cambridge_igcse_mathematics__core_and_extended_coursebook_with_cd_rom___cambridge_education___cambri_samples.pdf

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Mathematics

Core and Extended Coursebook

Morrison

Cambridge University Press and Cambridge International Examinations unique partnership enables thousands of students to successfully pass their exams through the provision of comprehensive, high‐quality and endorsed resources. Cambridge University Press works closely with CIE, and experienced authors, to produce high‐quality and endorsed textbooks and software that support Cambridge teachers and encourage Cambridge learners.

Cambridge IGCSE

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The author has substantial experience of teaching and developing physics resources for a range of markets and is a member of the Institute of Physics. — Student’s CD‐ROM offers full support for independent learning and a dedicated miniwebsite. — Topics are all introduced in terms of their to add relevance toBack life in Cover the 21stCopy century. — Example examination questions and worked answers with commentary.

Karen Morrison and Nick Hamshaw

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Physics IGCSE is a resource that has been written and developed to provide full support for the IGCSE syllabus offered by CIE (0625). The book is in full colour and includes a free CDROM containing a range of supporting activities. The materials are fully endorsed by CIE.

Cambridge IGCSE Mathematics Second edition

Cambridge IGCSE Mathematics Second edition

To find out more about Cambridge University Press visit www.cambridge.org/cie

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University of Cambridge International Examinations

© Cambridge University Press 2012

Karen Morrison and Nick Hamshaw

Cambridge IGCSE®

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Mathematics Core and Extended

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Coursebook

© Cambridge University Press 2012

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cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107606272 © Cambridge University Press 2012 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 2012 Printed in the United Kingdom at the University Press, Cambridge

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A catalogue record for this publication is available from the British Library ISBN-13 978-1-107-60627-2 Paperback with CD-ROM Cover image: Seamus Ditmeyer/Alamy

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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 are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

notice to teachers References to Activities contained in these resources are provided ‘as is’ and information provided is on the understanding that teachers and technicians shall undertake a thorough and appropriate risk assessment before undertaking any of the Activities listed. Cambridge University Press makes no warranties, representations or claims of any kind concerning the Activities. To the extent permitted by law, Cambridge University Press will not be liable for any loss, injury, claim, liability or damage of any kind resulting from the use of the Activities. IGCSE® is the registered trademark of University of Cambridge International Examinations.

© Cambridge University Press 2012

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Contents Introduction

v

Acknowledgements

vii

Unit 1 2 3 6 10 13 14 18

Chapter 2: Making sense of algebra 2.1 Using letters to represent unknown values 2.2 Substitution 2.3 Simplifying expressions 2.4 Working with brackets 2.5 Indices

23 25 27 31 33

Unit 2

Chapter 3: Lines, angles and shapes 3.1 Lines and angles 3.2 Triangles 3.3 Quadrilaterals 3.4 Polygons 3.5 Circles 3.6 Construction

44 53 57 59 62 63

Chapter 4: Collecting, organising and displaying data 4.1 Collecting and classifying data 4.2 Organising data 4.3 Using charts to display data

74 76 83

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Chapter 1: Reviewing number concepts 1.1 Different types of numbers 1.2 Multiples and factors 1.3 Prime numbers 1.4 Powers and roots 1.5 Working with directed numbers 1.6 Order of operations 1.7 Rounding numbers

99 100 105 110 114 115

Chapter 6: Equations and transforming formulae 6.1 Further expansions of brackets 6.2 Solving linear equations 6.3 Factorising algebraic expressions 6.4 Transformation of a formula

119 121 123 124

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Chapter 5: Fractions 5.1 Equivalent fractions 5.2 Operations on fractions 5.3 Percentages 5.4 Standard form 5.5 Your calculator and standard form 5.6 Estimation

Chapter 7: Perimeter, area and volume 7.1 Perimeter and area in two-dimensions 7.2 Three-dimensional objects 7.3 Surface areas and volumes of solids Chapter 8: Introduction to probability 8.1 Basic probability 8.2 Theoretical probability 8.3 The probability that an event does not happen 8.4 Possibility diagrams 8.5 Combining independent and mutually exclusive events

129 141 143

154 154 156 158 160

Unit 3

Chapter 9: Sequences and sets 9.1 Sequences 9.2 Rational and irrational numbers 9.3 Sets

165 170 172

Chapter 10: Straight lines and quadratic equations 10.1 Straight lines 184 10.2 Quadratic expressions 198 Chapter 11: Pythagoras’ theorem and similar shapes 11.1 Pythagoras’ theorem 11.2 Understanding similar triangles

207 211

11.3 Understanding similar shapes 11.4 Understanding congruence

216 224

Chapter 12: Averages and measures of spread 12.1 Different types of average 12.2 Making comparisons using averages and ranges 12.3 Calculating averages and ranges for frequency data 12.4 Calculating averages and ranges for grouped continuous data 12.5 Percentiles and quartiles

Examination practice: structured question for Unit 1-3

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231 233 235 237 240 245

Contents

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Unit 4 Chapter 13: Understanding measurement 13.1 Understanding units 13.2 Time 13.3 Upper and lower bounds 13.4 Conversion graphs 13.5 More money Chapter 14: Further solving of equations and inequalities 14.1 Simultaneous linear equations 14.2 Linear inequalities 14.3 Regions in a plane 14.4 Linear programming 14.5 Completing the square 14.6 Quadratic formula 14.7 Factorising quadratics where the coefficient of ‘x2‘s is not 1 14.8 Algebraic fractions

251 253 257 262 264

268 275 279 284 286 287

Chapter 15: Scale drawings, bearings and trigonometry 15.1 Scale Drawings 15.2 Bearings 15.3 Understanding the tangent, cosine and sine ratios 15.4 Solving problems using trigonometry 15.5 Angles between 90° and 180° 15.6 The sine and cosine rules 15.7 Area of a triangle 15.8 Trigonometry in three-dimensions

318 322 323 331 334

Chapter 16: Scatter diagrams and correlation 16.1 Introduction to bivariate data

342

Chapter 19: Symmetry and loci 19.1 Symmetry in two-dimensions 19.2 Symmetry in three-dimensions 19.3 Symmetry properties of circles 19.4 Angle relationships in circles 19.5 Locus

398 401 404 407 412

300 302 303

289 291

Unit 5

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Chapter 17: Managing money 17.1 Earning money 17.2 Borrowing and investing money 17.3 Buying and selling

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Chapter 18: Curved graphs 18.1 Plotting quadratic graphs (the parabola) 18.2 Plotting reciprocal graphs (the hyperbola) 18.3 Using graphs to solve quadratic equations 18.4 Using graphs to solve simultaneous linear and non-linear equations 18.5 Other non-linear graphs 18.6 Finding the gradient of a curve

352 357 363

371 375 377

379 381 390

Chapter 20: Histograms and frequency distribution diagrams 20.1 Histograms 420 20.2 Cumulative frequency 426

Unit 6

Chapter 21: Ratio, rate and proportion 21.1 Working with ratio 21.2 Ratio and scale 21.3 Rates 21.4 Kinematic graphs 21.5 Proportion 21.6 Direct and inverse proportion in algebraic terms 21.7 Increasing and decreasing amounts by a given ratio

Chapter 22: More equations, formulae and functions 22.1 Setting up equations to solve problems 22.2 Using and transforming formulae 22.3 Functions and function notation

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442 446 450 452 460 463 467

Chapter 23: Transformations and matrices 23.1 Simple plane transformations 23.2 Vectors 23.3 Further transformations 23.4 Matrices and matrix transformation 23.5 Matrices and transformations

488 500 512 517 525

Chapter 24: Probability using tree diagrams 24.1 Using tree diagrams to show outcomes 24.2 Calculating probability from tree diagrams

534 535

472 474 477

Examination practice: structured question for Unit 4-6

542

Exercise Answers

552

Glossary

615

Index

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Introduction

REWIND

You learned how to plot lines from equations in chapter 10. W

FAST FORWARD

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You will learn much more about sets in chapter 9. For now, just think of a set as a list of numbers or other items that are often placed inside curly brackets. X

This highly illustrated coursebook covers the complete Cambridge IGCSE Mathematics (0580) syllabus. Core and Extended material is combined in one book, offering a one-stop-shop for all students and teachers. Useful hints are included in the margins for students needing more support, leaving the narrative clear and to the point. The material required for the Extended course is clearly marked using colour panels and these students are given access to the parts of the Core syllabus they need without having to use an additional book. The coursebook has been written with a clear progression from start to finish, with some later chapters requiring knowledge learnt in earlier chapters. There are useful signposts throughout the coursebook that link the content of the chapters, allowing the individual to follow their own course through the book: where the content in one chapter might require knowledge from a previous chapter, a comment is included in a ‘Rewind’ box; and where content will be covered in more detail later on in the coursebook, a comment is included in a ‘Fast forward’ box. Examples of both are included here. Worked examples are used throughout to demonstrate each method using typical workings and thought processes. These present the methods to the students in a practical and easy-to-follow way that minimises the need for lengthy explanations. There is plenty of practice offered via ‘drill’ exercises throughout each chapter. The exercises are progressive questions which allow the student to practise methods that have just been introduced. At the end of each chapter there are ‘Exam-style’ questions and ‘Past paper’ questions. The exam-style questions have been written by the authors in the style of questions on exam papers. The past paper questions are real questions taken from past exam papers. These end of chapter questions typically reflect the ‘short’ Paper 1 (Core) and Paper 2 (Extended) questions, though you will find some more structured ones in there as well. The answers to all of these questions are supplied at the back of the book, allowing self- and/or class- assessment. Students can assess their progress as they go along, choosing to do more or less practise as required. The ‘suggested’ progression through the coursebook is for Units 1-3 to be covered in the first year of both courses, and Units 4-6 to be covered in the second year of both courses. On this basis, there is mixed exam practice at the end of Unit 3 and the end of Unit 6. This is however, only a suggested structure and the course can be taught in various different ways; the signposting throughout the coursebook means that it can be used alongside any order of teaching. The end of Unit questions represent the longer answer ‘structured’ questions of Paper 3 (Core) and Paper 4 (Extended) exam papers and will use a combination of methods from across all relevant chapters. As with the end of chapter questions, these are a mixture of ‘Exam-style’ and ‘Past paper’ questions. The answers to these questions are on the Teacher’s resource so that they can be used in classroom tests or for homework, if desired. The coursebook also comes with a glossary to provide a definition for important / tricky terms. Helpful guides in the margin of the book include: Hints: these are general comments to remind you of important or key information that is useful to tackle an exercise, or simply useful to know. They often provide extra information or support in potentially tricky topics.

Remember ‘coefficient’ is the number in the term.

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Introduction

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Tip It is essential that you remember to work out both unknowns. Every pair of simultaneous linear equations will have a pair of solutions.

