Cambridge IGCSE®

Maths STUDENT’S BOOK Also for Cambridge IGCSE® (9–1)

Chris Pearce

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CONTENTS How to use this book

Key E = Extended

5

Number Chapter 1: Number 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Square numbers and cube numbers Multiples of whole numbers Factors of whole numbers Prime numbers Prime factorisation More about HCF and LCM Real numbers

Chapter 2: Fractions and percentages 2.1 2.2 E 2.3 2.4 2.5 2.6

Equivalent fractions Fractions and decimals Recurring decimals Percentages, fractions and decimals Calculating a percentage Increasing or decreasing quantities by a percentage 2.7 One quantity as a percentage of another 2.8 Simple interest and compound interest 2.9 A formula for compound interest E 2.10 Reverse percentage

Chapter 3: The four rules 3.1 3.2 3.3 3.4 3.5

Order of operations Choosing the correct operation Finding a fraction of a quantity Adding and subtracting fractions Multiplying and dividing fractions

Chapter 4: Directed numbers 4.1 4.2 4.3 4.4 4.5

Introduction to directed numbers Everyday use of directed numbers The number line Adding and subtracting directed numbers Multiplying and dividing directed numbers

Chapter 5: Powers and roots 5.1 5.2 5.3 E 5.4

Squares and square roots Cubes and cube roots More powers and roots Exponential growth and decay

6 8 12 13 15 16 18 20 22 24 26 27 30 34 36 40 43 45 47 50 52 54 55 57 60 64 66 67 68 70 73 76 78 79 81 82

Chapter 6: Ordering and set notation 6.1 6.2 E 6.3

Inequalities Sets and Venn diagrams More about Venn diagrams

86 88 90 94

Chapter 7: Ratio, proportion and rate 100 7.1 E 7.2 7.3 7.4 7.5 7.6

Ratio Increases and decreases using ratios Speed Rates Direct proportion Inverse proportion

Chapter 8: Estimation and limits of accuracy 8.1 8.2 8.3 8.4 E 8.5

Rounding whole numbers Rounding decimals Rounding to significant figures Upper and lower bounds Upper and lower bounds for calculations

Chapter 9: Standard form 9.1 9.2

Standard form Calculating with standard form

102 108 110 113 116 117

120 122 123 125 126 128

132 134 136

Chapter 10: Applying number and using calculators

140

10.1 Units of measurement 10.2 Converting between metric units 10.3 Time 10.4 Currency conversions 10.5 Using a calculator efficiently Examination questions: Number

142 143 145 147 149 151

Algebra Chapter 11: Algebraic representation and formulae

160

11.1 11.2 11.3 E 11.4

162 165 167 169

The language of algebra Substitution into formulae Rearranging formulae More complicated formulae

Chapter 12: Algebraic manipulation 12.1 Simplifying expressions 12.2 Expanding brackets

172 174 178

2

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12.3 12.4 12.5 E 12.6 E 12.7 E 12.8

Factorisation Multiplying two brackets: 1 Multiplying two brackets: 2 Expanding three brackets Quadratic factorisation Algebraic fractions

Chapter 13: Solutions of equations and inequalities 13.1 Solving linear equations 13.2 Setting up equations E 13.3 Solving quadratic equations by factorisation E 13.4 Solving quadratic equations by the quadratic formula E 13.5 Solving quadratic equations by completing the square 13.6 Simultaneous equations E 13.7 Linear and non-linear simultaneous equations E 13.8 Solving inequalities

182 184 187 190 192 197

332

202

334 339

204 210

Chapter 20: Linear programming

342

212

E 20.1 Graphical inequalities E 20.2 More than one inequality E 20.3 Linear programming

344 347 349

217

Chapter 21: Functions

352

E E E E

354 355 357 359

219 222 229 232

236

Conversion graphs Travel graphs Speed–time graphs Curved graphs

238 242 246 251

Chapter 15: Straight-line graphs

256

15.1 15.2 E 15.3 15.4 15.5 E 15.6 E 15.7

258 261 265 267 270 272 274

E E E E

16.1 16.2 16.3 16.4 16.5 16.6

Quadratic graphs Turning points on a quadratic graph Reciprocal graphs More graphs Exponential graphs Estimating gradients

