Cambridge IGCSE Mathematics Coursebook PDF
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Original material © Cambridge University Press 2018
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Cambridge IGCSE
Mathematics Core and Extended Coursebook Second edition
Original material © Cambridge University Press 2018
Universi University ty Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, 06 -04/06, Singapore 079906 Cambridge University Press is part o the University o Cambridge. It urthers the Universi University’s ty’s mission mission by disseminating knowledge in the pursuit o education, learning and research at the highest international levels o excellence. www.cambridge.org Inormation on this title: education.cambridge.org/9781108437189 education.cambridge.org/9781108437189 © Cambridge University Press 2018 Tis publication is in copyright. Subject to statutory exception and to the provision provisionss o relevant collective licensing agreements, no reproduction o any part may take place without the written permission o Cambridge C ambridge University Press. First published 2012 Revised Edition First Published 2015 Printed in Dubai by Oriental Press
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Original material © Cambridge University Press 2018
Contents Introduction Acknowledgements
v viii
Unit 1
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
1 2 3 6 10 13 15 18
Chapter 2: Making sense of algebra 2.1 Using letters to represent unknown values 2.2 Substitution 2.3 Simplifying expres expressions sions 2.4 Working with brackets 2.5 Indices
23
24 26 29 33 35
Chapter 3: Lines, angles and shapes 3.1 Lines and angles 3.2 Triangles 3.3 Quadrilater Quadrilaterals als 3.4 Polygons 3.5 Circles 3.6 Construction
45 46 55 59 61 64 65
Chapter 4: Collecting, organising and displaying data 4.1 Collecting and classifying data 4.2 Organising data 4.3 Using charts to display data
71 74 76 86
Unit 2
Chapter 5: Fractions and standard form 5.1 Equivalent fractions 5.2 Operations on fractions 5.3 Percentages 5.4 Standard form 5.5 Your calcul calculator ator and standard form 5.6 Estimation
101 103 104 109 114 118 119
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
123 124 125 128 129
Chapter 7: Perimeter Perimeter,, area and volume 7.1 Perimeter and area in two dimensions 7.2 Three-dimensio Three-dimensional nal objects 7.3 Surface areas and volumes of solids
133 135 148 150
Chapter 8: Introduction to probability 8.1 Basic probability 8.2 Theoret Theoretical ical probability 8.3 The probabili probability ty that an event does not happen 8.4 Possibility diagrams 8.5 Combining independent and mutually exclusive events
160 161 162
11.3 Understanding similar shapes 11.4 Understand Understanding ing congruence
236 244
Chapter 12: Averages Averages and measures of spread 12.1 Different types of average 12.2 Making comparisons using average averagess and ranges 12.3 Calculating averages and ranges for frequency data 12.4 Calculating averages and ranges ranges for grouped continuous data 12.5 Percen Percentiles tiles and quartiles 12.6 Box-and-whisker plots
253 254
164 165 167
Unit 3
Chapter 9: Sequences and sets 9.1 Sequences 9.2 Rational and irrationa irrationall numbers 9.3 Sets
173 174 182 185
Chapter 10: Straight lines and quadratic equations 198 10.1 Straight lines 200 10.2 Quadratic (and other) expressions 216 Chapter 11: Pythagoras’ theorem and similar shapes 11.1 Pythagoras Pythagoras’’ theorem 11.2 Understanding similar triangles
226 227 231
257 258 262 265 269 277
Examination practice: practice: structured questions for Units 1–3
Contents
iii
Original material © Cambridge University Press 2018
Cambridge IGCSE Mathematics
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 x 2 is not 1 14.8 Algebraic fractions
281 283 285 289 294 297
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
301 303 310 314 319 321 322
15.5 trigonometry Sines, cosines and tangents tangents of angles more than 90° 15.6 The sine and cosine rules 15.7 Area of a triangle 15.8 Trigonometry in three dimensions
359 363 371 374
Chapter 16: Scatter diagrams and correlation 16.1 Introduction to bivariate data
383 384
Chapter 19: Symmetry 19.1 Symmetry in two dimensions 19.2 Symmetry in three dimensions 19.3 Symmetry properties of circles 19.4 Angle relationsh relationships ips in circles
459 461 464 467 470
324 326
335 336 339
340 355
Unit 5
Chapter 17: Managing money 17.1 Earning money 17.2 Borrowing and investing money 17.3 Buying and selling
394 395 401 409
Chapter 18: Curved graphs 18.1 Drawing quadratic graphs (the parabola) 18.2 Drawing 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 18.7 Derived functions
415 416 424 428
Chapter 20: Histograms and frequency distribution diagrams 483 20.1 Histogra Histograms ms 485 20.2 Cumulative frequency 492
429 431 441 443
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 transfor transforming ming formulae 22.3 Functions and function notation
iv
Contents
506 507 512 515 517 525
528 532
Chapter 23: Vectors and transformations 23.1 Simple plane transfor transformations mations 23.2 Vectors 23.3 Further transformations
556 557 570 582
Chapter 24: Probability using tree tree diagrams and Venn diagrams 24.1 Using tree diagrams to show outcomes 24.2 Calculating probability from tree diagrams 24.3 Calculating probability from Venn diagrams 24.4 Conditional probability
595 597 598 600 604
536 537 543 546
Examination practice: structured questions for Units 4–6 Answers
611 617
Glossary
688
Index
694
Original material © Cambridge University Press 2018
Chapter 5: 5: Fractions and standard form Key words
• Fraction • Vulgar fraction • Numerator • Denominator • Equivalent fraction • Simplest form • Lowest terms • Mixed number • Common denominator • Reciprocal • Percentage increase •• Percentage Percentage decrease • Reverse percentage • Standard form • Estimate In this chapter you will learn how to:
• find equivalent fractions • simplify fractions • add, subtract, multiply and divide fractions and mixed numbers
D E D N E T X E
• find fractions of numbers one number as a • find percentage of another a percentage of a • find number percentage • calculate increases and decreases and decrease by a • increase given percentage reverse percentages • handle (undoing increases and decreases) with standard form make estimations without a • work calculator.
