c 134 cm2 50 cm2 42 cm2 a 253 m2 a 8 cm c 160.5 cm2 50.27 mm2 a 5.66 m a 32 cm2 c 6.87 cm2 a c e a c e

b d f b d f

4 cm2 72.2 m2 40.4 m2 71.7 m2 70 mm2 21 cm2

2

3 4

5

a c e a c e a a c e a

1950 m2 3800 m2 7992 m2 1500 m2 7500 m2 3150 m2 1767 m2 345 m2 297 m2 6038 m2 2211.5 m2

1 2 3 4

5 6 7 8 9 10 11 12

b d f

b

b b

b b

1 2

4 5 6 7 8

$5060 6 cm 80 m2 8

b

4400 m3

b d f

5832 cm3 450 m3 3079 mm3

b b

2513.3 m3 48 m2

b

180 m3

9 10 11

1200 m2 7350 m2 1767 m2 1312.5 m2 7210 m2 1100 m2 37 m 2046 m2 3350 m2 48.38 m2 41 m

6 7 8 9

b

3

Four cans with 300 mL remaining. 300 mL b 20 60 d 30 4 b 2 70 d 34 900 f 0.5 43 h 30 000 103 j 7000 5 l 8 000 000 200 mL 720 000 L 60 mL 3L a 96 000 000 mL b 0.04 mL c 5.65 mL d 270 000 000 mL e 147 000 000 mL f 471.24 mL 421.88 mL a 12.6 m2 b 31.4 m3 c 31 kL a 217.5 m3 b 218 kL

B B C

2 5 8

D B A

3 D 6 B 9 B

Section II 1 2 3

a c

35.00 38.47 10 mm a 26.2 cm c 69.1 cm e 21.4 mm

b d f b

17.5 m2 47.0 cm2 3.0 m2 0.16 m2

b b

49 m 144 m3

b

3217 mm3

6L

4

5

6 7 8 9 10 11 12 13 14 15

3 2 b 14 7 1 5 c d 7 14 a 0.118 b 0.294 c 0.412 d 0.176 a 10.5% b 21.0% c 14.5% d 17.0% e 21.5% f 15.5% a 20% b 30% c 10% d 15% e 25% 1 5 a b 3 6 1 1 c d 8 2 a 10% b 75% c 37.5% d 80% 6.25% 0.96 71 134 Computer application a 4 b 0.25 c 20 Investigation Investigation Investigation Investigation a

Exercise 9B

Section I 1 4 7

27.0 m2 125.4 cm2 28.0 mm2 11.52 m2 72 tiles 5720 m2 1331 mm3 24 mm3 201 mm2

Exercise 9A

376 cm2

a c a c e g i k

a c e a c a a c a

Chapter 9

1

Review b d f b d f b b d f b

5

2

82.8 cm2 31.8 cm2 120 m2

36 cm2

96 m3 112 m3 165 mm3 750 m3 a 108 m3 c 240 mm3 e 1583 mm3 307.1 cm3 1570.80 cm3 a 314.2 m2 a 600 m3 c 480 m3 a 15 m2 14 726.2 cm3 38.971 cm3 a 280 cm3 a c

Exercise 8F

3

Exercise 8D 1

4

Exercise 8E

b

15.65

b d f

26.8 cm 30.8 m 40.0 m

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1

a b

2

a b

3

a c

6 {RD-GD-GS, RD-GS-GD, GD-RD-GS, GD-GS-RD, GS-RD-GD, GS-GD-RD} 24 {DHSC, DHCS, DSHC, DSCH, DCSH, DCHS, HDSC, HDCS, HSDC, HSCD, HCSD, HCDS, SHDC, SHCD, SCHD, SCDH, SDHC, SDCH, CDHS, CDSH, CHDS, CHSD, CSDH, CSHD} 362 880 b 120 24 Cambridge University Press

Answers

5

489

Answers

490

4 5 6 7

8 9 10 11 12 13 14

Preliminary Mathematics General

120 a

b

3 628 800

36 a b

8 {HHH, HHT, HTH, HTT, TTT, TTH, THT, THH} c 16 {LM, LN, ML, MN, NM, NL} a 6 b {BR, BP, BG, RP, RG, PG} a 60 b 80 a 20 b 60 35 a 42 b 210 a 90 b 126

Exercise 9C 1

2

3

E1

6

F

G

X

XD

XE

XF

XG

Y

YD

YE

YF

YG

B

G

B

BB

BG

G

GB

GG

2×2=4

b a

Y2

Y3

G1

Y1

Y1Y1

Y1Y2

Y1Y3

Y1G1

Y2

Y2Y1

Y2Y2

Y2Y3

Y2G1

Y3

Y3Y1

Y3Y2

Y3Y3

Y3G1

G1

8

1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

10

5

6

7

8

9

10

11

6

7

b a

6 × 6 = 36

8

9

10

11

G1Y1

M4

E1M1

E1M2

E1M3

E1M4

E2M2

E2M3

E2M4

E3

E3M1

E3M2

E3M3

E3M4

D

E

A

AD

AE

B

BD

BE

C

CD

b a b

G1Y2

G1Y3

Q AQ

K

Q

G1G1 J

K

Q

J

K

KK

KQ

KJ

Q

QK

QQ

QJ

J

JK

JQ

JJ

15

b a

4 × 3 = 12 Leader

10

a b a b a

Q

KQ

J

KJ

A

QA

K

QK

J

QJ

A

JA

K

JK

Q

JQ

C AC

A

D AD A BA C BC

B

D BD A CA

Y

YY

N Y

YN NY

B CB

C

D CD A DA B DB

D

C DC

NN

{TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF} 8 c 2×2×2=8 {12, 13, 21, 23, 31, 32} 6 c 3×2=6 1st