Tip: these are tips that relate to the exam. They cover common pitfalls based on the authors’ experiences of their students, and give you things to be wary of or to remember in order to score marks in the exam. Please note that this advice is not from the University of Cambridge International Examinations Syndicate and they bear no responsibility for any such advice given. The accompanying student CD-ROM at the back of the coursebook includes:

• •

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A ‘coverage grid’ to map the contents of the syllabus to the topics and chapters in the coursebook. A ‘Calculator support’ chapter. This chapter covers the main uses of calculators that students seem to struggle with, and includes some worksheets to provide practice at using your calculator in these situations. Revision Core revision worksheets (and answers) provide extra exercises for each chapter of the book. These worksheets contain only content from the Core syllabus. Extended revision worksheets (and answers) provide extra exercises for each chapter of the book. These worksheets contain the same questions as the Core worksheets, in addition to some more challenging questions, and questions to cover content unique to the Extended syllabus. Students are encouraged to do some (if not all) of the ‘Core’ questions on these worksheets, as well as the Extended ones (shaded) in order to fully revise the course. If time is limited, you might find it easier to pick two or three ‘Core’ questions to do before moving on to the ‘Extended’ questions. Quick revision tests – these are interactive questions in the form of multiple choice, drag and drop, or hide and reveal. They are quick-fire questions to test yourself in a different medium to pen and paper, and to get you thinking on the spot. They cover the Core content, with only a few additional screens being specific to the Extended course. There is at least one activity for each chapter. Students are recommended to use the Revision worksheets for a more comprehensive revision exercise. Worked solutions – these are interactive hide and reveal screens showing worked solutions to some of the end of chapter examination practice questions. Some of these will be ‘Exam-style’ and some will be ‘Past paper’ questions but all will be taken from the end of the chapter. There will be at least one for each chapter. The screen includes the question, and the answer, but also includes a series of ‘Clue’ or ‘Tip’ boxes. The ‘Clue’ boxes can be clicked on to reveal a clue to help the student if they are struggling with how to approach the question. The ‘Tip’ boxes contain tips relating to the exam, just like the ‘Tip’ boxes in the coursebook.

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Also in the Cambridge IGCSE Mathematics series are two Practice Books – one for Core and one for Extended – to offer students targeted practice. These follow the chapters and topics of the coursebook, including additional exercises for those who want more practice. These too, include ‘Hints’ and ‘Tips’ to help with tricky topics.

vi

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Acknowledgements

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The authors and publishers acknowledge the following sources of copyright material and are grateful for the permissions granted. Past paper examination questions are reproduced by permission of University of Cambridge International Examinations Cover image Seamus Ditmeyer/Alamy; p. 1 © sanderderwilde.com; p. 13 Dmitry Lavruhin/Shutterstock; p. 13 Hadriann/Shutterstock; p. 13 Jason Cox/Shutterstock; p. 13 Ruslan Nabiyev/Shutterstock; p. 46 Ivangott/Shutterstock; p. 47 Claudio Baldini/ Shutterstock; p. 63 akiyoko/Shutterstock; p. 73 INSAGO/Shutterstock; p. 98 © The Trustees of the British Museum; p. 119 Wikipedia; p. 128 Francesco Dazzi/Shutterstock; p. 164 North Wind Picture Archive/Alamy; p. 148 Paolo Gianti/Shutterstock; pp. 153, 341, 487, 533 iStockphoto/ Thinkstock; p. 155 Opachevsky Irina/Shutterstock; p. 156 Chad Littlejohn/Shutterstock; p. 159 sahua d/Shutterstock; p. 206 Photos. com/Thinkstock; pp. 223, 419, 428 Mike van der Wold; p. 235 Picsfive/Shutterstock; p. 250 Vladislav Gurfinkel/Shutterstock; p. 259 Mike Tan C.T./Shutterstock; p. 261 Suzan/Shutterstock; p. 261 R-studio/Shutterstock; p. 264 Galyna Andrushko/Shutterstock; p. 299 Gustavo Miguel Fernandes/Shutterstock; p. 351 Stephanie Fray/Shutterstock; p. 375 Kristina Postnikova/Shutterstock; p. 397 Connors Bros./Shutterstock; p. 441 Philippe Wojazer/AP/Press Association Images

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1

Reviewing number concepts

Key words Natural number Integer Prime number Symbol Multiple Factor Composite numbers Prime factor Square root Cube Directed numbers BODMAS

In this chapter you will learn how to:

• • • • •

identify and classify different types of numbers

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• •

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• • • • • • • • • • • •

find common factors and common multiples of numbers

write numbers as products of their prime factors

calculate squares, square roots, cubes and cube roots of numbers work with integers used in real-life situations revise the basic rules for operating with numbers perform basic calculations using mental methods and with a calculator.

This statue is a replica of a 22 000-year-old bone found in the Congo. The real bone is only 10 cm long and it is carved with groups of notches that represent numbers. One column lists the prime numbers from 10 to 20. It is one of the earliest examples of a number system using tallies.

Our modern number system is called the Hindu-Arabic system because it was developed by Hindus and spread by Arab traders who brought it with them when they moved to different places in the world. The Hindu-Arabic system is decimal. This means it uses place value based on powers of ten. Any number at all, including decimals and fractions, can be written using place value and the digits from 0 to 9.

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1 Reviewing number concepts

1.1 Different types of numbers Make sure you know the correct mathematical words for the types of numbers in the table. Number

You should already be familiar with most of the concepts in this chapter. It is included here so that you can revise the concepts and check that you remember them. FAST FORWARD

You will learn about the difference between rational and irrational numbers in chapter 9. X

Exercise 1.1 FAST FORWARD

Example

Natural number

Any whole number from 1 to infinity, sometimes called ‘counting numbers’. 0 is not included.

1, 2, 3, 4, 5, . . .

Odd number

A whole number that cannot be divided exactly by 2.

1, 3, 5, 7, . . .

Even number

A whole number that can be divided exactly by 2.

2, 4, 6, 8, . . .

Integer

Any of the negative and positive whole numbers, including zero.

. . . −3, −2, −1, 0, 1, 2, 3, . . .

Prime number

A whole number greater than 1 which has only two factors: the number itself and 1.

2, 3, 5, 7, 11, . . .

Square number

The product obtained when an integer is multiplied by itself.

1, 4, 9, 16, . . .

A number representing parts of a whole number, can be written as a common (vulgar) fraction in the form of ab or as a decimal using the decimal point.

1 1 1 1 13 2 4 3 8 3

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Find the ‘product’ means ‘multiply’. So, the product of 3 and 4 is 12, i.e. 3 × 4 = 12.

Definition

, , , , , 2 12

0.5, 0.2, 0.08, 1.7

1 Here is a set of numbers: {−4, −1, 0, 12 , 0.75, 3, 4, 6, 11, 16, 19, 25} List the numbers from this set that are: (a) natural numbers (b) even numbers (d) integers (e) negative integers (g) square numbers (h) prime numbers

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Fraction

(c) odd numbers (f) fractions (i) neither square nor prime.

2 List: (a) (b) (c) (d) (e) (f)

the next four odd numbers after 107 four consecutive even numbers between 2008 and 2030 all odd numbers between 993 and 1007 the first five square numbers four decimal fractions that are smaller than 0.5 four vulgar fractions that are greater than 12 but smaller than 34 .

3 State whether the following will be odd or even: Remember that a 'sum' is the result of an addition. The term is often used for any calculation in early mathematics but its meaning is very specific at this level.

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(a) (b) (c) (d) (e) (f)

the sum of two odd numbers the sum of two even numbers the sum of an odd and an even number the square of an odd number the square of an even number an odd number multiplied by an even number.

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1 Reviewing number concepts Living maths 4 There are many other types of numbers. Find out what these numbers are and give an example of each. (a) Perfect numbers. (b) Palindromic numbers. (c) Narcissistic numbers. (In other words, numbers that love themselves!)

Using symbols to link numbers Mathematicians use numbers and symbols to write mathematical information in the shortest, clearest way possible. You have used the operation symbols +, −, × and ÷ since you started school. Now you will also use the symbols given in the margin below to write mathematical statements.

Exercise 1.2

(a) 19 is less than 45 (b) 12 plus 18 is equal to 30 (c) 0.5 is equal to 12 (d) 0.8 is not equal to 8.0 (e) −34 is less than 2 times −16 (f) therefore the number x equals the square root of 72 (g) a number (x) is less than or equal to negative 45 (h) π is approximately equal to 3.14 (i) 5.1 is greater than 5.01 (j) the sum of 3 and 4 is not equal to the product of 3 and 4 (k) the difference between 12 and −12 is greater than 12 (l) the sum of −12 and −24 is less than 0 (m) the product of 12 and a number (x) is approximately −40

= is equal to ≠ is not equal to

≤ is less than or equal to > is greater than ≥ is greater than or equal to ∴ therefore the square root of

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≈ is approximately equal to < is less than

1 Rewrite each of these statements using mathematical symbols.

2 Say whether these mathematical statements are true or false. (a) 0.599 > 6.0 (c) 8 1 = 8 101 (e) 20 × 9 ≥ 21 × 8 (g) −12 > −4 (i) 1000 > 199 × 5 (k) 35 × 5 × 2 ≠ 350 (m) 20 − 4 ≠ 4 − 20

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Remember that the 'difference' between two numbers is the result of a subtraction. The order of the subtraction matters.

(b) (d) (f) (h) (j) (l) (n)

5 × 1999 ≈ 10 000 6.2 + 4.3 = 4.3 + 6.2 6.0 = 6 19.9 ≤ 20 16 = 4 20 ÷ 4 = 5 ÷ 20 20 × 4 ≠ 4 × 20

3 Work with a partner. (a) Look at the symbols used on the keys of your calculator. Say what each one means in words. (b) List any symbols that you do not know. Try to find out what each one means.

1.2 Multiples and factors You can think of the multiples of a number as the ‘times table’ for that number. For example, the multiples of 3 are 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9 and so on.

Multiples A multiple of a number is found when you multiply that number by a positive integer. The first multiple of any number is the number itself (the number multiplied by 1).

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1 Reviewing number concepts

Worked example 1 (a) What are the first three multiples of 12? (b) Is 300 a multiple of 12? (a)

12, 24, 36

To find these multiply 12 by 1, 2 and then 3. 12 × 1 = 12 12 × 2 = 24 12 × 3 = 36

(b)

Yes, 300 is a multiple of 12.

To find out, divide 300 by 12. If it goes exactly, then 300 is a multiple of 12. 300 ÷ 12 = 25

Exercise 1.3

1 List the first five multiples of: (a) 2 (e) 9

(b) 3 (f) 10

(c) 5 (g) 12

(d) 8 (h) 100

2 Use a calculator to find and list the first ten multiples of: (b) 44 (f) 350

(c) 75 (g) 1012

(d) 114 (h) 9123

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(a) 29 (e) 299

3 List:

(a) the multiples of 4 between 29 and 53 (b) the multiples of 50 less than 400 (c) the multiples of 100 between 4000 and 5000.

4 Here are five numbers: 576, 396, 354, 792, 1164. Which of these are multiples of 12?

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5 Which of the following numbers are not multiples of 27? (a) 324 (b) 783 (c) 816 (d) 837 (e) 1116

The lowest common multiple (LCM) The lowest common multiple of two or more numbers is the smallest number that is a multiple of all the given numbers.

Worked example 2

Find the lowest common multiple of 4 and 7. M4 = 4, 8, 12, 16, 20, 24, 28 , 32 M7 = 7, 14, 21, 28 , 35, 42 LCM = 28

Exercise 1.4 FAST FORWARD

Later in this chapter you will see how prime factors can be used to find LCMs. X

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List several multiples of 4. (Note: M4 means multiples of 4.) List several multiples of 7. Find the lowest number that appears in both sets. This is the LCM.