278 280 285 286 288 292 296

Chapter 17: Number sequences

300

17.1 17.2 17.3 E 17.4

302 304 309 314

Patterns in number sequences The nth term of a sequence General rules from patterns Further sequences

320 322 324 327

Using indices Negative indices Multiplying and dividing with indices Fractional indices

E 19.1 Direct proportion E 19.2 Inverse proportion

14.1 14.2 E 14.3 E 14.4

Chapter 16: Graphs of functions

318

18.1 18.2 18.3 E 18.4

Chapter 19: Proportion

Chapter 14: Graphs in practical situations

Drawing straight-line graphs The equation y = mx + c More about straight-line graphs Solving equations graphically Parallel lines Points and lines Perpendicular lines

Chapter 18: Indices

21.1 21.2 21.3 21.4

Function notation Inverse functions Composite functions More about composite functions

Chapter 22: Differentiation

362

E 22.1 The gradient of a curve E 22.2 More complex curves E 22.3 Turning points Examination questions: Algebra

364 366 369 372

Geometry and trigonometry Chapter 23: Angle properties

382

23.1 Angle facts 23.2 Parallel lines 23.3 Angles in a triangle 23.4 Angles in a quadrilateral 23.5 Regular polygons 23.6 Irregular polygons 23.7 Tangents and diameters 23.8 Angles in a circle 23.9 Cyclic quadrilaterals 23.10 Alternate segment theorem

384 386 390 392 395 398 400 402 405 408

Chapter 24: Geometrical terms and relationships

412

24.1 24.2 24.3 24.4 E 24.5 24.6 E 24.7 E 24.8

414 417 420 423 424 427 430 433

E E E E

Measuring and drawing angles Bearings Nets Congruent shapes Congruent triangles Similar shapes Areas of similar triangles Areas and volumes of similar shapes

3

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Chapter 25: Geometrical constructions 25.1 Constructing shapes 25.2 Scale drawings

Chapter 26: Trigonometry 26.1 26.2 26.3 26.4

E E E E E E

Pythagoras’ theorem Trigonometric ratios Calculating angles Using sine, cosine and tangent functions 26.5 Which ratio to use 26.6 Applications of trigonometric ratios 26.7 Problems in three dimensions 26.8 Sine and cosine of obtuse angles 26.9 The sine rule and the cosine rule 26.10 Using sine to find the area of a triangle 26.11 Sine, cosine and tangent of any angle

438 440 442

446 448 452 454 455 459 462 466 468 470 477 479

Chapter 27: Mensuration

486

27.1 Perimeter and area of a rectangle 27.2 Area of a triangle 27.3 Area of a parallelogram 27.4 Area of a trapezium 27.5 Circumference and area of a circle 27.6 Surface area and volume of a cuboid 27.7 Volume and surface area of a prism 27.8 Volume and surface area of a cylinder 27.9 Sectors and arcs: 1 E 27.10 Sectors and arcs: 2 27.11 Volume of a pyramid 27.12 Volume and surface area of a cone 27.13 Volume and surface area of a sphere

488 491 494 495 498 501 503 506 508 510 512 514 516

Chapter 28: Symmetry

518

28.1 Lines of symmetry 520 28.2 Rotational symmetry 522 28.3 Symmetry of special two-dimensional shapes 523 E 28.4 Symmetry of three-dimensional shapes 525 E 28.5 Symmetry in circles 526

Chapter 29: Vectors

530

29.1 Introduction to vectors E 29.2 Using vectors E 29.3 The magnitude of a vector

532 535 540

Chapter 30: Transformations

542

30.1 Translations 30.2 Reflections: 1

E 30.3 Reflections: 2 30.4 Rotations: 1 E 30.5 Rotations: 2 30.6 Enlargements: 1 E 30.7 Enlargements: 2 E 30.8 Combined transformations Examination questions: Geometry

548 550 553 554 559 561 564

Statistics and probability Chapter 31: Statistical representation

576

31.1 31.2 31.3 31.4 31.5 31.6 E 31.7

578 581 583 587 591 596

Frequency tables Pictograms Bar charts Pie charts Scatter diagrams Histograms Histograms with bars of unequal width