AJ KA

Deputy

2nd

1st

J A

B AB

3×3=9 {HH, HT, TH, TT} 4 c 2×2=4

N

b 16

4 × 3 = 12 Tens

Units

3

2nd R

RR

G

RG

R

GR

G

GG

R

5

G

3 × 4 = 12

b a

Second card K AK

A

N

12

M3

First card

Y

11

M2

a

4 × 4 = 16

b a

9

12

M1

14

a

a

E2M1

13

b a

Coin

CE T

3×2=6 2×4=8 b

7

2×2=4

H

b 5

E

Y1

7

E2

4

D

720

2×3=6

Spinner R

HR

A

HA

G

HG

R

TR

A

TA

G

TG

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Exercise 9D 1

a c e

1 4 1 2 1 6

b d f

2 7 5 26 1 5

Cambridge University Press

Answers

a c

3

a c

5

a c

7

b

3 8

a c e g i

b d

13 20 4 5

1 3 2 9 1 52 1 2 1 13 5 13 1 26

b d b d f h

4 9 2 3 1 4 3 26 1 52 3 13

Disagree. Depends on the location

9

a

e 10

a c e

11

a c

12

a c e

13

a c

14

2

3

8

c

Exercise 9E 1

1 2 1 3

4

6

5 8 5 8 1 5 3 20

1 2 5 16 15 16 1 8 1 2 1 1 24 23 24 1 17 1 51 4 17 1 4 1 2

b d f b d f b d b d f b d

7 8 1 4 1 4 3 8 1 4 1 4 1 24 23 24 1 17 4 51 11 51 3 4 1 24

False. Not all outcomes are equally likely.

4 5 6 7

8

9 10 11 12

3 10 c 0 7 e 30 57 a 100 7 c 20 5 a 8 a 0.25 c 0.40 15% a 1 c 0.6 23 a 65 11 c 30 203 e 390 a 0.51 c 0.50 e 0.26 1 3 33 56 13 20 3 a i 5 1 iii 5 4 v 5 b i 2 iii 1 v 2 c Parramatta a

b

a b c d e

f

0.1

d

3:4

d

f

0.55

h

3

a

0.71

b

4

0.4 4 15

4:7 71 100

b

1 15 1

2 25

b d

3 8 0.15 1

b

0.7

b

b d f b d f

7 18 337 585 869 1170 0.49 0.47 0.53

e

6

7

ii iv vi ii iv

Selecting a red card from a normal pack of cards. Not winning first prize in Lotto. Throwing an odd number when a die is rolled. Obtaining a head when a coin is tossed. Drawing a club, diamond or heart from a normal pack of playing cards. Choosing a blue or red ball from a bag containing a blue, red and green ball.

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

a

3 8

b

0.375

c

37.5% 3 4 7 9 17 36 137 396 12 13 6 7 0.9 0.09 1 0.25 92% 90% 18% 13 16

a c e g

8

a

9

a

10

a c e g a c e

11

3 10 3 5 1 5 4 1

a c

5

Exercise 9F 1

b

g

4 5 38% 8 11 62.5%

2

12

a

b d f h b d f

9 11 40 99 259 396 25 44 25 26 5 7 0.85 1 1 0.75 93% 17% 75%

b

6

b d f h b b

Answers

2

491

Review Section I 1 4 7 10

C D C C

2 5 8

C C A

3 C 6 C 9 D

Section II 1 2 3 4 5 6 7 8 9

a c

0.20 b 0.24 0.32 d 0.24 17% 0.58 No. The applicants may not be equally good. 132 {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF} 1728 10 000 1 1 a b 50 2 Cambridge University Press

Answers

492

Preliminary Mathematics General

c e 10 11

12

1 2

1 4

c

1

a

a c

4 9 7 9 1 8 0

1 2 2 a 5 1 c 5 a 0.125 c 0.875 e 0.625 32 a 35 2 c 7 12 a 13 25 c 26 68.75% e

14

15

16

17

18

d f

2 5 7 50

9 10

b

b

b d f b

1 2

1 3

1 4 1 4 1 2

1

2 3 4 5 6 7 8 9

2 3 4

5 6 7 8

67.20 $915.20

a c e g i

$78 880 $79 280 $75 795 $71 480 $49 440 $64 378 a $3200 a $89 164 c $88 024 a $2210 a $58 240 a $64 642 a $72 280 c $824 a $99 580 c $1974.60

b d f h

$74 429 $59 896 $66 406 $60 110

b b

$72 280 $1140

b b b b d b d

$61 410 $56 780 $60 322 $301 $71 456 $234.60 $97 605.40

b d f b d b

0.5 0.125 0.375 3 35 4 5 3 4

1

2 3 4

5 6

Blue shirt – $165, Black trouser – $330, Belt – $45, Tie – $68, Dry cleaning – $128, Allowable deduction – $736 $1591 $4614 a $2456.40 b $896 c $2656.40 d $1357.20 e $2142.45 f $1972.60 $5107.20 a $24 000 b $48 000 c $72 000 d $96 000 $1890 a $775.50 b $519.59 c $348.12