1 Find the LCM of: (a) (d) (g) (j)

2 and 5 3 and 9 2, 4 and 8 1, 3 and 7

(b) (e) (h) (k)

8 and 10 35 and 55 4, 5 and 6 4, 5 and 8

(c) (f) (i) (l)

6 and 4 6 and 11 6, 8 and 9 3, 4 and 18

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1 Reviewing number concepts 2 Is it possible to find the highest common multiple of two or more numbers? Give a reason for your answer.

Factors A factor is a number that divides exactly into another number with no remainder. For example, 2 is a factor of 16 because it goes into 16 exactly 8 times. 1 is a factor of every number. The largest factor of any number is the number itself.

Worked example 3 Find the factors of:

F12 means the factors of 12.

(a) 12

(c) 110

(a)

F12 = 1, 2, 3, 4, 6, 12

Find pairs of numbers that multiply to give 12: 1 × 12 2×6 3×4 Write the factors in numerical order.

(b)

F25 = 1, 5, 25

1 × 25 5×5 Do not repeat the 5.

F110 = 1, 2, 5, 10, 11, 22, 55, 110

1 × 110 2 × 55 5 × 22 10 × 11

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To list the factors in numerical order go down the left side and then up the right side of the factor pairs. Remember not to repeat factors.

(b) 25

(c)

Exercise 1.5

1 List all the factors of:

(b) 5 (g) 35 (l) 132

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(a) 4 (f) 12 (k) 100

(c) 8 (h) 40 (m) 160

(d) 11 (i) 57 (n) 153

(e) 18 (j) 90 (o) 360

2 Which number in each set is not a factor of the given number? (a) (b) (c) (d) (e)

FAST FORWARD

Later in this chapter you will learn more about divisibility tests and how to use these to decide whether or not one number is a factor of another. X

14 15 21 33 42

{1, 2, 4, 7, 14} {1, 3, 5, 15, 45} {1, 3, 7, 14, 21} {1, 3, 11, 22, 33} {3, 6, 7, 8, 14}

3 State true or false in each case. (a) (c) (e) (g)

3 is a factor of 313 3 is a factor of 300 2 is a factor of 122 488 210 is a factor of 210

(b) (d) (f) (h)

9 is a factor of 99 2 is a factor of 300 12 is a factor of 60 8 is a factor of 420

4 What is the smallest factor and the largest factor of any number?

© Cambridge University Press 2012

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1 Reviewing number concepts The highest common factor (HCF) The highest common factor of two or more numbers is the highest number that is a factor of all the given numbers.

Worked example 4 Find the HCF of 8 and 24. F8 = 1, 2, 4, 8 F24 = 1, 2, 3, 4, 6, 8, 12, 24 HCF = 8

List the factors of each number. Underline factors that appear in both sets. Pick out the highest underlined factor (HCF).

Exercise 1.6 FAST FORWARD

You will learn how to find HCFs by using prime factors later in the chapter. X

1 Find the HCF of each pair of numbers. (a) 3 and 6 (b) 24 and 16 (c) 15 and 40 (e) 32 and 36 (f) 26 and 36 (g) 22 and 44 2 Find the HCF of each group of numbers. (a) 3, 9 and 15 (b) 36, 63 and 84

(d) 42 and 70 (h) 42 and 48

(c) 22, 33 and 121

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3 Not including the factor provided, find two numbers that have: (a) an HCF of 2 (b) an HCF of 6

4 What is the HCF of two different prime numbers? Give a reason for your answer.

Word problems involving HCF usually involve splitting things into smaller pieces or arranging things in equal groups or rows.

Living maths

5 Simeon has two lengths of rope. One piece is 72 metres long and the other is 90 metres long. He wants to cut both lengths of rope into the longest pieces of equal length possible. How long should the pieces be?

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6 Ms Sanchez has 40 canvases and 100 tubes of paint to give to the students in her art group. What is the largest number of students she can have if she gives each student an equal number of canvasses and an equal number of tubes of paint? 7 Indira has 300 blue beads, 750 red beads and 900 silver beads. She threads these beads to make wire bracelets. Each bracelet must have the same number and colour of beads. What is the maximum number of bracelets she can make with these beads?

1.3 Prime numbers Prime numbers have exactly two factors: one and the number itself. Composite numbers have more than two factors. The number 1 has only one factor so it is not prime and it is not composite.

Finding prime numbers Over 2000 years ago, a Greek mathematician called Eratosthenes made a simple tool for sorting out prime numbers. This tool is called the ‘Sieve of Eratosthenes’ and the figure on page 7 shows how it works for prime numbers up to 100.

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Unit 1: Number

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1 Reviewing number concepts 1 11 21 31 41 51 61 71 81 91

2 12 22 32 42 52 62 72 82 92

3 13 23 33 43 53 63 73 83 93

4 14 24 34 44 54 64 74 84 94

5 15 25 35 45 55 65 75 85 95

6 16 26 36 46 56 66 76 86 96

7 17 27 37 47 57 67 77 87 97

8 18 28 38 48 58 68 78 88 98

9 19 29 39 49 59 69 79 89 99

10 20 30 40 50 60 70 80 90 100

Cross out 1, it is not prime. Circle 2, then cross out other multiples of 2. Circle 3, then cross out other multiples of 3. Circle the next available number then cross out all its multiples. Repeat until all the numbers in the table are either circled or crossed out. The circled numbers are the primes.

Other mathematicians over the years have developed ways of finding larger and larger prime numbers. Until 1955, the largest known prime number had less than 1000 digits. Since the 1970s and the invention of more and more powerful computers, more and more prime numbers have been found. The graph below shows the number of digits in the largest known primes since 1955.

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You should try to memorise which numbers between 1 and 100 are prime.

Number of digits in largest known prime number against year found

100 000 000 10 000 000 1 000 000

Number of digits

100 000

D

10 000 1000

100 1955

1965

1975

1985

1995

2005

2015

Year

Today anyone can join the Great Internet Mersenne Prime Search. This project links thousands of home computers to search continuously for larger and larger prime numbers while the computer processors have spare capacity.

Exercise 1.7 FAST FORWARD

A good knowledge of primes can help when factorising quadratics in chapter 10. X

1 Which is the only even prime number? 2 How many odd prime numbers are there less than 50? 3 (a) List the composite numbers greater than four, but less than 30. (b) Try to write each composite number on your list as the sum of two prime numbers. For example: 6 = 3 + 3 and 8 = 3 + 5. 4 Twin primes are pairs of prime numbers that differ by two. List the twin prime pairs up to 100.

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Unit 1: Number

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1 Reviewing number concepts 5 Is 149 a prime number? Explain how you decided. 6 Super-prime numbers are prime numbers that stay prime each time you remove a digit (starting with the units). So, 59 is a super-prime because when you remove 9 you are left with 5, which is also prime. 239 is also a super-prime because when you remove 9 you are left with 23 which is prime, and when you remove 3 you are left with 2 which is prime. (a) Find two three-digit super-prime numbers less than 400. (b) Can you find a four-digit super-prime number less than 3000? (c) Sondra’s telephone number is the prime number 987-6413. Is her phone number a super-prime?

Tip Whilst super-prime numbers are interesting, they are not on the syllabus.

Prime factors Remember a product is the answer to a multiplication. So if you write a number as the product of its prime factors you are writing it using multiplication signs like this: 12 = 2 × 2 × 3.

Prime factors are the factors of a number that are also prime numbers. Every composite whole number can be broken down and written as the product of its prime factors. You can do this using tree diagrams or using division. Both methods are shown in worked example 5.

Worked example 5

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Write the following numbers as the product of prime factors. (a) 36

(b) 48

Using a factor tree

2

4

3

D

Prime numbers only have two factors: 1 and the number itself. As 1 is not a prime number, do not include it when expressing a number as a product of prime factors.

4

12

3

Write the number as two factors.

48

36

2

36 = 2 × 2 × 3 × 3

If a factor is a prime number, circle it.

12

2

If a factor is a composite number, split it into two factors.

4

3

Keep splitting until you end up with two primes.

2

2 48 = 2 × 2 × 2 × 2 × 3

2

Write the primes in ascending order with × signs.

Using division 2 36 2 18 3 9 3 3 1 36 = 2 × 2 × 3 × 3

2 48 2 24 2 12 2 6 3 3 1 48 = 2 × 2 × 2 × 2 × 3

Choose the method that works best for you and stick to it. Always show your method when using prime factors.

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Unit 1: Number

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Divide by the smallest prime number that will go into the number exactly. Continue dividing, using the smallest prime number that will go into your new answer each time. Stop when you reach 1. Write the prime factors in ascending order with × signs.

© Cambridge University Press 2012

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1 Reviewing number concepts

Exercise 1.8

1 Express the following numbers as the product of prime factors. (a) 30 (f) 504

When you write your number as a product of primes, group all occurrences of the same prime number together.

(b) 24 (g) 650

(c) 100 (h) 1125

(d) 225 (i) 756

(e) 360 (j) 9240

Using prime factors to find the HCF and LCM FAST FORWARD

You can also use prime factors to find the square and cube roots of numbers if you don’t have a calculator. You will deal with this in more detail on page 11. X

When you are working with larger numbers you can determine the HCF or LCM by expressing each number as a product of its prime factors.

Worked example 6 Find the HCF of 168 and 180. First express each number as a product of prime factors. Use tree diagrams or division to do this. Underline the factors common to both numbers. Multiply these out to find the HCF.

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168 = 2 × 2 × 2 × 3 × 7 180 = 2 × 2 × 3 × 3 × 5 2 × 2 × 3 = 12 HCF = 12

Worked example 7

Find the LCM of 72 and 120.

D

72 = 2 × 2 × 2 × 3 × 3 120 = 2 × 2 × 2 × 3 × 5 2 × 2 × 2 × 3 × 3 × 5 = 360 LCM = 360

Exercise 1.9

First express each number as a product of prime factors. Use tree diagrams or division to do this. Underline the largest set of multiples of each factor. List these and multiply them out to find the LCM.

1 Find the HCF of these numbers by means of prime factors. (a) 48 and 108 (e) 100 and 125

(b) 120 and 216 (f) 154 and 88

(c) 72 and 90 (g) 546 and 624

(d) 52 and 78 (h) 95 and 120

2 Use prime factorisation to determine the LCM of: (a) 54 and 60 (e) 120 and 180

(b) 54 and 72 (f) 95 and 150

(c) 60 and 72 (g) 54 and 90

(d) 48 and 60 (h) 90 and 120

3 Determine both the HCF and LCM of the following numbers. (a) 72 and 108

(b) 25 and 200

(c) 95 and 120

(d) 84 and 60

Living maths Word problems involving LCM usually include repeating events. You may be asked how many items you need to ‘have enough’ or when something will happen again at the same time.

4 A radio station runs a phone-in competition for listeners. Every 30th caller gets a free airtime voucher and every 120th caller gets a free mobile phone. How many listeners must phone in before one receives both an airtime voucher and a free phone? 5 Lee runs round a track in 12 minutes. James runs round the same track in 18 minutes. If they start in the same place, at the same time, how many minutes will pass before they both cross the start line together again?

© Cambridge University Press 2012

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Unit 1: Number

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1 Reviewing number concepts

Divisibility tests to find factors easily Sometimes you want to know if a smaller number will divide into a larger one with no remainder. In other words, is the larger number divisible by the smaller one? These simple divisibility tests are useful for working this out:

Tip Divisibility tests are not part of the syllabus. They are just useful to know when you work with factors and prime numbers.