Te Rhind Mathematical Papyrus is one of the earliest examples of a mathematical document. It is thought to have been written sometime between 1600 and 1700 BC by an Egyptian scribe called calle d Ahmes, though it may be a copy of an older document. Te first section secti on of it is devoted to work with fractions.
Fractions are not only useul or improving your arithmetic skills. You You use them, on an almost daily basis, ofen without realising it. How ar can you travel travel on hal a tank t ank o petrol? I your share o do a pizza is two-thirds will you still be hungry? o your journey complete how ar you still have to travel? A hairdresser needsItothree-fifhs mix her dyes by the correctis amount and a nurse needs the correct dilution o a drug or a patient.
Unit 2: Number
101 10 1
Original material © Cambridge University Press 2018
Cambridge IGCSE Mathematics
RECAP
You should already be familiar with the following fractions work: work: Equivalent fractions
Find equivalent fractions by multiplying or dividing the numerator and denominator by the same number. ×
=
and
are equivalent
÷
= and are equivalent To simplify a fraction you divide the numerator and denominator de nominator by the same number.
Mixed numbers
Convert between mixed numbers and improper fractions: Number o data in that group, not individual values.
Calculating with fractions
To add or subtract fractions make sure they have the same denominators. Class intervals are equal and should not overlap.
To multiply fractions, multiply numerators by numerators and denominators by denominators. Write the answer in simplest form. Multiply to find a fraction of an amount. The word ‘of’ means multiply.
To divide by a fraction you multiply by its reciprocal.
Percentages
The symbol % means per cent or per hundred. Percentages can be written as fractions and decimals. 45% =
=
45% = 45 ÷ 100 = 0.45 Calculating percentages
To find a percentage perc entage of an amount: use fractions and cancel or 25% of 60 = 1
60 ×
1
= 15
41
102
or
use a calculator
15
25 100
use decimals
Unit 2: Number
0.25 × 60 = 15
2
5
÷
×
6 0
=
15
Original material © Cambridge University Press 2018
5 Fractions and and standard form
5.1 REWIND
Before reading this next section you should remind yourself about Highest Common Factors (HCFs) in chapter 1. 1.
Equivalent fractions A fraction is part o a whole number. Common ractions (also called vulgar fractions) are written in the orm . Te number on the top, a, can be any number and is called the t he numerator . Te number on the bottom, b, can be any number except 0 and is called the t he denominator. Te numerator and the denominator are separated by a horizontal line. I you multiply or dividethe both theamount numerator and the denominator byraction. the sameTe number, number the new raction still represents same o the whole as the original new ,raction is known as an equivalent fraction. For example,
REWIND
You have come across simplifying in chapter 2 in the context of algebra. algebra.
LINK
and
.
Notice in the second example that the original raction
has been divided to smaller terms
and that as 5 and 7 have no common actor other than 1, the raction cannot be divided any urther. Te raction is now expressed in its simplest form (sometimes called the lowest terms). So, simpliying a raction means expressing it using the lowest possible terms.
Worked example 1
Percentages are
Express each of the following in the simplest form possible.
particularly important when we deal with money. How ofen have you been in a shop where the signs tell you that prices are reduced by 10%? Have you considered a bank account and how money is added. The study o inancial ideas orms the greater part o Economics.
a
Notice that in each case you y ou divide the numerator and the denominator by the HCF of both.
b
c
d
a
b
c
d
is already in its simplest form (5 and 8 have no common factors other than 1).
Worked example 2 Which two of ,
and
are equivalent fractions?
Simplify each of the other fractions:
is already in its simplest form.
You could have written: 5
15 18
5 =
6
6
This is called cancelling cancelling and and is a shorter way of showing what you have done.
So
and
are equivalent.
Unit 2: Number
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Original material © Cambridge University Press 2018
Cambridge IGCSE Mathematics
Exercise 5.1
1
By multiplying or dividing both the numerato numeratorr and denominator by the same number, number, find three equivalent ractions or each o the ollowing. a
b
c
d
e
2 Express each o the ollowing ractions in its simplest orm. a
5.2
b
c
d
e
f
g
Operations on fractions Multiplying fractions When multiplying two or more ractions together you can simply multiply the numerators numerators and then multiply the denominators. Sometimes you will then need to simpliy your answer answer.. It can be aster to cancel the ractions beore you multiply.
Worked example 3 Calculate: a
b
c
a
Notice that you can also cancel before multiplying:
Multiply the numerators to get the new numerator value. Then do the same with the denominators. Then express the fraction in its simplest form.
1
3 4
2 ×
7
=
3×1 2×7
Divide the denominator of the first fraction, and the numerator of the second fraction, by two.
3 =
14
2
To multiply a fraction by an integer you only multiply the numerator by the integer. For example,
15 and 7 do not have a common factor other than 1 and so cannot be simplified.
b
. c
Here, you have a mixed number ( ). This needs to be changed to an improper fraction (sometimes called a top heavy fraction), which is a fraction where the numerator is larger than the denominator. This allows you to complete the multiplication.