7 8 9

a c e g i a a a c e g i a a c e g i

$345 b $1328 $609 d $704 $1017 f $3009 $2555 h $237 $1360 $846 b $12 366 $1872 b $38 392 $600 b $900 $1200 d $1500 $1800 f $2100 $2400 h $2700 $3000 $1778 b $23 938 $20 000 b $45 000 $150 000 d $112 620 $59 206 f $36 214 $140 368 h $75 412 $24 516 Computer application $79 560 $52 136

2

3 4

a c e g i a c e

B C E C A $1500 $17 100 $81 100 $3300 $31 000

8 9

10 11

12 13

14

15 cents $3600.30 $18 600.40 $733.50 b $16 789.80 $5944.20 d $32 960 $65 993.50 f $10 623 $377 owing $7273.60 owing $28 420 b $3363 $1577 $50 500 b $9750 $8320 Pay another $1430 in tax. $44 873.60 b 30.8% $25 480 b $29 308 Pay another $3828 in tax. 27.4% Computer application a $21 660 b $19 524 c Refund of $2136 d 23.7% a $1275 b $7582.50

1

2 3 4 5

6

7 8 9 10

a c e g i a a a c a c e g a c e a

$3.60 b $14.00 $17.00 d $0.32 $49.00 f $4.20 $4.29 h $260 $37.00 $62 b $682 $78 b $858 £26.25 b £14 £3.50 d £1.40 8820 b 2940 7140 d 5250 7560 f 2100 5880 h 2730 $16 b $22 $90 d $35 $14 f $180 $540 b $347 Computer application $329.09 $1681.82

Exercise 10F

Exercise 10D 1

7

a b c a c e a b a c a c d a a c d

Exercise 10E

Exercise 10C

Exercise 10A a b c d e

5 b d

6

1

Chapter 10

1

$47 160 $115.20 $54.72

a c

Exercise 10B

a

c 13

19 50 1 5

b d f h

A D C D

b d f

$4350 $35 640 $18 600

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

1

2

3 4 5

a c e a c e

$0 $750 $9000 $60 000 $40 000 $89 000 $80 000 a 0 c 0.30 a $8000 c $22 600

b d f b d f b d b d

$25 500 $16 500 $21 000 $100 000 $50 000 $67 000

0.15 0.45 $40 000 and over $2600

Cambridge University Press

Answers

Tax payable (thousands of dollars)

60

a c e

40

2 5

D C B

3 B 6 B

A B

Section II 1 2 3 4 5 6 7 8 9

$5773 $77 520 a $1270.50 a $96 720 c $3230 a $53 738 c $785.37 $40 532 $690 a $28.60 a $5000 b $2500 c 0.25

2 b b d b d

$2579.50 $493 $93 490 $52 358 $13 650

b

$314.60

Exercise 11A

2

3 4

5 6 7 8

9

a c e g a c e g a c a c e g a c a a c a c e a c

9 34 2 1 7 20.5 120 50 10 16 6 9 8 9 20 14 27 17 26 25 22 22 16 3

3 4

5 6 7 8 9

Chapter 11 1

1

7 1100 100 10 16 12 3 10.5 19

b d f h b d b b d b d

25 7 5 9.5 1 6 9 7 24 19 4

b d

13 14

1

Freq ( f )

fx

3

66

8

4

23

3

69

10

7

12

6

24

4

96

14

3

25

2

50

26

5

130

a

b

30

22

x = 24.18

a c e g a c e g a c a c e g

9 8 7 11 14.0 9.5 11.0 10.2 1406 69.95 10 2 13 9

d b d f h b d f h b

28 5 42 14 7.0 9.5 2.2 8.8 70.30

b d f h

6 22 5 39

2 3

12 7.89 b 7 4 Computer application 1.96 15

a

Freq ( f )

fx

4

3

12

Freq ( f )

fx

2

5

10

3

6

18

4

7

28

5

6

30

6

5

30

x = 4.00 15.36 6.00

a c

Score (x)

Score (x)

a b a

Exercise 11C b d f h b d f h b

Score (x) 22

Exercise 11B

Section I 1 4 7

c

Frequency

Taxable income (thousands of dollars)

Review

8 10

Size

20

11

b d

14 6

Text

Freq ( f )

fx

0

4

0

1

3

3

2

5

10

3

5

15

4

7

28

5

4

20

b 4 5

2.7 30.89

5

5

25

6

2

12

7

2

14

Class

Class Centre

f

fx

8

3

24

15−19

17

20

340

20−24

22

29

638

25−29

27

22

594

30−34

32

18

576

35−39

37

32

1184

x = 5.80 b

a

Score (x)

Freq ( f )