A number is exactly divisible by: 2 3 4 5 6 8 9 10

if it ends with 0, 2, 4, 6 or 8 (in other words is even) if the sum of its digits is a multiple of 3 (can be divided by 3) if the last two digits can be divided by 4 if it ends with 0 or 5 if it is divisible by both 2 and 3 if the last three digits are divisible by 8 if the sum of the digits is a multiple of 9 (can be divided by 9) if the number ends in 0.

There is no simple test for divisibility by 7, although multiples of 7 do have some interesting properties that you can investigate on the internet. 23

65

92

10

104

70

500

21

64

798

1223

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Exercise 1.10

1 Look at the box of numbers above. Which of these numbers are: (a) divisible by 5? (b) divisible by 8? (c) divisible by 3?

2 Say whether the following are true or false. (a) 625 is divisible by 5 (b) 88 is divisible by 3 (c) 640 is divisible by 6 (d) 346 is divisible by 4 (e) 476 is divisible by 8 (f) 2340 is divisible by 9 (g) 2890 is divisible by 6 (h) 4562 is divisible by 3 (i) 40 090 is divisible by 5 (j) 123 456 is divisible by 9

D

3 Can $34.07 be divided equally among: (b) three people? (a) two people?

(c) nine people?

4 A stadium has 202 008 seats. Can these be divided equally into: (a) five blocks? (b) six blocks? (c) nine blocks? 5 (a) If a number is divisible by 12, what other numbers must it be divisible by? (b) If a number is divisible by 36, what other numbers must it be divisible by? (c) How could you test if a number is divisible by 12, 15 or 24?

1.4 Powers and roots REWIND

In section 1.1 you learned that the product obtained when an integer is multiplied by itself is a square number. W

Square numbers and square roots A number is squared when it is multiplied by itself. For example, the square of 5 is 5 × 5 = 25. The symbol for squared is 2. So, 5 × 5 can also be written as 52. The square root of a number is the number that was multiplied by itself to get the square number. The symbol for square root is . You know that 25 = 52, so 25 = 5.

Cube numbers and cube roots A number is cubed when it is multiplied by itself and then multiplied by itself again. For example, the cube of 2 is 2 × 2 × 2 = 8. The symbol for cubed is 3. So 2 × 2 × 2 can also be written as 23.

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Unit 1: Number

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© Cambridge University Press 2012

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1 Reviewing number concepts The cube root of a number is the number that was multiplied by itself to get the cube number. The symbol for cube root is 3 . You know that 8 = 23, so 3 8 = 2. 2

2

2 2 2

a) Square numbers can be arranged to form a square shape. This is 22.

b) Cube numbers can be arranged to form a solid cube shape. This is 23.

Finding powers and roots Not all calculators have exactly the xy and same buttons. x◻ ∧ all mean the same thing on

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different calculators.

You can use your calculator to square or cube numbers quickly using the x2 and x3 keys or the x ◻ key. Use the or 3 keys to find the roots. If you don’t have a calculator, you can use the product of prime factors method to find square and cube roots of numbers. Both methods are shown in the worked examples below.

Worked example 8

Use your calculator to find: (a) 132

(b) 53

(c)

324

(d)

3

512

=

(a)

132 = 169

Enter

1

3

x2

(b)

53 = 125

Enter

5

x3

=

. If you do not have a

3

=

; for this key you have to enter the power.

3

2

4

=

5

1

2

=

5

324 = 18

Enter

512 = 8

Enter

D

(c)

(d)

3

x

3

◻

x3

button then enter

Worked example 9

If you do not have a calculator, you can write the integer as a product of primes and group the prime factors into pairs or threes. Look again at parts (c) and (d) of worked example 8: 3

(c)

324

(c)

2×3 3×3 3    324 = 2 3 3 2

(d)

512

2 × 3 × 3 = 18 324 = 18 (d)

Group the factors into pairs, and write down the square root of each pair. Multiply the roots together to give you the square root of 324.

2  2×2 2 2×2 2  2      512 = 2 2 2 2

Group the factors into threes, and write the cube root of each threesome.

2×2×2=8 3 512 = 8

Multiply together to get the cube root of 512.

© Cambridge University Press 2012

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Unit 1: Number

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1 Reviewing number concepts

Exercise 1.11

1 Calculate: (a) 32 (f) 192

(b) 72 (g) 322

(c) 112 (h) 1002

(d) 122 (i) 142

(e) 212 (j) 682

(b) 33 (g) 1003

(c) 43 (h) 183

(d) 63 (i) 303

(e) 93 (j) 2003

2 Calculate: (a) 13 (f) 103 Learn the squares of all integers between 1 and 20 inclusive. You will need to recognise these quickly.

3 Find a value of x to make each of these statements true. (a) x × x = 25 (b) x × x × x = 8 (c) x × x = 121 (d) x × x × x = 729 (e) x × x = 324 (f) x × x = 400 (g) x × x × x = 8000 (h) x × x = 225 (i) x × x × x = 1 (j) x = 9 (k) 1 = x (l) x = 81 (m) 3 x = 2

(n)

3

x =1

(o)

3

64 = x

4 Use a calculator to find the following roots. (a)

(b) (g) (l) (q)

64 3

(c) 1 (h) 400 (m) 3 27 (r) 3 729

81 1

(d) (i) (n) (s)

4 3

1296 64

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9 (f) 0 (k) 3 8 (p) 3 216

3

512

3

1728

(e) 100 (j) 1764 (o) 3 1000 (t) 3 5832

5 Use the product of prime factors given below to find the square root of each number. Show your working. (a) 324 = 2 × 2 × 3 × 3 × 3 × 3 (c) 784 = 2 × 2 × 2 × 2 × 7 × 7 (e) 19 600 = 2 × 2 × 2 × 2 × 5 × 5 × 7 × 7

(b) 225 = 3 × 3 × 5 × 5 (d) 2025 = 3 × 3 × 3 × 3 × 5 × 5 (f) 250 000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5 × 5

6 Use the product of prime factors to find the cube root of each number. Show your working. 27 = 3 × 3 × 3 (b) 729 = 3 × 3 × 3 × 3 × 3 × 3 2197 = 13 × 13 × 13 (d) 1000 = 2 × 2 × 2 × 5 × 5 × 5 15 625 = 5 × 5 × 5 × 5 × 5 × 5 32 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

D

(a) (c) (e) (f)

Brackets act as grouping symbols. Work out any calculations inside brackets before doing the calculations outside the brackets. Root signs work in the same way as a bracket. If you have 25 + 9 , you must add 25 and 9 before finding the root.

7 Calculate:

)2 (a) ( (e) 9 16 100 − 36 (i)

(b) ( )2 (f) 9 16 (j) 100 − 36

(m)

(n)

9 4

9

4

(c) ( 3 )3 (g) 36 + 64 (k) 25 × 4 36 (o) 4

(d) ( 3 )3 (h) 36 + 64 (l) 25 × 4 36 (p) 4

8 Find the length of the edge of a cube with a volume of: (a) 1000 cm3

(b) 19 683 cm3

(c) 68 921 mm3

(d) 64 000 cm3

9 If the symbol * means ‘add the square of the first number to the cube of the second number’, calculate: (a) 2 * 3 (f) 4 * 2

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(b) 3 * 2 (g) 1 * 9

(c) 1 * 4 (h) 9 * 1

(d) 4 * 1 (i) 5 * 2

(e) 2 * 4 (j) 2 * 5

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1 Reviewing number concepts

1.5 Working with directed numbers Once a direction is chosen to be positive, the opposite direction is taken to be negative. So:

• if up is positive, down is negative • if right is positive, left is negative north is positive, south is • ifnegative above 0 is positive, below 0 is • ifnegative.

A negative sign is used to indicate that values are less than zero. For example, on a thermometer, on a bank statement or in an elevator.

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When you use numbers to represent real-life situations like temperatures, altitude, depth below sea level, profit or loss and directions (on a grid), you sometimes need to use the negative sign to indicate the direction of the number. For example, a temperature of three degrees below zero can be shown as −3 °C. Numbers like these, which have direction, are called directed numbers. So if a point 25 m above sea level is at +25 m, then a point 25 m below sea level is at −25 m. 1 Express each of these situations using a directed number. (a) a profit of $100 (b) 25 km below sea level (c) a drop of 10 marks (d) a gain of 2 kg (e) a loss of 1.5 kg (f) 8000 m above sea level (g) a temperature of 10 °C below zero (h) a fall of 24 m (i) a debt of $2000 (j) an increase of $250 (k) a time two hours behind GMT (l) a height of 400 m (m) a bank balance of $450.00

D

Exercise 1.12

Comparing and ordering directed numbers

FAST FORWARD

You will use similar number lines when solving linear inequalities in chapter 14. X

In mathematics, directed numbers are also known as integers. You can represent the set of integers on a number line like this: –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

1

2

3

4

5

6

7

8

9 10

The further to the right a number is on the number line, the greater its value.

Exercise 1.13 It is important that you understand how to work with directed numbers early in your IGCSE course. Many topics depend upon them!

1 Copy the numbers and fill in < or > to make a true statement. (a) (d) (g) (j) (m)

2F8 6F 4 −2 F −11 −2 F 2 0F 3

(b) (e) (h) (k) (n)

4F9 −7 F 4 −12 F −20 −12 F −4 −3 F 11

(c) (f) (i) (l) (o)

12 F 3 −2 F 4 −8 F 0 −32 F −3 12 F −89

2 Arrange each set of numbers in ascending order. (a) −8, 7, 10, −1, −12 (c) −11, −5, −7, 7, 0, −12

(b) 4, −3, −4, −10, 9, −8 (d) −94, −50, −83, −90, 0

© Cambridge University Press 2012

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Unit 1: Number

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1 Reviewing number concepts Living maths 3 Study the temperature graph carefully. 10 8 6 4 Temperature (°C) 2 0 –2 –4 Sunday M 14

W

T

F

S Sunday M T 21 Day of the week

W

T

F

S Sunday 28

What was the temperature on Sunday 14 January? By how much did the temperature drop from Sunday 14 to Monday 15? What was the lowest temperature recorded? What is the difference between the highest and lowest temperatures? On Monday 29 January the temperature changed by −12 degrees. What was the temperature on that day?

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(a) (b) (c) (d) (e)

T

The difference between the highest and lowest temperature is also called the range of temperatures.

4 Matt has a bank balance of $45.50. He deposits $15.00 and then withdraws $32.00. What is his new balance?

5 Mr Singh’s bank account is $420 overdrawn. (a) Express this as a directed number. (b) How much money will he need to deposit to get his account to have a balance of $500? (c) He deposits $200. What will his new balance be?

D

6 A diver 27 m below the surface of the water rises 16 m. At what depth is she then? 7 On a cold day in New York, the temperature at 6 a.m. was −5 °C. By noon, the temperature had risen to 8 °C. By 7 p.m. the temperature had dropped by 11 °C from its value at noon. What was the temperature at 7 p.m.? 8 Local time in Abu Dhabi is four hours ahead of Greenwich Mean Time. Local time in Rio de Janeiro is three hours behind Greenwich Mean Time. (a) If it is 4 p.m. at Greenwich, what time is it in Abu Dhabi? (b) If it is 3 a.m. in Greenwich, what time is it in Rio de Janiero? (c) If it is 3 p.m. in Rio de Janeiro, what time is it in Abu Dhabi? (d) If it is 8 a.m. in Abu Dhabi, what time is it in Rio de Janeiro?