To change a mixed number to a vulgar fraction, multiply the whole number part (in this case 4) by the denominator and add it to the numerator. So:
Exercise 5.2
Unit 2: Number
Notice that the word ‘of’ is replaced with the × sign.
Evaluate each o the ollowing. 1 a
b
c
d
2
a
b
c
d
e
f
g
h
104
Original material © Cambridge University Press 2018
5 Fractions and and standard form
Adding and subtracting fractions REWIND
You will need to use the lowest common multiple (LCM) in this section. You met this in chapter 1. 1.
You can only add or subtract ractions that are the t he same type. In other words, they must have the same denominator. denominator. Tis is called call ed a common denominator. You must use what you know about equivalent ractions to help you make sure ractions have a common denominator. denominator. Te ollowing worked example shows how you can use the LCM o both denominators as the common denominator.
Worked example 4 Write each of the following as a single fraction in its simplest form. a
b
a
c
Find the common denominator.
Notice that, once you have a common denominator, you only add the numerators. Never add add the denominators!
The LCM of 2 and 4 is 4. Use this this as the common denominator and find the equivalent fractions. Then add the numerators. b
You will sometimes find that two fractions added together can result in an improper fraction (sometimes called a top-heavy fraction). Usually you will re-write this as a mixed number.
Find the common denominator. The LCM of 4 and 6 is 12. 12. Use this as the common denominator and find the equivalent fractions. Add the numerators. Change an improper fraction to a mixed number. c
The same rules apply for subtracting fractions as adding them.
Change mixed numbers to improper fractions to make them easier to handle. The LCM of 4 and 7 is 28, 28, so this is the common denominator. Find the equivalent fractions. Subtract one numerator from the other.
Change an improper fraction to a mixed number.
Egyptian fractions
Tip Egyptian ractions are a good example o manipulating manipula ting ractions but they are not in the syllabus.
Exercise 5.3
An Egyptian raction is the sum o any number o different ractions ractions (different denominators) denominators) each with numerato numeratorr one. For example
is the Egyptian raction that represents . Ancient
Egyptians used to represent single ractions in this way but in modern times we tend to preer the single raction that results rom finding a common denominator. denominator. Evaluate the ollowing. 1
a
b
c
d
e
f
g
h
105
Original material © Cambridge University Press 2018
Cambridge IGCSE Mathematics
2
Remember to use BODMAS here. Think which two fractions with a numerator of 1 might have an LCM equal to the denominator given.
FAST FORWARD
The multiplication, division, addition and subtraction of fractions will be revisited in chapter 14 when algebraic fractions fractions are considered.
a
b
c
d
e
f
g
h
i
j
k
l
3 Find Egyptian ractions or each o the ollowing.
a
b
c
d
Dividing fractions Beore describing how to divide two ractions, the reciprocal needs to be introduced. Te reciprocal o any raction can be obtained by swapping the numerator and the denominator. denominator. So, the reciprocal o
is
Also the reciprocal o
and the reciprocal o
is
is
.
or just 2 and the reciprocal o 5 is
.
I any raction is multiplied by its reciprocal then the result is always 1. For example: ,
and
o divide one raction ract ion by another raction, you simply multiply the first raction by the reciprocal o the second. Look at the example below:
Now multiply both the numerator and denominator by bd and and cancel:
Worked example 5 Evaluate each of the following. a
b
a
3 4
2
d
1
1 ÷
c
3 =
4 2
2 ×
1
3 =
2
=1
1
Multiply by the reciprocal of . Use the rules you
2
have learned about multiplying fractions.
106
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
b
3
1
4
÷
2
1 3
7 ÷
4 17
=
4
Convert the mixed fractions to improper fractions.
7
=
3
Multiply by the reciprocal of .
3 ×
71
3 =
c 5 8
÷
4
Write 2 as an improper fraction.
5 2 2= ÷ 8 1 5 =
8
Multiply by the reciprocal of .
1 ×
2
5 =
d
To divide a fraction by an integer you can either just multiply the denominator by the integer, or divide the numerator by the same integer.
Exercise 5.4
6 7
16
2
6
÷
3=
7 2
=
1 ×
3 1
7
Evaluate each o the ollowing. 1 2
3
4
5
7
8
6
E
9 Evaluate a
2 1 − 1 2 ÷ 1 1 3 5 3
b 2
1 2 1 1 1 3 5 3 −
÷
Fractions with decimals Sometimes you will find that either the numerator or the denominator denominator,, or even both, b oth, are not whole whole numbers! o express these ractions in their simplest orms you need to
Worked example 6
• •
make sure both the numerator and denominator are converted to an integer by finding an equivalent raction check that the equivalent raction has been simplified.
Simplify each of the following fractions. a
b
c
a
Multiply 0.1 by 10 to convert 0.1 to an integer. To To make sure the fraction is equivalent, you need to do the same to the numerator and the denominator, so multiply 3 by 10 as well.
b
Multiply both the numerator and denominator by 10 to get integers. 13 and 24 do not have a HCF other than 1 so cannot be simplified.
c
Multiply 0.12 by 100 to produce an integer. Remember to also multiply the numerator by 100, so the fraction is equivalent. The final fraction can be simplified by cancelling.
Unit 2: Number
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107 10 7
Cambridge IGCSE Mathematics
Exercise 5.5 Remember that any fraction that contains a decimal in either its numerator or denominator will not be considered to be simplified. What fraction can be used to represent 0.3?
Simpliy each o the ollowing ractions. 1
2
3
4
5
6
7
8
Further calculations with fractions You can use ractions to help you solve problems. Remember or example, that
and that, although this may seem trivial, this simple act
can help you to solve problems easily.