fx

15

2

30

16

3

48

17

4

68

40−44

42

18

756

18

2

36

45−49

47

23

1081

19

3

57

162

5169

x = 17.07

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

b

162

c

32

Cambridge University Press

Answers

10

6

493

Answers

494

6

Preliminary Mathematics General

a

4

a

1.01

5

a

x = 14 and σ n = 2.97

Class

Class Centre

f

fx

980−984

982

2

1964

985−989

987

16

15792

990−994

992

130

128960

995−999

997

352

350944

1000−1004

1002

353

353706

1005−1009

1007

128

128896

Exercise 11E

1010−1014

1012

19

19228

1

1015−1019 b 7

8

a c

1017

1 c

1001

b

160−169

Height range

Class Centre

f

fx

140–149

144.5

2

289

150–159

154.5

8

1236

160–169

164.5

6

987

170–179

174.5

3

523.5

180–189

184.5

1

184.5

d

161

a

65−69, 70−74, 75−79, 80−84, 85−89, 90−94 67, 72, 77, 82, 87, 92

b c

f

40%

f

fx

65–69

67

3

201

70–74

72

4

288

75–79

77

7

539

80–84

82

8

656

85–89

87

3

261

90–94

92

5

460

30 80−84

e g

7 8

3

4

5%

Class Centre

Class

d f

e

6

2

3

5

6

7

80.2 65−69 and 85−89

Exercise 11D 1

2 3

a c e g a a

2.8 8.0 0.0 2.4 0.37 1.7

b d f h b b

2.0 2.5 10.2 49.9 0.43 2.2

1.32

x =12.1 and σ n = 4.68 Andrew’s results are more consistent as the standard deviation is smaller. a 4.3 b 7.8 c 3.4 d 44.9 e 3.6 f 3.8 a 4.15 b 6.53 21.5 b c

1017

999.5

b

8 9

a d

8.3 b 8.0 c 7.0 Both mean and median are central and typical of the data. The mode is the first score (7). a Mean = 12.5, median = 7.4, mode = 2.3 b Median is central and typical of the data. The mode is the first score (7) and the mean is distorted by the larger scores (40.2). a Mean = 1.2, median = 0, mode = 0 b Mean is central and typical of the data. The mode and median are affected by the first score (0). a Mean = 1.6, median = 1, mode = 0 b Median is central and typical of the data. The mode is the first score (0) and the mean is distorted by the outliers. a mean = 21.4; median = 18 b mean = 17.1; median = 18; Removing the outlier has affected the mean but has not affected the median. a Mean = 85.1, median = 95.5, mode = 96 b True but misleading. Molly’s score is only above-average because the mean is affected by the outlier. She’s below the median. a i 3.5 ii 3.6 iii 2.0 iv 1.6 b i 3.0 ii 3.4 iii 3.0 iv 1.7 c Median for Sample A is larger (3.5 against 3.0). Sample A has a larger mean (3.6 against 3.4) and sample B has a larger mode (3 against 2). Sample B has a larger standard deviation (1.7 against 1.6). Computer application Median is better measure to analyse real estate prices as it is not distorted by outliers.

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Review Section I 1 4 7 10

2 5 8

D B A B

3 B 6 B 9 C

C B C

Section II 1 2 3 4 5 6 7 8 9 10 11 12

13

14

a a a a a

11 12 16 7.5 21.5

b b b b b

40 16 27 c 7.3 c 22.0

4 22.0

7 a

8.5 2.03 a 3.4 a 2.5 d 3.1 11

b e

b

2.0

b

16.9 c 0 f 4

2.9 3.0

a

Class

Class Centre

f

fx

10–12

11

9

99

13–15

14

3

42

16–18

17

4

68

19–21

20

1

20

22–24

23

1

23

b c d a

11, 14, 17, 20, 23 10−12 14 Mean = 74.4, median = 78.5, mode = 90 b Both mean and median are central and typical of the data. The mode occurs at the end of the data. Median is central and typical of the data. The mode is the first score (0) and the mean is distorted by the outlier.

Chapter 12 Exercise 12A 1 2

3

A–1, B–9, C–4, D–3, E–2, F–6, G–7, H–5, I–8 1 a 2 b 4 1 c 4 d 3 1 e 1.5 f 3 a 2 b 3 c 15 d 30 Cambridge University Press

495

Answers

a b

5 6 7 8

9

a b

Regular pentagons have equal angles (108°). 4 2 or 6 3 i

2:1

ii

2:1

2 Dimensions of the screen are 108 cm by 135 cm. a False b True a Three angles of one triangle equal three angles of the second triangle. b 3 c m = 5 cm d n = 6 cm a Three angles of one triangle equal three angles of the second triangle. (common angle A and a right angle) x+ y a b = = b b z y

Exercise 12B 1 2

3 4 5 6 7 8 9 10 11 12 13

4

5 6

7 8 9

10

11

12

a c e a a c e

200 m b 100 m 50 m d 10 m 34 m f 80 m 37.5 km b 675 km 10 mm b 16 mm 20 mm d 24 mm 30 mm f 48 mm Map distance is 7 mm. a 1125 mm or 1.125 m b 0.04 m or 40 mm a 1 mm to 1.16 m (approximation). b Height of the antenna is approximately 57 m. a 10 mm b 24 mm c 40 mm d 600 mm e 80 mm f 480 mm a 1 mm to 1 m or 1:1000 b 30 m c 15 m d 15 m a 1 to 1000 b 200 m c 150 m d 500 m

x=6 b x = 16 x = 4.2 d x = 29.4 a = 4, b = 6 b x = 10, y = 8 p = 7.2, q = 15 c = 16, d = 5 Height of tree is 7 m. Height of building is 12 m. The tower is 40 m in height. Block of units is 13.5 m in height. Flagpole has a height of 4.5 m. Height of building is 21.6 m. Height of the light pole is 2.6 m. a x = 1.5, y = 2 b z = 21 Lucas is 2 m tall. a Height of the lighthouse is 12 m. b Height of the wall is 4 m. a c a c d

1

2

3

a b c d e f a b c a b

a

c d

4m 32 m b

Three angles of one triangle equal three angles of the second triangle. c Height of the tree is 8 m. Feet of the ladder are 187 cm apart.