1.6 Order of operations At this level of mathematics you are expected to do more complicated calculations involving more than one operation (+, −, × and ÷). When you are carrying out more complicated calculations you have to follow a sequence of rules so that there is no confusion about what operations you should do first. The rules governing the order of operations are:

• • • 14

Unit 1: Number

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complete operations in grouping symbols first (see page 15) do division and multiplication next, working from left to right do addition and subtractions last, working from left to right.

© Cambridge University Press 2012

3/8/2012 5:08:07 PM

1 Reviewing number concepts Many people use the letters BODMAS to remember the order of operations. The letters stand for:

Brackets Of

(Sometimes, ‘I’ for ‘indices’ is used instead of ‘O’ for ‘of ’)

Divide Multiply Add Subtract BODMAS indicates that powers are considered after brackets but before all other operations.

Grouping symbols The most common grouping symbols in mathematics are brackets. Here are some examples of the different kinds of brackets used in mathematics: (4 + 9) × (10 ÷ 2)

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[2(4 + 9) − 4(3) − 12]

{2 − [4(2 − 7) − 4(3 + 8)] − 2 × 8}

When you have more than one set of brackets in a calculation, you work out the innermost set first. Other symbols used to group operations are: 5 12 fraction bars, e.g. 3 8 root signs, such as square roots and cube roots, e.g. 9 16 powers, e.g. 52 or 43

• • •

D

Worked example 10 Simplify: (a)

7 × (3 + 4)

(b)

(10 − 4) × (4 + 9)

(c)

45 − [20 × (4 − 3)]

(a)

7 × 7 = 49

(b)

6 × 13 = 78

(c)

45 − [20 × 1] = 45 − 20 = 25

Worked example p 11 Calculate: (a)

3 + 82

(b)

4 28 17 − 9

(a)

3 (8 × 8 ) = 3 64 = 67

(b)

( ) ( = 32 ÷ 8 =4

© Cambridge University Press 2012

IGCSE_Maths_Ch01_p1-21.indd 15

)

(c)

36 ÷ 4 + 100 − 36

(c)

36 ÷ 4 + 100 − 36 = 9 64 =3 8 = 11

Unit 1: Number

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1 Reviewing number concepts

Exercise 1.14

1 Calculate. Show the steps in your working. (a) (4 + 7) × 3 (e) (4 + 7) × 4 (i) 40 ÷ (12 − 4)

(b) (20 − 4) ÷ 4 (f) (100 − 40) × 3 (j) 100 ÷ (4 + 16)

(c) 50 ÷ (20 + 5) (g) 16 + (25 ÷ 5) (k) 121 ÷ (33 ÷ 3)

(d) 6 × (2 + 9) (h) 19 − (12 + 2) (l) 15 × (15 − 15)

2 Calculate: (a) (4 + 8) × (16 − 7) (d) (33 + 17) ÷ (10 − 5) (g) (105 − 85) ÷ (16 ÷ 4)

(b) (12 − 4) × (6 + 3) (e) (4 × 2) + (8 × 3) (h) (12 + 13) ÷ 52

(c) (9 + 4) − (4 + 6) (f) (9 × 7) ÷ (27 − 20) (i) (56 − 62) × (4 + 3)

3 Simplify. Remember to work from the innermost grouping symbols to the outermost. A bracket ‘type’ is always twinned with another bracket of the same type/shape. This helps mathematicians to understand the order of calculations even more easily.

(a) (c) (e) (g) (i)

4 + [12 − (8 − 5)] 8 + [60 − (2 + 8)] 200{100 − [4 × (2 + 8)]} [(30 + 12) − (7 + 9)] × 10 1000 − [6 × (4 + 20) − 4 × (3 + 0)]

(b) (d) (f) (h)

6 + [2 − (2 × 0)] 200 − [(4 + 12) − (6 + 2)] {6 + [5 × (2 + 30)]} × 10 6 × [(20 ÷ 4) − (6 − 3) + 2]

4 Calculate: (a) 6 + 72

(c) 8 × 42 100 − 40 (f) 5 4

(h)

(i)

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(d) 20 − 4 ÷ 2

(b) 29 − 23 31 − 10 (e) 14 − 7

(g)

100 − 36

8 8

90 − 9

5 Insert brackets into the following calculations to make them true. (a) (d) (g) (j) (m) (p)

(b) (e) (h) (k) (n) (q)

25 − 15 × 9 = 90 12 + 3 ÷ 5 = 3 3 + 8 × 15 − 9 = 66 6÷3+3×5=5 8 + 5 − 3 × 2 = 20 40 ÷ 4 + 1 = 11

(c) (f) (i) (l) (o) (r)

40 − 10 × 3 = 90 19 − 9 × 15 = 150 9 − 4 × 7 + 2 = 45 15 − 6 ÷ 2 = 12 36 ÷ 3 × 3 − 3 = 6 6 + 2 × 8 + 2 = 24

Working in the correct order

D

FAST FORWARD

3 × 4 + 6 = 30 14 − 9 × 2 = 10 10 + 10 ÷ 6 − 2 = 5 10 − 4 × 5 = 30 1 + 4 × 20 ÷ 5 = 20 3×4−2÷6=1

You will apply the order of operation rules to fractions, decimals and algebraic expressions as you progress through the course. X

Exercise 1.15

Now that you know what to do with grouping symbols, you are going to apply the rules for order of operations to perform calculations with numbers.

1 Simplify. Show the steps in your working. (a) 5 × 10 + 3 (d) (2 + 10) × 3 15 − 5 (g) 2 5 (j) 17 + 3 × 21 (m) 15 + 30 ÷ 3 + 6

(b) 5 × (10 + 3) (e) 23 + 7 × 2

(k) 48 − (2 + 3) × 2 (n) 20 − 6 ÷ 3 + 3

(c) 2 + 10 × 3 (f) 6 × 2 ÷ (3 + 3) 16 − 4 (i) 4 1 (l) 12 × 4 − 4 × 8 (o) 10 − 4 × 2 ÷ 2

(b) 14 − (21 ÷ 3) (e) 5 + 36 ÷ 6 − 8

(c) 24 ÷ 8 × (6 − 5) (f) (8 + 3) × (30 ÷ 3) ÷ 11

(h) (17 + 1 ) ÷ 9 + 2

2 Simplify: (a) 18 − 4 × 2 − 3 (d) 42 ÷ 6 − 3 − 4

3 State whether the following are true or false. (a) (1 + 4) × 20 + 5 = 1 + (4 × 20) + 5 (c) 8 + (5 − 3) × 2 < 8 + 5 − (3 × 2)

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(b) 6 × (4 + 2) × 3 > (6 × 4) ÷ 2 × 3 (d) 100 + 10 ÷ 10 > (100 + 10) ÷ 10

© Cambridge University Press 2012

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1 Reviewing number concepts 4 Place the given numbers in the correct spaces to make a correct number sentence. (a) (b) (c) (d)

In this section you will use your calculator to perform operations in the correct order. However, you will need to remember the order of operations rules and apply them throughout the book as you do more complicated examples using your calculator. Experiment with your calculator by making several calculations with and without brackets. For example: 3 × 2 + 6 and 3 × (2 + 6). Do you understand why these are different?

F F F F

0, 2, 5, 10 9, 11, 13, 18 1, 3, 8, 14, 16 4, 5, 6, 9, 12

F F F F F F−F F=F (F F) − (F F) = F

Using your calculator A calculator with algebraic logic will apply the rules for order of operations automatically. So, if you enter 2 + 3 × 4, your calculator will do the multiplication first and give you an answer of 14. (Check that your calculator does this!). When the calculation contains brackets you must enter these to make sure your calculator does the grouped sections first.

Worked example 12 Use a calculator to find: (a) 3 + 2 × 9

(b) (3 + 8) × 4

(c) (3 × 8 − 4) − (2 × 5 + 1)

21

Enter

3

+

2

×

9

=

Your calculator might only have one

(b)

44

Enter

(

3

+

8

)

×

4

type of bracket ( and ) . If there are two different shaped brackets in the calculation (such as [4 × (2 – 3)], enter the calculator bracket symbol for each type.

(c)

9

Enter

(

3

×

8

−

4

)

−

(

2

×

5

+

1

)

=

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(a)

Exercise 1.16 Some calculators have two ‘−’

1 Use a calculator to find the correct answer. (a) 10 − 4 × 5 (c) 3 + 4 × 5 − 10 (e) 5 − 3 × 8 − 6 ÷ 2 (g) (1 + 4) × 20 ÷ 5 (i) (8 + 8) − 6 × 2 (k) 24 ÷ (7 + 5) × 6 (m) [(12 + 6) ÷ 9] × 4 (o) 4 × [25 ÷ (12 − 7)]

(b) (d) (f) (h) (j) (l) (n)

D

buttons: − and (−) . The first means ‘subtract’ and is used to subtract one number from another. The second means ‘make negative’. Experiment with the buttons and make sure that your calculator is doing what you expect it to do!

=

12 + 6 ÷ 7 − 4 18 ÷ 3 × 5 − 3 + 2 7+3÷4+1 36 ÷ 6 × (3 − 3) 100 − 30 × (4 − 3) [(60 − 40) − (53 − 43)] × 2 [100 ÷ (4 + 16)] × 3

2 Use your calculator to check whether the following answers are correct. If the answer is incorrect, work out the correct answer. (a) 12 × 4 + 76 = 124 (c) 12 × 18 − 4 × 23 = 124 (e) (82 − 36) × (2 + 6) = 16 The more effectively you are able to use your calculator, the faster and more accurate your calculations are likely to be. If you have difficulty with this you will find advice and practice exercises on the CD-ROM.

(b) 8 + 75 × 8 = 698 (d) (16 ÷ 4) × (7 + 3 × 4) = 76 (f) (3 × 7 − 4) − (4 + 6 ÷ 2) = 12

3 Each * represents a missing operation. Work out what it is. (a) 12 * (28 * 24) = 3 (c) 3 * 7(0.7 * 1.3) = 17 (e) 40 * 5 * (7 * 5) = 4

(b) 84 * 10 * 8 = 4 (d) 23 * 11 * 22 * 11 = 11 (f) 9 * 15 * (3 * 2) = 12

4 Calculate: (a)

7 16 23 7 2 − 1

(b)

52

4 1 62 − 12

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(c)

52

2 32 4 × 10 − 25

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1 Reviewing number concepts

(d)

62 11 2(17 2 4)

(e)

32 3 2 81

(g)

36 − 3 16 15 − 32 3

(h)

−30 + [18 ÷ (3 12) 24] 5 8 − 32

(f)

32

5+6 4 5

5 Use a calculator to find the answer. (a) FAST FORWARD

0.345 1.34 4.2 7

(b)

12.32 × 0.0378 16 + 8 05

(c)

16 × 0.087 22 5.098

(d)

19.23 × 0.087 2.452 1.032

6 Use your calculator to evaluate.

In this chapter you are only dealing with square and cube numbers, and the roots of square and cube numbers. When you work with indices and standard form in chapter 5, you will need to apply these skills and use your calculator effectively to solve problems involving any powers or roots. X

64 × 125

(b)

23 32 × 6

82 192

(d)

(e)

3.22 11.17 173

(f)

412 − 362 3 1.453 − 00.13 132

(g)

1 1 1 1 + + 4 4 4 4

(h)

(a) (c)

3

3

2.752 − 12 × 1 73

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1.7 Rounding numbers

In many calculations, particularly with decimals, you will not need to find an exact answer. Instead, you will be asked to give an answer to a stated level of accuracy. For example, you may be asked to give an answer correct to 2 decimal places, or an answer correct to 3 significant figures. To round a number to a given decimal place you look at the value of the digit to the right of the specified place. If it is 5 or greater, you round up; if it less than 5, you round down.