Worked example 7 Suppose that
of the students in a school are are girls. If the school has 600 students, how
many girls are there? 120
2 2 2 600 of 600 = × 600 = × = 2 ×120 = 240 girls 5 5 5 1
Remember in worked example 3,
1
×
you saw that ‘of’ is replaced by .
Worked example 8 Now imagine that that
of the students in another school are boys, and that there are 360
boys. How many students are there in the whole school? of the total is 360, so
of the total must be half of this, 180. This means that
of
the population, that is all of it, is 5 × 180 = 900 students in total.
Exercise 5.6
1
o the people at an auction bought an item. I there are 120 people at the auction, how many bought something?
2 An essay contains 420 sentences. 80 o these t hese sentences contain typing errors. What raction (given in its simplest orm) o the sentences contain errors?
3 28 is o which number? 4 I
o the people in a theatre buy a snack during during the interval, interval, and o those who buy a snack buy ice cream, what raction o the people in the theatre buy ice cream?
5 Andrew Andrew,, Bashir and Candy C andy are trying to save money or a birthday party. I Andrew saves o the total needed, Bashir saves
and Candy saves
, what raction o the cost o the
party is lef to pay? 1 6 cups o cooked rice or a recipe o Nasi Goreng. I 2 cups o uncooked 6 Joseph needs 1 2 1 rice with 2 cups o water make 4 cups o cooked rice, how many cups o uncooked 2 3 rice does Joseph need or his recipe ? How much water should he add ?
E
108
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
5.3
Percentages A percentage is a raction with a denominator o 100. Te symbol used to represent percentage is %. o find 40% o 25, you simply need to find
o 25. Using Using what you know about multiplying
ractions: 2
40
40
100
×
25 =
100
1
5
25 5
2 =
1
5
×
2 =
∴
25 ×
1
1 5
×
1
=
10
40% o 25 = 10
Equivalent forms A percentage can be converted into a decimal by dividing by 100 (notice that the digits d igits move two places to the right). So, 45%
45 =
=
100
0.45 and 3.1%
3 .1 =
=
100
0.031.
A decimal can be converted to a percentage by multiplying multiplying by 100 (notice that the digits d igits move two places to the lef). So, 0.65
65 =
=
65% and
0.7 × 100 = 70.
100
Converting percentages to vulgar ractions (and vice versa) involves a ew more stages.
Worked example 9 Convert each of the following percentages to fractions in their simplest form. a
Remember that a fraction that contains a decimal is not in its simplest form.
25%
b
30%
c
3.5%
a
Write as a fraction with a denominator of 100, then simplify.
b
Write as a fraction with a denominator of 100, then simplify.
c
Write as a fraction with a denominator of 100, then simplify.
Worked example 10 Convert each of the following fractions into percentages. a a
b
Find the equivalent fraction with a denominator of 100. (Remember to do the same thing to both the numerator and denominator).
Unit 2: Number
Original material © Cambridge University Press 2018
109
Cambridge IGCSE Mathematics
Find the equivalent fraction with a denominator of 100. (Remember to do the same thing to both the numerator and denominator).
b
Although it is not always easy to find an equivalent raction with a denominator o 100, any raction can be converted into a percentage by multiplying by 100 and cancelling.
Worked example 11 Convert the following fractions into percentages: a
b
a
Exercise 5.7
Later in the chapter you will see that a percentage can be greater than 100.
1
b
Convert each o the ollowing percentages into ractions in their simplest orm. a 70%
b 75%
c 20%
d 36%
g 215%
h 132%
i 117.5% j 108.4%
e 15%
f 2.5%
k 0.25%
l 0.002%
e
f
2 Express the ollowing ractions as percentages. p ercentages. a
b
c
d
Finding one number as a percentage of another o write one number as a percentage o another number, you start by writing the first number as a raction o the second number then multiply by 100.
Worked example 12 a
Express 16 as a percentage of 48. First, write 16 as a fraction of 48, then multiply by 100. This may be easier if you write the fraction in its simplest form first.
b
Express 15 as a percentage of 75. Write 15 15 as a fraction of 75, then simplify and multiply by 100. You know that 100 divided by 5 is 20, so you don’t need a calculator.
c
Express 18 as a percentage of 23. 23. You need to calculate
, but this is not easy using basic fractions because
you it further,Simply and 23type: does not divide neatly into 100. 100. Fortunately, you cannot can usesimplify your calculator. 1
8 ÷
2
3
×
1 0 0
=
78.26%
110 11 0
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
Exercise 5.8
Where appropria appropriate, te, give your answer to 3 significant figures.
1 Express 14 as a percentage o 35. 2 Express 3.5 as a percentage o 14. 3 Express 17 as a percentage o 63. 4 36 people live in a block o flats. 28 o these people jog around the park each morning. What percentage o the people living in the block o flats go jogging jog ging around the park? marks did Jack Jack get? 5 Jack scores in a test. What percentage o the marks 6 Express 1.3 as a percentage o 5.2.
7 Express 0.13 as a percentage o 520.
Percentage increases and decreases Suppose the cost o a book increases rom $12 to $15. Te actual increase is $3. As a raction o the original value, the increase is
. Tis is the ractional change and you can write
this raction as 25%. In this example, the value o the book bo ok has increased by 25% o the original value. Tis is called the percentage increase. I the value had reduced (or example i something was on sale in a shop) then it would have been a percentage decrease. Note careully : whenever increases or decreases are stated as percentages, they are stated as percentages o the original value. value.