e

f

Exercise 12C 1

2

3

a c e a c e a

5

a b

6

a

b

Exercise 12D

1m

14

c

2m b 3.4 m d 8.5 m f 80 mm b 1.6 mm d 2 mm f 1:2 b 1:50

h = 10, o = 8, a = 6 h = 13, o = 12, a = 5 h = 5, o = 3, a = 4 h = 39, o = 15, a = 36 h = 15, o = 9, a = 12 h = 30, o = 24, a = 18 h = z, o = x, a = y h = c, o = a, a = b h = f, o = e, a = d 4 3 sin θ = , cos θ = , tan θ = 5 5 12 5 sin θ = , cos θ = , 13 13 12 tan θ = 5 3 4 sin θ = , cos θ = , tan θ = 5 5 5 12 sin θ = , cos θ = , 13 13 5 tan θ = 12 3 4 sin θ = , cos θ = , 5 5 3 tan θ = 4 4 3 sin θ = , cos θ = , 5 5 4 3 x sin θ = , cos θ = z a sin θ = , cos θ = c tan θ = a b

c

4 3

3 4

d

7

a

b

8

a

3

tan θ =

1m 2.8 m 4.9 m 30 mm 140 mm 55 mm c 1:300 000

4

a b

e d , cos θ = , f f e tan θ = d i sin θ ii cos θ iii tan θ i cos θ ii tan θ iii sin θ 11 60 i sin θ = , cos θ = , 61 61 11 tan θ = 60 60 11 ii sin φ = , cos φ = , 61 61 60 tan φ = 11 3 4 i sin θ = , cos θ = , 5 5 3 tan θ = 4 4 3 ii sin φ = , cos φ = , 5 5 4 tan φ = 3 8 15 i sin θ = , cos θ = , 17 17 8 tan θ = 15 15 8 ii sin φ = , cos φ = , 17 17 15 tan φ = 8 q p i sin θ = , cos θ = , r r q tan θ = p p q ii sin φ = , cos φ = , r r p tan φ = q 15 8 sin A = , cos A = , 17 17 8 15 sin B = , cos B = 17 17 3 4 sin A = , cos A = , 5 5 4 3 sin B = , cos B = 5 5 sin θ =

5 4

x y , tan θ = y z b , c

θ

i 5 3 4 ii sin θ = , cos θ = 8

5

b 8

5

10 θ 6

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Cambridge University Press

Answers

4

Answers

496

Preliminary Mathematics General

i 6 4 3 ii cos θ = , tan θ =

3

5

8

c

12 13 14

25 2

θ 7 i 24 24 24 ii sin θ = , tan θ =

9

a

25 7 Hypotenuse is 4 and 6

b

cos 60° = 0.50

c

sin 60° = 0.87

3

4

5

Exercise 12E 1

2

3

4

5

6

7

8 9

10

11

a c e g a c e g a c e g a c e g a c e g a c e a c e a c a c e g a c e a c e

60′ 1200′ 30′ 12′ 2° 1° 0.5° 0.75° 0.34 2.14 0.23 0.97 0.45 0.61 0.39 2.48 3.5 4.8 4.0 1.7 35 16 30 21°16′ 81°46′ 36°52′ 2.04 55.80° 5.96 13.24 5.09 7.25 60° 68° 30° 40°54′ 10°33′ 11°25′

b d f h b d f h b d f h b d f h b d f h b d f b d f b

420′ 3600′ 20′ 300′ 8° 10° 0.25° 0.33° 0.73 0.31 0.99 1.11 0.65 0.82 1.36 0.41 4.0 5.9 5.9 6.6 81 37 51 40°13′ 75°31′ 53°8′ 1.85

b d f h b d f b d f

1.12 4.48 1.57 21.80 24° 59° 45° 25°40′ 22°12′ 25°14′

9° 24° 64°32′

b 24° b 0.9 b 0.903

c c c

68° 0.44 0.4299

Exercise 12F 1

24

a a a

a d g a c e a c e a c e a c

7.73 b 17.09 c 14.24 8.39 e 65.82 f 12.37 51.31 h 30.18 i 7.88 24.38 b 33.16 30.55 d 16.68 16.67 f 4.75 50.81 b 6.73 141.38 d 61.58 95.78 f 15.84 68.4 b 90.9 126.8 d 25.5 66.7 f 88.7 175.918 b 105.541 10.251

Exercise 12I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Exercise 12G 1

2

3

4 5 6

a c e g i a c e a c e a c a c a b c

47° b 47° 25° d 67° 52° f 40° 49° h 56° 55° 51° b 51° 50° d 58° 47° f 43° 51°19′ b 57°22′ 48°39′ d 47°15′ 23°49′ f 32°35′ 50° b 32° 61° 53° b 62° 44° θ = 67° 23′ φ = 22° 37′ θ = 28° 04′ φ = 61° 56′ θ = 36° 52′ φ = 53° 08′

Exercise 12H 1 2 3 4 5 6 7 8 9 10 11 12

Height is 15.0 m Pole is 4.50 m high River is 40 m wide Depth is 40.2 m Angle is 34° Angle is 44° a Horizontal distance is 3.8 km b Height is 1.3 km Ramp is 8.77 m Angle is 1°26′ Length of rope is 6.6 m a Ladders reach 3.83 m b Angle is 33° Angle is 38°

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18 19

Height is 752 m Height of the tower is 116.5 m Boat is 107.23 m from the base of the cliff. Plane was 17 326 m from the airport Height of the tree is 55.6 m Depth of the shaft is 74 m Angle of depression is 9° Angle of elevation is 23° Angle of elevation is 34° Angle of depression is 2.5° Angle of elevation is 3° Angle of elevation is 34° Angle of depression is 47° Launching pad is 45 m in height. Height of the tree is 10 m. Angle of depression is 20°58′ a Height of hill is 210 m. b Height of hill and tower is 281 m. c Height of tower 71 m. a x = 53 b y = 113 c Man is 60 m from the boat. Boat travels 500 m towards the cliff.