Worked example 13

D

Round 64.839906 to:

(a) the nearest whole number (a)

(b)

(c)

18

(b) 1 decimal place

(c) 3 decimal places

64.839906

4 is in the units place.

64.839906

The next digit is 8, so you will round up to get 5.

= 65 (to nearest whole number)

To the nearest whole number.

64.839906

8 is in the first decimal place.

64.839906

The next digit is 3, so the 8 will remain unchanged.

= 64.8 (1 d.p.)

Correct to 1 decimal place.

64.839906

9 is in the third decimal place.

64.839906

The next digit is 9, so you need to round up. When you round 9 up, you get 10, so carry one to the previous digit and write 0 in the place of the 9.

= 64.840 (3 d.p.)

Correct to 3 decimal places.

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1 Reviewing number concepts To round to 3 significant figures, find the third significant digit and look at the value of the digit to the right of it. If it is 5 or greater, add one to the third significant digit and lose all of the other digits to the right. If it is less than 5, leave the third significant digit unchanged and lose all the other digits to the right as before. To round to a different number of significant figures, use the same method but find the appropriate significant digit to start with; the fourth for 4sf, the seventh for 7sf etc. If you are rounding to a whole number, write the appropriate number of zeros after the last significant digit as place holders to keep the number the same size.

The first significant digit of a number is the first non-zero digit, when reading from left to right. The next digit is the second significant digit, the next the third significant and so on. All zeros after the first significant digit are considered significant.

Worked example 14 Round: (a) 1.076 to 3 significant figures (a)

1.076

The third significant figure is the 7. The next digit is 6, so round 7 up to get 8.

= 1.08 (3sf)

Correct to 3 significant figures.

0.00736

The first significant figure is the 7. The next digit is 3, so 7 will not change.

= 0.007 (1sf)

Correct to 1 significant figure.

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(b)

(b) 0.00736 to 1 significant figure

Exercise 1.17 Remember, the first significant digit in a number is the first nonzero digit, reading from left to right. Once you have read past the first non-zero digit, all zeros then become significant.

FAST FORWARD

(i) 4 significant figures (a) 4512 (e) 25.716 (i) 0.00998

(ii) 3 significant figures

(b) 12 305 (f) 0.000765 (j) 0.02814

(c) 65 238 (g) 1.0087 (k) 31.0077

(e) 0.999 (j) 0.4236 (o) 15.11579

(iii) 1 significant figure

(d) 320.55 (h) 7.34876 (l) 0.0064735

3 Change 2 5 to a decimal using your calculator. Express the answer correct to: 9

(a) 3 decimal places (d) 3 significant figures

(b) 2 decimal places (e) 2 significant figures

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(d) 2.149 (i) 8.299 (n) 12.0164

2 Express each number correct to:

D

You will use rounding to a given number of decimal places and significant figures in almost all of your work this year. You will also apply these skills to estimate answers. This is dealt with in more detail in chapter 5. X

1 Round each number to 2 decimal places. (a) 3.185 (b) 0.064 (c) 38.3456 (f) 0.0456 (g) 0.005 (h) 41.567 (k) 0.062 (l) 0.009 (m) 3.016

(c) 1 decimal place (f) 1 significant figure

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1 Reviewing number concepts

Summary Do you know the following?

• • • • • • •

Are you able to . . . ?

• • • • • • •

identify natural numbers, integers, square numbers and prime numbers find multiples and factors of numbers and identify the LCM and HCF write numbers as products of their prime factors using division and factor trees calculate squares, square roots, cubes and cube roots of numbers work with integers used in real-life situations apply the basic rules for operating with numbers perform basic calculations using mental methods and with a calculator.

D

•

Numbers can be classified as natural numbers, integers, prime numbers and square numbers. When you multiply an integer by itself you get a square number (x2). If you multiply it by itself again you get a cube number (x3). The number you multiply to get a square is called the square root and the number you multiply to get a cube is called the cube root. The symbol for square root is . The symbol for cube root is 3 . A multiple is obtained by multiplying a number by a natural number. The LCM of two or more numbers is the lowest multiple found in all the sets of multiples. A factor of a number divides into it exactly. The HCF of two or more numbers is the highest factor found in all the sets of factors. Prime numbers have only two factors, 1 and the number itself. The number 1 is not a prime number. A prime factor is a number that is both a factor and a prime number. All natural numbers that are not prime can be expressed as a product of prime factors. Integers are also called directed numbers. The sign of an integer (− or +) indicates whether its value is above or below 0. Mathematicians apply a standard set of rules to decide the order in which operations must be carried out. Operations in grouping symbols are worked out first, then division and multiplication, then addition and subtraction.

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• •

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1 Reviewing number concepts

Examination practice Exam-style questions 1

Here is a set of numbers: {−4, −1, 0, 3, 4, 6, 9, 15, 16, 19, 20} Which of these numbers are: (a) natural numbers? (d) prime numbers?

(b) square numbers? (e) multiples of two?

2

(a) List all the factors of 12.

3

Find the HCF of 64 and 144.

4

List the first five multiples of: (a) 12

(b) 18

(c) negative integers? (f) factors of 80?

(b) List all the factors of 24.

(c) 30

(c) Find the HCF of 12 and 24.

(d) 80

Find the LCM of 24 and 36.

6

List all the prime numbers from 0 to 40.

7

(a) Use a factor tree to express 400 as a product of prime factors. (b) Use the division method to express 1080 as a product of prime factors. (c) Use your answers to find: (i)

the LCM of 400 and 1080

(iii) 8

400

Calculate: (a) 262

9

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5

(ii) the HCF of 400 and 1080

(iv) whether 1080 is a cube number; how can you tell?

(b) 433

What is the smallest number greater than 100 that is: (a) divisible by two?

(b) divisible by ten?

(c) divisible by four?

D

10 At noon one day the outside temperature is 4 °C. By midnight the temperature is 8 degrees lower. What temperature is it at midnight? 11 Simplify:

(a) 6 × 2 + 4 × 5

(b) 4 × (100 − 15)

(c) (5 + 6) × 2 + (15 − 3 × 2) − 6

12 Add brackets to this statement to make it true. 7 + 14 ÷ 4 − 1 × 2 = 14

Past paper questions 1

Insert brackets to make this statement correct. 2×3−4+5=3

[1] [Question 2, Cambridge IGCSE Mathematics 0580 Paper 1 (Core) October/November 2006]

2

Calculate: (a)

0.0548 1.65 5.2 7

(b)

0.0763 1.85 4.7 8

[2] [Question 2a, Cambridge IGCSE Mathematics 0580 Paper 1 (Core) May/June 2009]

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7 Key words Perimeter Area Irrational number Sector Arc Semi-circle Solid Net Vertices Face Surface area Volume Apex Slant height

In this chapter you will learn how to:

EXTENDED

EXTENDED

• • • • • •

calculate areas and perimeters of twodimensional shapes

D

•

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• • • • • • • • • • • • • •

Perimeter, area and volume

calculate areas and perimeters of shapes that can be separated into two or more simpler polygons calculate areas and circumferences of circles calculate perimeters and areas of circular sectors

understand nets for threedimensional solids calculate volumes and surface areas of solids calculate volumes and surface area of pyramids, cones and spheres.

The glass pyramid at the entrance to the Louvre Art Gallery in Paris. Reaching to a height of 20.6 m, it is a beautiful example of a three-dimensional object. A smaller pyramid – suspended upside down – acts as a skylight in an underground mall in front of the museum.

When runners begin a race around a track they do not start in the same place because their routes are not the same length. Being able to calculate the perimeters of the various lanes allows the officials to stagger the start so that each runner covers the same distance. A can of paint will state how much area it should cover, so being able to calculate the areas of walls and doors is very useful to make sure you buy the correct size can. How much water do you use when you take a bath instead of a shower? As more households are metered for their water, being able to work out the volume used will help to control the budget.

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7 Perimeter, area and volume

7.1 Perimeter and area in two dimensions Polygons A polygon is a flat (two-dimensional) shape with three or more straight sides. The perimeter of a polygon is the sum of the lengths of its sides. The perimeter measures the total distance around the outside of the polygon. The area of a polygon measures how much space is contained inside it. Two-dimensional shapes

Formula for area

Quadrilaterals with parallel sides

h h

Triangles

h

b rectangle

b parallelogram

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b rhombus

h

h

b

b

Trapezium

D h

a

1 Area = ( 2

h

b

regular hexagon

b)h or

(

b)h 2

b

Here are some examples of other two-dimensional shapes.

It is possible to find areas of other polygons such as those on the left by dividing the shape into triangles and quadrilaterals.

irregular pentagon

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1 bh Area = bh or 2 2

b

a

kite

Area = bh

h

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7 Perimeter, area and volume

Units of area

Tip

If the dimensions of your shape are given in cm, then the units of area are square centimetres and this is written cm2. For metres, m2 is used and for kilometres, km2 is used and so on. Area is always given in square units.

You should always give units for a final answer if it is appropriate to do so. It can, however, be confusing if you include units throughout your working.

The formula for the area of a triangle can be written in different ways: 1 bh × b× h = 2 2 ⎛1 ⎞ OR = b ×h ⎝2 ⎠ ⎛1 ⎞ h OR = b × ⎝2 ⎠

(a) Calculate the area of the shape shown in the diagram.

Choose the way that works best for you, but make sure you write it down as part of your method.

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Worked example 1

5 cm

6 cm

7 cm

This shape can be divided into two simple polygons: a rectangle and a triangle. Work out the area of each shape and then add them together.

5 cm

5 cm

6 cm

D

You do not usually have to redraw the separate shapes, but you might find it helpful.

7 cm rectangle

triangle

Area of rectangle = bh = 7 × 5 = 35 cm (substitute values in place of b and h) 1 1 1 Area of triangle = bh = × 5 6 = × 30 = 15 cm2 2 2 2 Total area = 35 + 15 = 50 cm2

REWIND

(b) The area of a triangle is 40 cm2. If the base of the triangle is 5 cm, find the height.

At this point you may need to remind yourself of the work you did on transformation of formulae in chapter 6. W

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2

1 b×h 2 1 40 = × 5 × h 2 ⇒ 40 × 2 5 × h 40 × 2 80 ⇒h= = = 16 cm 5 5 A

Use the formula for the area of a triangle. Substitute all values that you know.

Rearrange the formula to make h the subject.

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7 Perimeter, area and volume

Exercise 7.1

1 By measuring the lengths of each side and adding them together, find the perimeter of each of the following shapes. (a)

(b)

(c)

(d)

2 Calculate the perimeter of each of the following shapes. (a)

(b)

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2.5 cm

5.5 cm

(c)

(d)

7 cm

D

4 cm

5 cm

3 cm

4 cm

2 cm

4 cm

10 cm

10 cm

2 cm 9 cm

(e)

2.5 m

(f)

2.8 m 1.9 m

8 km

9 km

8.4 m

3 km

3 km

7.2 m

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7 Perimeter, area and volume 3 Calculate the area of each of the following shapes. (a)

(b) 3m 11 cm 5m

5 cm

(c)

(d) 3.2 cm

4m

1.4 cm

5m

(f)

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(e) 2m

2.8 cm

8m

(g)

(h)

6 cm

6m

6m

D

5 cm

8m

10 cm

(i)

(j)

6 cm

4 cm 6 cm

4 cm 12 cm Draw the simpler shapes separately and then calculate the individual areas, as in worked example 1.