Worked example 13 a
The value of a house increases from $120 000 to $124 800 between August and December.. What percentage increase December in crease is this?
$124 800 − $120 000 = $4800
b
First calculate the increase. Write the increase as a fraction of the original and multiply by 100. Then do the calculation (either in your head or using a calculator).
The United States Census Census Bureau found the population of New Orleans, Louisiana to be 484 48 4 674 before Hurricane Katrina and 223 388 in the year following followin g Hurricane Katrina. What percentage decrease does this represent? 484 674 – 223 388 = 261 286
First calculate the decrease. Write the decrease as a fraction of the original, and multiply by 100. Use a calculator to find the answer. (This answer is rounded to 3sf)
Exercise 5.9 Applying your skills Where appropriate, appropriate, give your answer to the nearest whole percent.
1 Over a five-year period, p eriod, the population o the t he state o Louisiana in the Unit United ed States o America decreased rom 4 468 976 to 4 287 768. Find the percentage decrease in the population o Louisiana in this period. 2
Barney bought 38 CDs one year and 46 the next year. Find the percentage increase.
Unit 2: Number
Original material © Cambridge University Press 2018
111 11 1
Cambridge IGCSE Mathematics
3 A theatre has enough seats or 450 audience members. Afer renovation it is expected expec ted that this number will increase to 480. Find the percentage increase.
4 Sally works in an electrical component actory. On Monday she makes a total o 363 components but on uesday uesday she makes 432. Calculate Cal culate the percentage increase.
5 Inter Polation Polation Airlines carried a total o 383 402 402 passengers one year and 287 431 the ollowing year. year. Calculate the t he percentage decrease in passengers carried by the airline.
6 A liquid evaporates steadily. In one hour the mass o liquid in a laboratory container decreases rom 0.32 kg to 0.18 kg. Calculate the percentage decrease.
Increasing and decreasing by a given percentage I you know what percentage you want to increase or decrease an amount by, you can find the actual increase or decrease by finding a percentage o the original. I you want to know the new value you either add the increase to or subtract subtract the decrease rom the original value.
Worked example 14 Increase 56 by: Remember that you are always considering a percentage of the original value.
a
10% 10%
a
b
15% 15%
c
4%
First of all, you need to calculate 10% of 56 to work out the size of the increase. To increase the original by 10% you need to add this To to 56.
56 + 5.6 = 61.6
If you don’t need to know the actual increase but just the final value, you can use this method: If you consider the original to be 100% then adding 10% to this will give If gi ve 110% of the original. So multiply 56 by b
, which gives 61.6.
A 15% increase will lead to 115% of the original.
c
A 4% increase will lead to 104% of the original. orig inal.
Worked example 15 In a sale all items are reduced by 15%. If the normal selling price for a bicycle is $120 calculate the sale price. 100 − 15 = 85 × $120 = $102
Exercise 5.10
Note that reducing a number by 15% leaves you with 85% of the original. So you simply find 85% of the original value.
1 Increase 40 by: a 10% 2 Increase 53 by: a 50%
b 15%
c 25%
d 5%
e 4%
b 84%
c 13.6%
d 112%
e
112 11 2
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
3 Decrease 124 by: a 10%
b 15%
c 30%
d 4%
e 7%
b 35.4%
c 0.3%
d 100%
e
4 Decrease 36.2 by: a 90%
Applying your skills 5 Shajeen usually works 30 hours per week but decides that he needs to increase this by 10% to be sure that he can save enough or a holiday. How many many hours per week will Shajeen need to work?
6 12% sales tax is applied to all items o clothing sold in a certain cert ain shop. I a -shirt -shirt is advertised or $12 (beore tax) what will be the cost o the -shirt once tax is added?
7 Te Oyler Teatre steps up its advertising campaign and manages to increase its audiences by 23% during the year. I 21 300 people watched plays at the Oyler Teatre during the previous year, how many many people watched plays in the year o the campaign?
8 Te population o rigville was 153 000 at the end o a year. Following a flood, 17% o the residents o rigville moved away. What was the population o rigville afer the flood?
9 Anthea decides that she is watching too much television. I Anthea watched 12 hours o television in one week and then decreased this by 12% the next week, how much time did Anthea spend watching television in the second week? Give your answer in hours and minutes.
Reverse percentages Sometimes you are given the value or amount o an item afer a a percentage increase or decrease has been applied to it and you need to know what the original value was. o solve this type o reverse percentage question it is important to remember that you are always dealing with percentages o the original values. values. Te method used in worked worked example 14 (b) and (c) is used to help us solve these type o problems.
Worked example 16 A store is holding a sale in which every item is reduced by 10%. A jacket in this sale is i s sold for $108. How can you find the original price of the Jacket? If an item is reduced by 10%, the new cost is 90% of the original (100–10). If x If x is is the original value of the jacket then you can write a formula using the new price.
108 108 original price = $120.
Notice that when the reciprocal,
was moved to the other side of the = sign it became its
.
Important : Undoing a 10% decrease is not is not the the same as increasing the reduced value by 10%. If you increase you increase the the sale price of $108 by 10% you will get
$108 = $118.80 which is a different (and incorrect ) answer.