Review Section I 1 4 7

2 5 8

B B D

D C D

3 A 6 A

Section II 1

2 3 4

5 6 7 8 9 10 11

a = 16, b = 5 2 b x = 9, y = 18 3 1 c m=5 ,n=4 3 d y = 8, h = 14 Height of building is 7 m. a 30 b 24 c 18 12 5 a sin θ = b cos θ = 13 13 12 c tan θ = 5 a 2.48 b 0.97 c 0.39 d 0.14 a 61° b 19° c 11° a 8.58 b 21.38 c 85.99 a 48° b 28° c 37° a 50°43′ b 69°5′ c 49°32′ Pole is 4 m high. Ship is 103.2 m from the base of the cliff. a

Cambridge University Press

Answers

l

Exercise 13A

60

3

4 5

b

6 7

c e g i a c a b c

b d

$1.17 $4.37 $16.37 i $29.00 i $600 i $0.25 80 $50.30 50 $3.77 11.34 am $0.86 $3.26 i $39.00 iii $4.37 i $69.00 iii $3.74 194 234 137 Casual $696 $5.28 i $2250 iii $500 i $133 iii $79 Pro plan

50

$2.77 $8.37

ii ii ii e b d f h b d ii

$2.06 $2.70 $3.10 1230 $5.46 $22.73 2 10 min 30 sec $1.82 $29.18 $3.02

ii

$2.60

d f h

428 39 Frequent

b d ii iv ii iv d

$151 800 MB $1200 $0 $94 $99 Normal plan

20

3

a

Phone charges

7

75 60 8 9

45 30 15

b d

$0.15 $0.30 $0.15

1

$0.15 $0.60

b c d e a c e f g

2 3 4 Time (min)

$0.15 $0.90 $40.95 See above $15 b 0.20 d c = 0.30t + 15 c = 0.20t + 18

5

0.30 0.10

10 11

1

Total monthly charge

30 Charge ($)

Phone charges

75

25 20 15

60

10

45

5

2

30 20 40 60 80 100 Monthly calls

$10 $30 80 calls 10 calls $0.50 $0.25

2

3 4 5 Time (min) b d f h j

$20 $35 30 calls 0.5 0.25

6

Exercise 13C 1

a c d e f

2048 B b 4096 KB 7 516 192 768 KB 3 145 728 MB 9216 GB 6 291 456 B

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

g h i j k l a c e g

8.796 × 1012 TB 5120 TB 4 294 967 296 GB 2 621 440 MB 4915.2 B 5 557 452.8 MB 1 KB b 2 GB 6 TB d 3 GB 5 GB f 8192 MB 7 KB h 3072 TB 2 097 152 KB a 20 971 b 10 240 c 16 d 2 097 152 a 2.8 GB b 3.7 GB c 6.4 GB a 3 KB b 4 TB c 2 MB d 0.0031 TB e 2.0078 MB f 7 MB a 201 050 MB b 47 450 MB c 1 044 826 MB d 6250 MB a 3296 MB b 1748 GB a 200 000 B, 2000 KB, 200 MB and 0.002 TB b 0.002 TB 200 MB, 2000 KB and 200 000 B a 5 days b 4767 days a 4 MB b No. A typical song has a file size larger than 4 MB.

Exercise 13D

35

1

a c e g i k

6

90

15

2

5

105

90 Cost (cents)

3 4 20 40 60 80 100 120 140 Monthly calls

Exercise 13B a c e f

2

30

10

4

1

40

Cost (cents)

2

a c e a b c d a c e g a c a

Change ($)

1

Total monthly charge

3 4 5

6

a b c d e f g h i j k l a c e g i

6000 Kbps 2 000 000 000 Kbps 3000 bps 5000 Gbps 9 000 000 Mbps 7 000 000 bps 7000 Gbps 8 000 000 000 bps 2 000 000 000 000 bps 2100 bps 7 300 000 Mbps 4 800 000 Mbps 9 Gbps b 9 Kbps 0.003 Gbps d 8 Tbps 4780 Mbps f 5.5 Gbps 6700 Kbps h 4.9 Mbps 3200 Tbps j 2.4 Kbps Research 7 000 000 Mbps a 4s b 2000 s c 0.000001 s d 8 000 000 s e 0.4 s f 8s a 700 000 bps, 7000 Kbps 700 Mbps and 0.007 Tbps. b 0.007 Tbps, 700 Mbps 7000 Kbps and 700 000 bps.

Cambridge University Press

Answers

Chapter 13

497

Preliminary Mathematics General

a c a c a c

b d b

2448 b 0.408 s 3 309 568 b 0.413 696 s 5 570 560 b 163.84 s 5033 s a 5 MB c 18 000 MB

8 9 10 11

Review

816 s 0.001224 s 1654.784 s

b d

111 411.2 s 0.139264 s

b

300 MB

a c e g a c d

2

3

1 4 7

1 2

92 b 82 82 d 73 49 f 24.808 24.872 h 458 29.47 s b 28 s 27 s and 28 s Median. Outlier of 57 affects the mean.