4 The following shapes can all be divided into simpler shapes. In each case find the total area. (a)

(b) 4m

5.1 m

7.2 m

1.2 m 8m

8m 2.1 m

4.5 m

5m

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7 Perimeter, area and volume (c)

(d)

4.9 cm

2.1 cm

5.3 cm 5.4 cm

8.2 cm

7.2 cm

7.2 cm

3.4 cm

7.8 cm

(e)

(f)

(g)

1.82 cm 3.71 cm

19.1 cm 8.53 cm

12 cm

12 cm

38.2 cm

18 cm

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7.84 cm

3.8 cm

2.4 cm

Write down the formula for the area in each case. Substitute into the formula the values that you already know and then rearrange it to find the unknown quantity.

5 For each of the following shapes you are given the area and one other measurement. Find the unknown length in each case. (a)

(b)

b

h

24 cm 2

289 cm 2

17 cm

(c)

D

8 cm

(d)

a

132 cm2

14 cm

(e)

15 cm

75

6 cm

h

b cm2

18 cm

200 cm2

6 cm

16 cm

6 How many 20 cm by 30 cm rectangular tiles would you need to tile the outdoor area shown below? 0.9 m 2.6 m

1.7 m 4.8 m

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7 Perimeter, area and volume 7 Sanjay has a square mirror measuring 10 cm by 10 cm. Silvie has a square mirror which covers twice the area of Sanjay’s mirror. Determine the dimensions of Silvie’s mirror correct to 2 decimal places. 8 For each of the following, draw rough sketches and give the dimensions: (a) two rectangles with the same perimeter but different areas (b) two rectangles with the same area but different perimeters (c) two parallelograms with the same perimeter but different areas (d) two parallelograms with the same area but different perimeters.

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Circles

‘Inscribing’ here means to draw a circle inside a polygon so that it just touches every edge. ‘Circumscribing’ means to draw a circle outside a polygon that touches every vertex.

Archimedes worked out the formula for the area of a circle by inscribing and circumscribing polygons with increasing numbers of sides.

REWIND

You learnt the names of the parts of a circle in chapter 3. The diagram on the right is a reminder of some of the parts. The diameter is the line that passes through a circle and splits it into two equal halves. W

umference circ ius rad

D

The circle seems to appear everywhere in our everyday lives. Whether driving a car, running on a race track or playing basketball, this is one of a number of shapes that are absolutely essential to us.

O is the centre

O diameter

Finding the circumference of a circle FAST FORWARD

π is an example of an irrational number. The properties of irrational numbers will be discussed later in chapter 9. X

Circumference is the word used to identify the perimeter of a circle. Note that the diameter = 2 × radius (2r). The Ancient Greeks knew that they could find the circumference of a circle by multiplying the diameter by a particular number. This number is now known as ‘π’ (which is the Greek letter ‘p’), pronounced ‘pi’ (like apple pie). π is equal to 3.141592654. . . The circumference of a circle can be found using a number of formulae that all mean the same thing. Circumference

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π diameter = πd (where d = diameter) = 2πr (where r = radius)

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7 Perimeter, area and volume Finding the area of a circle There is a simple formula for calculating the area of a circle. Here is a method that shows how the formula can be worked out: Consider the circle shown in the diagram below. It has been divided into 12 equal parts and these have been rearranged to give the diagram on the right.

r

height ≈ r length ≈

1 × 2π r = π r 2

Because the parts of the circle are narrow, the shape almost forms a rectangle with height equal to the radius of the circle and the length equal to half of the circumference. Now, the formula for the area of a rectangle is Area = bh so, (Using the values of b and h shown above)

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1 Area of a circle ≈ × 2πr × r 2 = πr 2

(Simplify)

If you try this yourself with a greater number of even narrower parts inside a circle, you will notice that the right-hand diagram will look even more like a rectangle. This indicates (but does not prove) that the area of a circle is given by: A = πr 2.

REWIND

BODMAS in chapter 1 tells you to calculate the square of the radius before multiplying by π. W

You will now look at some examples so that you can see how to apply these formulae.

Worked example 2

D

For each of the following circles calculate the circumference and the area. Give each answer to 3 significant figures.

Note that in (a), the diameter is given and in (b) only the radius is given. Make sure that you look carefully at which measurement you are given.

(a)

= π × diameter = π×8 = 25.1327...  25.1 mm

(a)

8 mm O

(b)

(b)

Circumference = π × diameter

5 cm O

= π × 10 ( d = 2 × ) = 31.415...  31.4 cm

Area = π × r

2

= π × 52 = π × 25 = 78.539...  78.5 cm2

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IGCSE_Maths_Ch07_p128-152.indd 135

d 2 = π × 42 = π × 16 = 50.265...

r=

 50.3 mm2

Tip Your calculator should have a π button. If it does not, use the approximation 3.142, but make sure you write this in your working. Make sure you record the final calculator answer before rounding and then state what level of accuracy you rounded to.

Area = π × r 2

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7 Perimeter, area and volume

Worked example 3 Calculate the area of the shaded region in the diagram. Shaded area = area of triangle – area of circle. 1 bh − πr 2 2 1 = × 18 × 20 − π × 2 52 2 = 160.365...

Area =

20 cm

Substitute in values of b, h and r. Round the answer. In this case it has been rounded to 3 significant figures.

 160.4 cm2

O 2.5 cm

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18 cm

Exercise 7.2

1 Calculate the area and circumference in each of the following. (a)

(b)

4m

3.1 mm

O

D

O

(c)

(d)

0.8 m 1 cm 2

O

In some cases you may find it helpful to find a decimal value for the radius and diameter before going any further, though you can enter exact values easily on most modern calculators. If you know how to do so, then this is a good way to avoid the introduction of rounding errors.

136

(e)

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IGCSE_Maths_Ch07_p128-152.indd 136

O

(f) 2 m π

2 km O

O

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7 Perimeter, area and volume 2 Calculate the area of the shaded region in each case. (a)

(b)

8 cm

8 cm

O

2 cm

8 cm

18 cm 8 cm

(c)

(d) 1m O 2m

12 m

15 cm 5 cm

O

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O

7m

10 cm

(e)

(f)

12 cm

5 cm

O

15 cm

12 cm

19 cm

D

3 cm

Living maths

3

3m

pond 10 m

12 m

The diagram shows a plan for a rectangular garden with a circular pond. The part of the garden not covered by the pond is to be covered by grass. One bag of grass seed covers five square metres of lawn. Calculate the number of bags of seed needed for the work to be done.

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7 Perimeter, area and volume 4

0.5 m

1.2 m

0.4 m

The diagram shows a road sign. If the triangle is to be painted white and the rest of the sign will be painted red, calculate the area covered by each colour. 5 Sixteen identical circles are to be cut from a square sheet of fabric whose sides are 0.4 m long. Find the area of the leftover fabric if the circles are made as large as possible.

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6 Anna and her friend usually order a large pizza to share. The large pizza has a diameter of 24 cm. This week they want to eat different things on their pizzas, so they decide to order two small pizzas. The small pizza has a diameter of 12 cm. They want to know if there is the same amount of pizza in two small pizzas as in one large. Work out the answer.

Arcs and sectors

arc len gt h

minor sector

r

The diagram shows a circle with two radii (plural of radius) drawn from the centre.

The region contained in-between the two radii is known as a sector. Notice that there is a major sector and a minor sector. A section of the circumference is known as an arc.

θ

D

r

O

The Greek letter θ represents the angle subtended at the centre.

major sector

Notice that the minor sector is a fraction of the full circle. It is Area of a circle is πr2. The sector is

θ of the circle. 360

θ of a circle, so replace ‘of ’ with ‘×’ to give: 360

θ × πr2 360 θ of a circle, then the length of the arc of a Circumference of a circle is 2πr. If the sector is 360 θ sector is of the circumference. So; 360 θ × 2πr Arc length = 360 Make sure that you remember the following two special cases: Sector area =

• • 138

If θ = 90° then you have a quarter of a circle. This is known as a quadrant. If θ = 180° then you have a half of a circle. This is known as a semi-circle.

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7 Perimeter, area and volume

Worked example 4 Find the area and perimeter of shapes (a) and (b), and the area of shape (c). Give your answer to 3 significant figures. (a)

Note that for the perimeter you need to add 5 cm twice. This happens because you need to include the two straight edges.

θ × 2π + 2r 360 30 = ×2×π×5+2×5 360 = 12.617...  12.6 m

θ × πr 2 360 30 = × π × 52 360 = 6.544...

Perimeter =

Area = 30° 5 m O

 6.54 m2

Note that the size of θ has not been given. You need to calculate it (θ = 360 − 65).

(b)

θ × πr 2 360 360 − 65 = × π × 42 360 295 = × π × 16 360 = 41.189...

Area = O 65° 4 cm

θ × 2π + 2r 360 295 = ×2×π×4+2×4 360 = 28.594...  28.6 cm

Perimeter =

ra ft

 41.2 cm2

FAST FORWARD

Total area = area of triangle + area of a semi-circle.

(c)

You will be able to find the perimeter of this third shape after completing the work on Pythagoras’ theorem in chapter 11. X

1 1 (Semi-circle is half of a circle bh + πr 2 2 2 so divide circle area by 2). 1 1 = × 8 × 6 + π × 42 2 2 = 49.132...

Area =

6 cm

O

 49.1 cm2

4 cm

D

Note that the base of the triangle is the diameter of the circle.

Exercise 7.3

1 Find Fi d the h area off the h coloured l d region i and d find d the h arc llength h l iin each h off the h ffollowing. ll i (a)

(b)

l

l

70° 18 cm 120° 8.2 cm O

O

(c)

l

(d) l

O 95°

3m O 175°

6.4 cm

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7 Perimeter, area and volume 2 For each of the following shapes find the area and perimeter. (a)

(b) 40° 6 cm

45° 8 cm

O

O

(c)

(d)

O 15°

5m

3.2 cm

75° O

(e)

(f) 15.4 m O 17.2 cm

ra ft

O

(g)

(h)

0.28 cm

O

O

(j)

15 cm 62°

6m

D

(i)

4.3 cm

62°

14 m

O OO

3 For each of the following find the area and perimeter of the coloured region. (b)

(a)

8 cm

10 cm 6 cm

5 cm

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9 cm

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7 Perimeter, area and volume

(c)

(d)

30°

(e)

8.4 cm

18 m

12 m 8.4 cm 18 m

30° 5m

4 Each of the following shapes can be split into simpler shapes. In each case find the perimeter and area. (a)

ra ft

(b)

28 cm

(d)

D

(c)

100° 3.2 cm

11 cm 7 cm

1.3 cm

3 cm

1.5 cm

7.2 Three-dimensional objects Tip

We now move into three dimensions but will use many of the formulae for two-dimensional shapes in our calculations. A three-dimensional object is called a solid.

You might be asked to count the number of vertices (corners), edges and faces that a solid has.

Nets of solids A net is a two-dimensional shape that can be drawn, cut out and folded to form a threedimensional solid.