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Unit 2: Number
Original material © Cambridge University Press 2018
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Cambridge IGCSE Mathematics
Exercise 5.11
E
1 I 20% o an amount is 35, what is 100%? 2 I 35% o an amount is 127, what is 100%? 3 245 is 12.5% o an amount. What is the total amount? 4 Te table gives the sale price and the % by which the price was reduced or a number o items. Copy the table, then complete it by calculating the original prices. Sale price ($)
% reduction
52.00 185.00 4700.00 2.90 24.50 10.00 12.50 9.75 199.50 99.00
10 10 5 5 12 8 7 15 20 25
Original price ($)
5 A shop keeper marks up goods by 22% beore selling them. Te selling price o ten items are given below. For each one, work out the cost price (the price beore the mark up). a $25.00 f $45.80
b g
$200.00 $29.75
c h
$14.50 $129.20
d i
$23.99 $0.99
e j
$15.80 $0.80
6 Seven students were absent rom a class on Monday. Monday. Tis is 17.5% o the class. total? a How many students are there in the class in total? b How many students were present on Monday?
7 A hat shop is holding a 10% sale. I Jack buys a hat or $18 in the t he sale, how much did the hat cost beore the sale?
8 Nick is training or a swimming race and reduces his weight by 5% over a 3-month period. I Nick now weighs 76 kg how much did he weigh beore he started training?
9 Te water in a pond evaporates at a rate o 12% per week. I the pond now contains 185 litres o water, approximately approximately how much water was in the pond a week ago?
5.4
Standard form When numbers are very small, like 0.0000362, or very large, like 358 000 000, calculations can be time consuming and it is easy to miss out some o the t he zeros. Standard form is used to express very small and very large numbers in a compact and and efficient way. way. In standard standard orm, numbers numbers are written as a number multiplied by 10 raised to a given power.
Remember that digits are in place order: 100 000s 0s 100 00ss 10s un unit itss 3
0
0
0
10t 0ths hs 100 00th thss 100 000t 0ths hs •
0
0
0
Standard form for large numbers Te key to standard orm or large numbers is to understand what happens when you multiply by powers o 10. Each time you multiply a number by 10 each digit within the t he number moves one place order to the lef (notice that this looks looks like like the decimal point has moved one place to the right). 3.2 3.2 × 10 = 32.0
Te digits have moved one place order to the lef.
2
3.2 × 10 = 3.2 × 100 = 320.0 Te digits have moved two places. 3.2 × 103 = 3.2 × 1000 = 3200.0 Te digits have moved three places. ... and so on. You You should see a pattern orming.
114 11 4
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
Any large number can be expressed in standard orm by writing it as a number between 1 and 10 multiplied by a suitable power o 10. o do this t his write the appropriate number between 1 and 10 first (using the non-zero digits o the original number) and then count the number o places you need to move the first digit to the t he lef. Te number o places tells you by what power p ower,, 10 should be multiplied.
Worked example 17 Write 320 000 in standard form. 3.2
5
3
Start by finding the number between 1 and 10 that has the same digits in the same order as the original number. Here, the extra 4 zero digits can be excluded because they do not change the size of your new number. 4
3
2 0 0
2
Now compare the position of the first digit in both numbers: ‘3’ has to move 5 place orders to the left to get from the new number to the original number.
1
0 0.0 3.2
320 000 = 3.2 × 105
The first digit, ‘3’, ‘3’, has moved five places. So, you multiply by 105.
Calculating using standard form REWIND
The laws of indices can be found in chapter 2.
Once you have converted large numbers into into standard orm, you can use the index laws to carry carr y out calculations involving multiplication and division.
Worked example 18 Solve and give your answer in standard form. a
b
c
d
Simplify by putting like terms together. Use the laws of indices where appropriate.
a
Write the number in standard form. You may be asked You asked to to convert convert your answe answerr to an an ordinary ordinary number number.. To To convert convert 6 × 1011 into an ordinary number, the ‘6’ needs to move 11 places to the left:
Although it is the place order that is changing; it looks like the decimal point moves to the right.
6.0
×
11 10 =
When you solve problems in standard form you need to check your results carefully. Always be sure to check that your final answer is in standard form. Check that all conditions are satisfied. Make sure that the number part is between 1 and 10.
b
c
11
10 9
8
7
6
5
6 0 0 0 0 0 0 0
4
3
2
0 0 0
1
0 .0
The answer 16 16 × 1010 is numerically correct but it is not in standard form because 16 is not between 1 and 10. You can change it to standard form by thinking of 16 as 1.6 × 10. Simplify by putting like terms together. Use the laws of indices.
Unit 2: Number
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Cambridge IGCSE Mathematics
To make it easier to add up the ordinary numbers make sure they are lined up so that the place values match:
d
When adding or subtracting numbers in standard form it is often easiest to re-write them both as ordinary numbers first, then convert the answer to standard form.
300 000 000 + 9 000 000
So
Exercise 5.12
1 Write each o the ollowing numbers in standard orm. a 380 e 20
When converting standard form back to an ordinary number, the power of 10 tells you how many places the first digit moves to the left (or decimal point moves to the right), not how many zeros there are.
Remember that you can write these as ordinary numbers before adding or subtracting.
b 4 200 000 f 10
c 45 600 000 000 g 10.3
d 65 400 000 000 000 h 5
2 Write each o the ollowing as an ordinary number. a
b
c
e 7.1 × 101
d
3 Simpliy each o the ollowing, leaving your answer in standard orm. a d g
b e h
c f i
j
k
l
4 Simpliy each o the ollowing, leaving your answer in standard orm. a d
b e
c
Standard form for small numbers You have seen that digits move place order to the lef when multiplying by powers o 10. I you divide divide by by powers o 10 move the digits in place order to the right and make the number smaller . Consider the ollowing pattern: 2300 2300 ÷ 10 = 230
. . . and so on. Te digits move place order to the right (notice that this looks lo oks like the decimal point is moving to the lef). You saw in chapter 1 that i a direction is taken to be positive, the opposite direction is taken to be negative. Since moving place order to the lef raises 10 to the power o a positive index, positive index, it ollows that moving place order to the right raises 10 to the power o a negative negative index. index. Also remember rom chapter 2 that you can write negative powers to indicate that you divide, and you saw above that with small numbers, you divide by 10 to express the number in standard orm.