3 4

5

a

b c d e a b c

4

5

a c a

6

Score

Freq.

Cum. Freq.

0

7

7

1

7

14

2

5

19

3

4

23

4

4

27

5

3

30

6

7

0 or 1 video files 2 2 23.3% i 296.4 ii 208.9 i 18.6 ii 17.3 Mean and standard deviation is higher for traditional than the online. Spread of the data is similar. 30.8 b 13.4 13.4

1 2 3 4 5

100

7 8

50

9

b d

20

30 File size

40

15 s c 146 s Time taken increases at a constant rate as the file size increases.

3 C 6 B

A D A

50

$3.89 $3.95 b 440 d $59 f $634 h $99 Plan j $99 a $0.20 b c $0.60 d e $0.40 f a 7000 bps b c 13 000 Kbps d 5 000 000 Kbps e 4 Tbps f g 9 Gbps h a 6 291 456 MB b 4608 GB c 3 221 225 472 B d 204.8 GB e f 6 GB g h 24 MB a 147.29 b c 47 a c e g i

$3.84 750 $99 $279 $59 Plan $0.20 $0.40 $0.20 2000 Gbps

1230 Mbps 2.8 Gbps

2

10

11 12 13

a c a c a c a c a c a c a a c a c a b c a c

16% 10% $29 500 $14 660 $5350 $15 785 $24 600 $56 160 $12 280 $30 280 $76 800 $42 300 $6400 $2700 $32 940 $87 000 $23 880 $26 000 $71 240 $23 400 $239 $321 $9026 Investigation

3 4 5 6

5 TB 2 GB 145

a d a b c d a

$432 b $1183 c $528 $396 e $1175 40% 40 and 60 age groups 10% Charge a higher excess. $474.60 b $1200 $1225.75 a $602 400 000 b $99 600 000 c $260 400 000 d $237 600 000 a Penrith b Yes c $770 d $540 e Brand A f $994.75 g $685.00 h $658.33 i $879.00 j $556.20

Exercise 14C 1

2

3 4

a d g a d g a c a c e

b e h b e h

$978 $414 $1735 $330 $1983 $1435 $355 $1071 $600 $1350 $10 000

$795 c $1350 $1068 f $717 $4025 i $5105 $745 c $1140 $1563 f $2508 $2172 i $2606 b $1160 d $861 b $2100 d $2600 f $64 000

5 3500 3000

Exercise 14A

6

10

1 2 5 8

Chapter 14

150

0

D B A

Section II

Exercise 13E 1

Exercise 14B

Section I

b d b d b d b d b

25% 12.3% $13 870 $53 050 $2886 $26 080 $26 880 $126 880 $18 000

b

$84 300

b b d b

160 weeks $30 240 $5940 $110 880

b d

$404 $101

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Stamp duty

7

Time (sec)

Answers

498

2500 2000 1500 1000 500 0

a c e

20000

40000 60000 80000 Market value

$750 $2300 $40 000

b d f

100000

$1500 $20 000 $97 500

Exercise 14D 1 2

3 4 5

a d a c e g a a a c

9.55 b 8.82 c 5.06 8.21 e 9.32 f 3.5 271.6 L b 73.2 L 65.8 L d 30.9 L 98.2 L f 342.9 L 404.9 L 17.453 L b $27.92 525 km b 8 L/100 km 444 km b 37.2 L $50.23

Cambridge University Press

Answers

480 km Once. Distance required is 772 km. Distance travelled on one tank of petrol is 625 km. a 35 L b 55 L c 45 L/100 km d 50 L/100 km e 70 km/h f 5L a $1320 b $1280 c $2600 d $2864 a $102.75 b $105.74 c $2.99 a 33.6 b 46.2 c Tyler 1747.2 L, Oscar 2402.4 L d $2446.08 e $1897.90 f 3 years g Research Research

9 10 11

12

Exercise 14E 1

a c e a c

$3040 $3040 $3040 $6950 $3550 $33 300 a $16 000 c $4000 e $2000 $49 950 $21 400 a 9 years a After 8 years a $6400 $2400 $3000 a $8000

2 3 4

5 6 7 8 9 10 11 12 c

1

2 3 4 5 6

7 8 9 10 11

8 9

a b c

$18 480.00 $15 523.20 $13 039.49 $4740 a $6532 b $10 968 $32 752 a $5046 a $4000 c 2.5 years e About $1250 f About $2600 17.27% a 27.143% 8.9% After 4 years a $5120

10

b b d

11 12

$14 854 $1000 3.5 years

13 14

b

$13 536 15 16 17

b

b d f b d

$12 160 $9120 $6080 $5250 $1850

b d f

$4000 After 2 years $14 000

Year

Current Depreciated value Depreciation value

1

$32 000

$5120

$26 880

2

$26 880

$4301

$22 579

3

$22 579

$3613

$18 966

4

$18 966

$3035

$15 931

5

$15 931

$2549

$13 382 b

c b b b

b

14 years After 9 years $19 200

30000 18 Value ($)

8

7

Exercise 14F

$32 000

20000

1

$82 000

$8000

$74 000

2

$74 000

$8000

$66 000

3

$66 000

$8000

$58 000

4

$58 000

$8000

$50 000

0

1

2

3 4 Years

Exercise 14G

d 1

Value ($)