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7 Perimeter, area and volume The following shape is the net of a solid that you should be quite familiar with. A

C

C

A

B

A

B

D

B

D

If you fold along the dotted lines and join the points with the same letters then you will form this cube: A

C

B

ra ft

D

You should try this yourself and look carefully at which edges (sides) and which vertices (the points or corners) join up. 1 The diagram shows a cuboid. Draw a net for the cuboid.

D

Exercise 7.4

c

b a

2 The diagram shows the net of a solid. (a) Describe the solid in as much detail as you can. (b) Which two points will join with point M when the net is folded? (c) Which edges are certainly equal in length to PQ?

S

P

R Q

T

U V

O

X N M

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W

Y Z

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7 Perimeter, area and volume 3 A teacher asked her class to draw the net of a cuboid cereal box. These are the diagrams that three students drew. Which of them is correct? (b)

(a)

(c)

ra ft

4 How could you make a cardboard model of this octahedral dice? Draw labelled sketches to show your solution.

7.3 Surface areas and volumes of solids It can be helpful to draw the net of a solid when trying to find its surface area.

The flat, two-dimensional surfaces on the outside of a solid are called faces. The area of each face can be found using the techniques from earlier in this chapter. The total area of the faces will give us the surface area of the solid.

The volume is the amount of space contained inside the solid. If the units given are cm, then the volume is measured in cubic centimetres (cm3) and so on. Some well known formulae for surface area and volume are shown below.

D

Cuboids

A cuboid has six rectangular faces, 12 edges and eight vertices. c

If the length, breadth and height of the cuboid are a, b and c (respectively) then the surface area can be found by thinking about the areas of each rectangular face.

b a

Notice that the surface area is exactly the same as the area of the cuboid’s net.

c

Surface area of cuboid = 2(ab + ac + bc) Volume of cuboid = a × b × c

b c

a

c c

b a×b

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7 Perimeter, area and volume The volume of a cuboid is its length × breadth × height. c

So, volume of cuboid = a × b × c.

b a

Prisms A prism is a solid whose cross-section is the same all along its length. (A cross-section is the surface formed when you cut parallel to a face.)

ra ft

length

cross-section

The cuboid is a special case of a prism with a rectangular cross-section. A triangular prism has a triangular cross-section.

length

length

cross-section

D

cross-section

The surface area of a prism is found by working out the area of each face and adding the areas together. There are two ends with area equal to the cross-sectional area. The remaining sides are all the same length, so their area is equal to the perimeter of the cross-section multiplied by the length: surface area of a prism = 2 × area of cross-section + perimeter of cross-section × length The volume of a prism is found by working out the area of the cross-section and multiplying this by the length. volume of a prism = area of cross-section × length

Cylinders r

A cylinder is another special case of a prism. It is a prism with a circular cross-section. h

r

h

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7 Perimeter, area and volume A cylinder can be ‘unwrapped’ to produce its net. The surface consists of two circular faces and a curved face that can be flattened to make a rectangle. Curved surface area of a cylinder = 2πrh and Volume = r 2h.

Exercise 7.5

2π r

h

1 Find the volume and surface area of the solid with the net shown in the diagram. 5 cm

3 cm 3 cm

5 cm

ra ft

4 cm

11 cm

3 cm

4 cm

5 cm

3 cm

D

5 cm

2 Find (i) the volume and (ii) the surface area of the cuboids with the following dimensions: (a) length = 5 cm, breadth = 8 cm, height = 18 cm (b) length = 1.2 mm, breadth = 2.4 mm, height = 4.8 mm

Living maths

3 The diagram shows a bottle crate. Find the volume of the crate.

80 cm

FIZZ 60 cm 90 cm

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7 Perimeter, area and volume 4 The diagram shows a pencil case in the shape of a triangular prism.

8 cm

10 cm 32

cm

6 cm

Calculate: (a) the volume and (b) the surface area of the pencil case.

ra ft

5 The diagram shows a cylindrical drain. Calculate the volume of the drain.

3m

1.2 m

Don’t forget to include the circular faces.

6 The diagram shows a tube containing chocolate sweets. Calculate the total surface area of the tube.

D

2.2 cm

10 cm

7 The diagram shows the solid glass case for a clock. The case is a cuboid with a cylinder removed (to fit the clock mechanism). Calculate the volume of glass required to make the clock case.

10 cm 5 cm

4 cm 8 cm

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7 Perimeter, area and volume 8 A storage company has a rectangular storage area 20 m long, 8 m wide and 2.8 m high. (a) Find the volume of the storage area. (b) How many cardboard boxes of dimensions 1 m × 0.5 m × 2.5 m can fit into this storage area? (c) What is the surface area of each cardboard box? 9 Vuyo is moving to Brazil for his new job. He has hired a shipping container of dimensions 3 m × 4 m × 4 m to move his belongings. (a) Calculate the volume of the container. (b) He is provided with crates to fit the dimensions of the container. He needs to move eight of these crates, each with a volume of 5 m3. Will they fit into one container?

Pyramids A pyramid is a solid with a polygon-shaped base and triangular faces that meet at a point called the apex.

h

If you find the area of the base and the area of each of the triangles, then you can add these up to find the total surface area of the pyramid. The volume can be found by using the following formula:

ra ft

The perpendicular height is the shortest distance from the base to the apex.

Volume =

1 × base area × perpendicular height 3

Cones

FAST FORWARD

l

D

The slant height can be calculated by using Pythagoras’ theorem, which you will meet in chapter 11. X

A cone is a special pyramid with a circular base. The length l is known as the slant height. h is the perpendicular height.

r

The curved surface of the cone can be opened out and flattened to form a sector of a circle. arc of

Curved surface area = πrl Volume =

length of circumf ere nc

length o f

and

r= cto se

1 2 πr h 3

of cone ase fb eo

If you are asked for the total surface area of a cone, you must work out the area of the circular base and add it to the curved surface area.

h

l l

h r

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7 Perimeter, area and volume

Spheres The diagram shows a sphere with radius r. Surface area = πr 2 r

and 4 Volume = πr 3 3

Exercise 7.6

1 The diagram shows a beach ball. (a) Find the surface area of the beach ball. (b) Find the volume of the beach ball.

ra ft

40 cm

2 The diagram shows a metal ball bearing that is completely submerged in a cylinder of water. Find the volume of water in the cylinder.

D

The volume of the water is the volume in the cylinder minus the displacement caused by the metal ball. The displacement is equal to the volume of the metal ball.

2 cm

15 cm

3 The Great Pyramid at Giza has a square base of side 230 m and perpendicular height 146 m.

Find the volume of the Pyramid.

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7 Perimeter, area and volume

13 m

4 The diagram shows a rocket that consists of a cone placed on top of a cylinder.

12

m

5m

(a) Find the surface area of the rocket.

25

m

(b) Find the volume of the rocket.

5 The diagram shows a child’s toy that consists of a hemisphere (half of a sphere) and a cone.

10 cm

(a) Find the volume of the toy.

ra ft

8 cm

(b) Find the surface area of the toy.

6 cm

D

6 The sphere and cone shown in the diagram have the same volume. Find the radius of the sphere.

7 The volume of the larger sphere (of radius R) is twice the volume of the smaller sphere (of radius r).

r 8.3 cm 2.4 cm

r

R

Find an equation connecting r to R.

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7 Perimeter, area and volume 8 A 32 cm long cardboard postage tube has a radius of 2.5 cm. (a) What is the volume of the tube? (b) For posting the tube is sealed at both ends. What is the surface area of the sealed tube? 9 A hollow metal tube is made using a 5 mm metal sheet. The tube is 35 cm long and has an exterior diameter of 10.4 cm. (a) Draw a rough sketch of the tube and add its dimensions (b) Write down all the calculations you will have to make to find the volume of metal in the tube. (c) Calculate the volume of metal in the tube. (d) How could you find the total surface area of the outside plus the inside of the tube?

Summary Do you know the following?

Are you able to . . . ?

•

• • • • • • • • • • • •

• • • •

150

recognise different two-dimensional shapes and find their areas give the units of the area calculate the areas of various two-dimensional shapes divide a shape into simpler shapes and find the area find unknown lengths when some lengths and an area are given calculate the area and circumference of a circle calculate the perimeter, arc length and area of a sector recognise nets of solids fold a net correctly to create its solid find the volumes and surface areas of a cuboid, prism and cylinder find the volumes of solids that can be broken into simpler shapes find the volumes and surface areas of a pyramid, cone and sphere.

ra ft

•

D

• •

The perimeter is the distance around the outside of a two-dimensional shape and the area is the space contained within the sides. Circumference is the name for the perimeter of a circle. If the units of length are given in cm then the units of area are cm2 and the units of volume are cm3. This is true for any unit of length. A sector of a circle is the region contained in-between two radii of a circle. This splits the circle into a minor sector and a major sector. An arc is a section of the circumference. Prisms, pyramids, spheres, cubes and cuboids are examples of three-dimensional objects (or solids). A net is a two-dimensional shape that can be folded to form a solid. The net of a solid can be useful when working out the surface area of the solid.

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7 Perimeter, area and volume

Examination practice Exam-style questions 1

A piece of rope is wound around a cylindrical pipe 18 times. If the diameter of the pipe is 600 mm, how long is the rope?

2

Find the perimeter and area of this shape. 1 cm

6 cm

1 cm

3

ra ft

9 cm

A cylindrical rainwater tank is 1.5 m tall with a diameter of 1.4 m. What is the maximum volume of rainwater it can hold?

30 mm

Past paper questions 1

An old Greek coin is a cylinder with a diameter of 30 millimetres and a thickness of 2 millimetres. Calculate, in cubic millimetres, the volume of the coin. [The volume of a cylinder, radius r, height h, is πr2h.]

2 mm

NOT TO SCALE

[2]

2

D

[Question 14, Cambridge IGCSE Mathematics 0588 Paper 1 (Core) June 2005]

4 cm

6 cm C

C

A

B

B

A

8 cm

The diagram shows a cuboid and its net. (a) Calculate the total surface area of the cuboid. (b) Calculate the volume of the cuboid.

[3] [2] [Question 8(a+b), Cambridge IGCSE Mathematics 0548 Paper 3 (Core) June 2005]

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7 Perimeter, area and volume

3

120 m

d

A 400 metre running track has two straight sections, each of length 120 metres, and two semicircular ends. (a) Calculate the total length of the curved sections of the track. (b) Calculate d, the distance between the parallel sections of the track.

[1] [2]

[Question 18, Cambridge IGCSE Mathematics 0548 Paper 1 (Core) Nov 2005]

A large conference table is made from four rectangular sections and four corner sections. Each rectangular section is 4 m long and 1.2 m wide. Each corner section is a quarter circle of radius 1.2 m.

ra ft

4

NOT TO SCALE

4

Each person sitting at the conference table requires one metre of its outside perimeter. Calculate the greatest number of people who can sit around the outside of the table. Show all your working.

[3]

5

D

[Question 11, Cambridge IGCSE Mathematics 0548 Paper 2 (Extended) Nov 2005]

A

12 cm

NOT TO SCALE

B

The largest possible circle is drawn inside a semi-circle, as shown in the diagram. The distance AB is 12 centimetres. (a) Find the shaded area. [4] (b) Find the perimeter of the shaded area. [2] [Question 23, Cambridge IGCSE Mathematics Paper 2 (Extended) June 2007]

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