116 11 6
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
Worked example 19 Write each of the following in standard form. a
0.004 0.004
b
0.000 000 34
a 1
0.0
2
3
0 4
4.0
c
Start with a number between 1 and 10, in this case 4. Compare the position of the first digit: ‘4’ needs to move 3 place orders to the right to get from the new number to the original number. In worked example 17 you saw that moving 5 places to the 5
–3 =
4 × 10
3
−
left meant meant multiplying by 10 , so it follows that moving 3 places to the right the right means means multiply by 10 . Notice also that the first non-zero digit in 0.004 0.0 04 is in the 3rd place after the decimal point and that the power of 10 is −3. Alternatively: you know that you need to divide by 10 three times, so you can change it to a fractional index and then a negative index.
b
1
2
3
4
5
6
7 –7
0. 0 0 0 0 0 0 3 4 = 3.4 × 10
Notice that the first non-zero digit dig it in 0.000 000 34 is in the 7th place after the decimal point and that the power of 10 is −7. c
Simplify by gathering like terms together. Use the laws of indices.
Exercise 5.13
1 Write each o the ollowing numbers in standard orm. a 0.004
When using standard form with negative indices, the power to which 10 is raised tells you the position of the first non-zero first non-zero digit digit after (to the right of) the decimal point. For some calculations, you might need to change a term into standard form before you multiply or divide.
b 0.00005
c 0.000032
d 0.0000000564
2 Write each o the ollowing as an ordinary number. a
b
c
d
e
3 Simpliy each o the ollowing, leaving your answer in standard orm. a
b
c
d
e
f
g
h
4 Simpliy each o the ollowing, leaving your answer in standard orm. Remember that you can write these as ordinary numbers before adding or subtracting.
a
b
c
d
Applying your skills 5 Find the number o seconds in a day, giving your answer in standard orm. 6 Te speed o light is approxi approximately mately metres per second. How ar will light travel in: a 10 seconds
b 20 seconds
c 102 seconds
Unit 2: Number
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Cambridge IGCSE Mathematics
7 Data storage (in computers) is measured in gigabytes. One gigabyte is 230 bytes. a Write 230 in standard orm correct to 1 significant figure. b Tere are 1024 gigabytes gigabytes in a terabyte. terabyte. How many many bytes is is this? Give your your answer in standard orm correct to one significant figure.
5.5 Standard form is also called scientific notation or exponential notation.
Your calculator and standard form On modern scientific calculators you can enter calculations in standard orm. Your calculator will also display numbers with too many digits or screen display in standard orm.
Keying in standard form calculations You will need to use the × 10 x button or the Exp or EE button on your calculator. Tese are known as the exponent keys. All Al l exponent keys work in the same way, so you can ollow the example below on your own calculator using whatever key you have and you will get the same result. When you use the exponent unction key o your calculator, you do NO enter the ‘× 10’ part o the calculation. Te calculator does that t hat part automatically as part o the unction.
Worked example 20 Using your calculator, calculate: a
Different calculators work in different ways and you need to understand how your own calculator works. Make sure you know what buttons to use to enter standard form calculations and how to interpret the display andinto convert your calculator answer decimal form.
2.134 × 104
b
3.124 × 10–6
a
2.134 × 104 340 = 21 340
Press: 2 . 1 3 4 × 10 x 4 This is the answer you will get.
b
3.124 × 10–6 = 0.000003123
Press: 3 . 1 2 3 Exp − 6 This is the answer you will get.
=
=
Making sense of the calculator calculator display Depending on your calculator, answers in scientific notation will be displayed on a line with an exponent like this:
Tis is 5.98 × 10–06
or on two lines with the calculation calc ulation and the answer, answer, like this:
Tis is 2.56 × 1024
I you are asked to give your answer in standard orm, all you need to do is interpret the display and write the answer correctly. I you are asked to give your answer as an ordinary number (decimal), then you need to apply the rules you already know to write wr ite the answer correctly. correctly.
Exercise 5.14
1 Enter each o these numbers into your calculator using the correct unction key and write down what appears on your calculator display display.. a 4.2 × 1012 –2
d 1.34 × 10 –4 g 3.102 × 10
b 1.8 × 10–5 –9
e 1.87 × 10 9 h 3.098 × 10
c 2.7 × 106 7
f 4.23 × 10 –23 i 2.076 × 10
118 11 8
Unit 2: Number
Original material © Cambridge University Press 2018
5 Fractions and and standard form
2 Here are ten calculator displays giving answers in standard orm. i
ii
iii
v
vi
vii
ix
x
iv
viii
a Write out each answer in standard orm. b Arrange the ten numbers numbers in order rom smallest to largest.