80000

60000

2

40000

3

20000

4 5 1

2 Years

3

Speed (km/h)

Stopping distance (m)

10

2.1

20

4.2

40

8.3

80

16.7

Yes. Table shows the stopping distance is close to doubling when the speed doubles. a

Speed (km/h)

Braking distance (m)

10

0.60

20

2.35

40

9.41

80

37.65

5 b

4

6

a c e a c e a c e

90 km/h 80 km/h 96 km/ 168 km 146 km 196 km 2h 2.5 h 6h 227.27 km/h a 70 km c 77.78 km a 45 minutes

b d f b d f b d f

97 km/h 84 km/h f 48 km/h 392 km 70 km 154 km 2.5 h 5h 8h

b

0.9 h

b

64 km/h

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

Yes. Table shows the braking distance is close to quadrupling when the speed doubles.

19

a

20

b c d a

b

25 m b 20 m 45 m 47 m b 49 m 12 m b 25 m 43 m d 66 m 94 m f 126 m No. Stopping distance is 15 m. Sarah would hit the child at a greater speed. 89.510 m b about 90 m 81 km/h b 63 km/h 48 km/h d 115 km/h 34 km b 2 km 91 km d 272 km 5 h 29 min b 50 min 5 h 6 min d 9 h 21 min 20 min f 6000 min 1224 km/h b 102 km 0.34 km d 48 min 169 km/h b 5 h 55 min

a a c a c a c e a c a a

10000

Current Depreciated Year value Depreciation value

0

a c a a c e a b

18  v2  d−  5V  170  0.6 s 1.8 s 3.0 s

t=

Reaction time (sec)

Reaction distance (m)

0.50

8

1.00

17

1.50

25

Reaction distance increases when the reaction time increases.

Cambridge University Press

Answers

6 7

499

Preliminary Mathematics General

4

Exercise 14H a c e a c e a c a c e a c e a

2

3 4

5

6

0.09 b 0.10 d 0.28 f 0.02 b 0.01 d 0.16 f 0.09 b about 6 hours 3 h 44 min b 5 h 48 min d 6 h 8 min f 4 h 56 min b 11 h 28 min d 3 h 20 min f

0.07 0.10 0.17 0.10 0.20 0.04 0.11

5 6

8 h 12 min 10 h 12 min 11 h 28 min 1 h 56 min 1h 52 min 20 min

a c d f h a c a c

2004.6 b 1786 1789, 1652, 1786, 2182, 2814 1225 e 877.5 507.8 g 454.2 28.1% 37 km/h b 36 km/h 36 km/h d 4.18 km/h 22 b 12.55 12.5

7 160 140 Distance (km)

1

80 60

BAC

8

0.1

a b

0.05

b

3 Drinks

4

5

6

Body Weight 115 kg

0.2

9

1

2

3

4

0.1

0.05

0

7

8 9 10 11

1

a c e a a a a b

2

3 Drinks

4

5

1.4 b 8.0 1.5 d 0.9 0.6 0.075 b 0.045 3 h 5 min b 5 h 45 min 3.2 b 6.8 Male – 4.7, Female – 7.6 Males are larger in size.

Exercise 14I 1 2

3

a c a c e a c

6

1h 5h 3.88 c/km $734.50 $250.00 4L $13.05

b d b d b

80 km 40 km/h Brand A $183.00 11 L

5 6 7 Time (h)

8

9

10

Mean = 1.23, Median = 0, Mode = 0 It depends. Mean has been affected by outlier of 6, but even without this outlier the average would still be 0.83 accidents/day. Median gives a good picture of a typical day, but conceals the fact that many days have multiple accidents.

a

0.15

a b

$765 $1600 $500 a 510 km b 38.25 L c $58.14 a $2322 b $1376 a $5046 b $14 854 36 m a 39 m b 29 m a 0.084 b 5.6 hours a Mean = 27.2 Median = 27.5 Mode = 23 b Mean and median are a better measure than the mode for the centre. Data is skewed. c Range = 27 IQR = 8 σn = 6.90

HSC Practice Paper 2

0.15

2

6 7 8 9 10 11

100

0

1

4 5

120

20

0.2

0

3

40

Body Weight 45 kg

BAC

Answers

500

1 4 7 10 13 16

C A A B B a

b

17

c a

Class

Class centre (x)

Freq. (f)

f×x

20–29

24.5

85

2082.5

b

30–39

34.5

72

2484

c

40–49

44.5

71

3159.5

50–59

54.5

55

2997.5

60–69

64.5

36

2322

b d

319 26.65%

c e

D

60

50 40

45

50 A d

2 5 8

A B D

3 C 6 C

Section II 1 2

i

41 28.53%

Section I A A C

B 3 C C 6 D A 9 C C 12 D B 15 D i $1306.60 ii 1 $681.10 2 $4573.10 i $59 000 ii $11 550 iii $885 iv $12 435 v Pay an additional $180 i −10a − 5 ii 5x5 i 9.7 m ii 11.7 m iii 13.9 m 7 12 i ii 13 13

C

Review 1 4 7

2 5 8 11 14

16.7% $61 800 $22 800

a b

© The Powers Family Trust 2013 ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

18

e a b c d

ii i ii

m2

7350 $3878.80 3.23% x=4 i 45 kg iii 19 i $3764.48 i 25.45 m3 2 i 3

ii iv ii ii

76 kg 300 Heater C 25 447 L

ii

7.5 cm

Cambridge University Press

B