3 Use your calculator. calculator. Give the answers in standard orm correct to 5 significant figures. a 42345 d 123 000 000 ÷ 0.00076
b 0.0008 ÷ 92003 e (97 × 876)4
g
h
c f
(1.009)5 (0.0098)4 × (0.0032)3
4 Use your calculator to find the answers correct to 4 significant figures.
5.6 REWIND
For this section you will need to remember how to round an answer to a specified number of significant figures. You covered this in chapter 1.
a 9.27 × (2.8 × 105)
b (4.23 × 10–2)3
d (3.2 × 10–4)2
e
g
h
231 × (1.5 × 10–6)
c
(3.2 × 107) ÷ (7.2 × 109)
f
(4.3 × 105) + (2.3 × 107)
i
Estimation It is important that you know whether or not an answer that you have obtained is at least roughly as you expected. Tis section demonstrates how you can produce an approxim approximate ate answer to a calculation easily e asily.. o estimate, the numbers you are using need to be rounded beore beore you you do the calculation. Although you can use any accuracy, usually the numbers in the calculation are rounded to one significant figure: 3.9 × 2.1 ≈ 4 × 2 = 8 Notice that 3.9 × 2.1 = 8.19, so the estimated value o 8 is not too ar rom the real value!
Worked example 21 Estimate the value of: a
Tip Note that the ‘≈’ symbol is only used used at the point p oint where an approximation is made. At other times you should use ‘=’ when two numbers are exactly equal.
b
a
Round the numbers to 1 significant figure.
Check the estimate:
(3sf)
Now if you use a calculator you will find the exact value and see that the estimate was good.
Unit 2: Number
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119 11 9
Cambridge IGCSE Mathematics
In this question you begin by rounding each value to one significant significant figure but it is worth noting that you can only easily take the square root of a square number! Round 35 up to 36 to get a square number.
b
A good starting point or the questions in the ollowing exercise will be to round the numbers to 1 significant figure. Remember that you can sometimes make your calculation even simpler by modiying your numbers again.
Exercise 5.15
1
Estimate the value o each o the ollowing. Show the rounded values that you use. a
b
c
d
e
f
g
h
i
(0.45 + 1.89)(6.5 – 1.9)
3 ×
j 2
l
k
(4.1)
4
(1.9)
Work out the actual answer or each part o question 1, using a calculator calculator..
Summary Do you know the following?
Are you able to. . . ?
•
find a raction o a number
•
Fractions can be added or subtracted, but you must make sure that you have a common denominator denominator first.
• • • •
• •
o multiply two ractions you multiply their numerators and multiply their denominato denominators. rs.
•
o divide by a raction you find its reciprocal and then multiply.
find a value beore a percentage change do calculations with numbers written in standard orm
•
find an estimate to a calculation.
• •
Percentages Percentag es are ractions with a denominato denominatorr o 100.
•
You can use reverse percentages to find the original value.
•
Standard orm can be used to write very large or very small numbers quickly.
•
Estimations can be made by rounding the numbers in a calculation to one significant figure.
An equivalent raction can be b e ound by multiplying or dividing the numerator and denominator by the same number.
Percentage increases and decreases are always Percentage percentages o the original value.
E
find a percentage o a number find one number as a percentage o another number calculate a percentage increase or decrease
E
120 12 0
Unit 2: Number
Original material © Cambridge University Press 2018
Examination practice Exam-style questions 1
Calculate
giving your answer as a raction in its lowest terms.
2
93 800 students took an examination. ex amination. 19% received grade A. 24% received grade B. 31% received grade C. 10% received grade D. 11% received grade E. Te rest received grade U.
a b c
3
During one summer there were 27 500 cases o Salmonella Salmonella poisoning poisoning in Britain. Te next summer there was an increase o 9% in the number o cases. Calculate how many cases there were in the second se cond year.
4
Abdul’ s height was 160 cm on his 15th birthday birthday.. It was 172 cm on his 16th birthday birthday.. What was the percentage increase in his height?
What percentage o the students received grade U? What raction o the students received grade B? Give your answer in its lowest terms. How many students received received grade A?
Past paper questions 1
Write 0.0000574 in standard orm.
[1] [Cambridge IGCSE Mathematics 0580 Paper 22 Q1 May/June 2016]
2
Do not use a calculator in this question question and show all the steps of your working. Give each answer as a raction in its lowest terms. Work out a
[2]
b
[2] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 11 Q21 October/November 2013]
3
Calculate 17.5% o 44 kg.
[2] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 11 Q10 October/November 2013]
4
Without using your calc calculator, ulator, work out
Give your answer as a raction ract ion in its lowest terms. You must show all your working.
[3] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 13 Q17 October/November 2012]
5
Samantha invests invests $600 at a rate o 2% per year simple interest. Calculate the interest Samantha earns in 8 years.
[2] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 13 Q5 October/November 2012]
Original material © Cambridge University Press 2018
Unit 2: Number
121 12 1
6
Show that Write down all the steps in your working.
[2] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 13 Q6 October/November 2012]
7
Maria pays $84 rent.
Te rent is increase increased d by 5%. Calcu Calculate late Maria’s new rent.
[2] [Cambridge IGCSE Mathematics Mathematics 0580 Paper 13 Q10 October/November 2012]
8
Huy borrowed $4500 rom a bank at a rate o 5% per year compound interest . He paid back the money and interest at the end o 2 years. How much interest did he pay?
[3]
[Cambridge IGCSE Mathematics Mathematics 0580 Paper 13 Q13 May/June 2013]
9
Jasijeet and her brother collect stamps.
When Jasjeet gives her brother 1% o her stamps, she has 2475 stamps lef.
Calculate how many stamps Jasjeet had originally
[3] [Cambridge IGCSE Mathematics 0580 Paper 22 Q14 October/November 2014]
10 Withou Withoutt using a calculator, work out 2 5 × 3 . 8 7 Show all your working and give your answer as a mixed number in its lowest terms.
[3]
[Cambridge IGCSE Mathema Mathematics tics 0580 Paper 22 Q14 May/J May/June une 2016]
122
Unit 2: Number
Original material © Cambridge University Press 2018
Original material © Cambridge University Press 2018
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