599

Chapter 32: Statistical measures

606

32.1 The mode 32.2 The median 32.3 The mean 32.4 The range 32.5 Which average to use 32.6 Stem-and-leaf diagrams 32.7 Using frequency tables E 32.8 Grouped data E 32.9 Cumulative frequency diagrams E 32.10 Box-and-whisker plots

608 610 612 615 618 620 624 628 631 638

Chapter 33: Probability

642

33.1 The probability scale 33.2 Calculating probabilities 33.3 Probability that an event will not happen 33.4 Probability in practice 33.5 Using Venn diagrams 33.6 Possibility diagrams 33.7 Tree diagrams E 33.8 Conditional probability Examination questions: Statistics and probability

644 646

Examination questions: Mixed type

686

Glossary Answers Index

692 704 766

649 651 654 657 661 665 672

544 546

4

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1 Number

C hapt er

Topics

Level

Key words

1 Square numbers and cube numbers

CORE

square, square number, square root, cube, cube number

2 Multiples of whole numbers

CORE

multiple

3 Factors of whole numbers

CORE

factor, factor pair, lowest common multiple, highest common factor

4 Prime numbers

CORE

prime number

5 Prime factorisation

CORE

product of prime factors, index (indices), prime factorisation

6 More about HCF and LCM

CORE

natural number, integer, real number, rational number, irrational number

7 Real numbers

CORE

natural number, integer, real number, rational number, irrational number, reciprocal

In this chapter you will learn how to: CORE ●

●

Identify and use: –

natural numbers

–

integers (positive, negative and zero)

–

prime numbers

–

square numbers

–

cube numbers

–

common factors and common multiples

–

rational and irrational numbers (e.g. π, 2)

–

real numbers

–

reciprocals

–

Express any number as a product of its prime factors

–

Find the lowest common multiple (LCM) and highest common factor (HCF) of two numbers. (C1.1 and E1.1)

Reason, interpret and communicate mathematically when solving problems.

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Why this chapter matters A pattern is an arrangement of repeated parts. You see patterns every day in clothes, art and home furnishings. Patterns can also occur in numbers. There are many mathematical problems that can be solved using patterns in numbers. Some numbers have fascinating features. Here is a pattern. 3 + 5 = 8 (5 miles ≈ 8 km) 5 + 8 = 13 (8 miles ≈ 13 km) 8 + 13 = 21 (13 miles ≈ 21 km) Approximately how many kilometres are there in 21 miles?

52 = 5

Note: ≈ means ‘approximately equal to’.

50 2 =

In the boxes are some more patterns. Can you work out the next line of each pattern?

50 0 2 =

Now look at these numbers and see why they are special.

×5=

50 × 5

50 0 ×

25

0 = 25

50 0 =

4096 = (4 + 09)6

00

250 0 0

0

81 = (8 + 1)2

= 10 0 10 × 10 10 0 0 × 10 = 0 1 × 10 10 0 0 0 × 10 = 0 1 × 0 10 × 1

Some patterns have special names. Can you pair up these patterns and the names? 4, 8, 12, 16, …

Prime numbers

1, 4, 9, 16, …

Multiples (of 4)

2, 3, 5, 7, …

Cube numbers

1, 8, 27, 64, …

Square numbers

You will look at these in more detail in this chapter.

1 1×1= = 121 11 × 11 321 11 = 12 111 × 1

Below are four sets of numbers. Think about which number links together all the other numbers in each set. (The mathematics that you cover in 1.3 ‘Factors of whole numbers’ will help you to work this out!)

1×1= 2×2=

10, 5, 2, 1

3×3=

18, 9, 6, 3, 2, 1

4×4=

25, 5, 1 32, 16, 8, 4, 2, 1

1 9 = 980 1089 × 901 9 = 98 × 9 8 9 10 89 901 ×9=9 9 8 9 9 10

1

1+3

1+3+

1+3+

5

5+7

Chapter 1: Number

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Chapter 1 . Topic 1

1.1 Square numbers and cube numbers What is the next number in this sequence? 1, 4, 9, 16, 25, … Write each number as: 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5, … These factors can be represented by square patterns of dots:

1×1

2×2

3×3

4×4

5×5

From these patterns, you can see that the next pair of factors must be 6 × 6 = 36, therefore 36 is the next number in the sequence. Because they form square patterns, the numbers 1, 4, 9, 16, 25, 36, … are called square numbers. When you multiply any number by itself, the answer is called the square of the number or the number squared. This is because the answer is a square number. For example: the square of 5 (or 5 squared) is 5 × 5 = 25 the square of 6 (or 6 squared) is 6 × 6 = 36 There is a short way to write the square of any number. For example: 5 squared (5 × 5) can be written as 52 13 squared (13 × 13) can be written as 132 So, the sequence of square numbers, 1, 4, 9, 16, 25, 36, … , can be written as: 12, 22, 32, 42, 52, 62, … The square root of n is the number of which the square is n. This can be written as n. For example, the square root of 16 (4) can be written as 16. Square numbers have exact square roots, for example: the square root of 9 is 3: 9 = 3 the square root of 25 is 5: 25 = 5 the square root of 100 is 10: 100 = 10

8

1.1 Square numbers and cube numbers

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1.1 EXERCISE 1A The square number pattern starts: 1

4

9

16 25

CORE

1

…

Copy and continue the pattern above until you have written down the first 20 square numbers. You may use your calculator for this. 2

Work out the answer to each of these number sentences. 1+3= 1+3+5= 1+3+5+7= Look carefully at the pattern of the three number sentences. Then write down the next three number sentences in the pattern and work them out.

3

4

Find the next three numbers in each of these number patterns. (They are all based on square numbers.) You may use your calculator. 1

4

9

16

25

36

49

64

81

a 2

5

10

17

26

37

…

…

…

b 2

8

18

32

50

72

…

…

…

c 3

6

11

18

27

38

…

…

…

d 0

3

8

15

24

35

…

…

…

Advice and Tips Look for the connection with the square numbers on the top line.

a Work out the values of both expressions in each pair. You may use your calculator. 32 + 42

and

52

52 + 122

and

132

72+ 242

and

252

92 + 402

and

412

b Describe what you notice about your answers to part a. This will help you communicate mathematically with others. 5

a 132 = 169. What is 169? b Find 25 c Find 81 d Find 121 e Find 400

6

4 and 81 are square numbers with a sum of 85. Find two different square numbers with a sum of 85.

Chapter 1: Number

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The following exercise will give you some practice on multiples, factors, square numbers and prime numbers.

CORE

EXERCISE 1B 1

Write out the first three numbers that are multiples of both of the numbers shown. a 3 and 4

2

b 4 and 5

c 3 and 5

d 6 and 9

e 5 and 7

Here are four numbers. 10

16

35

49

Copy and complete the table by putting each of the numbers in the correct box. Square number

Factor of 70

Even number Multiple of 7 3

Arrange these four number cards to make a square number.

1

4

6

7

4

An alarm flashes every 8 seconds and another alarm flashes every 12 seconds. If both alarms flash together, how many seconds will it be before they both flash together again?

5

A bell rings every 6 seconds. Another bell rings every 5 seconds. If they both ring together, how many seconds will it be before they both ring together again?

6

From this box, choose one number that fits each of these descriptions. a a multiple of 3 and a multiple of 4

13

12 8

b a square number and an odd number c a factor of 24 and a factor of 18 d a prime number and a factor of 39 e an odd factor of 30 and a multiple of 3 f

h a prime number that is one more than a squareÂ number

10

15 17

9

18 10

a number with 5 factors exactly

g a multiple of 5 and a factor of 20

21

14

6 16

1.1 Square numbers andÂ cube numbers

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1.1 Cube numbers What is the next number in this sequence? 1, 8, 27, … Write each number as: 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, … These factors can be represented by cube patterns of dots:

1×1×1 1×1

2×2×2 2×2×2

3×3×3 3×3×3

From these patterns, you can see that the next pair of factors must be 4 × 4 × 4 = 64, therefore 64 is the next number in the sequence. Because they form cubic patterns, the numbers 1, 8, 27, 64, … are called cube numbers. When you multiply any number by itself twice, the answer is called the cube of the number or the number cubed. This is because the answer is a cube number. For example: the cube of 5 (or 5 cubed) is 5 × 5 × 5 = 125. There is a short way to write the cube of any number. For example: 5 cubed (5 × 5 × 5) can be written as 53 10 cubed (10 × 10 × 10) can be written as 103 So, the sequence of cube numbers, 1, 8, 27, 64, … , can be written as: 13, 23, 33, 43, … You will learn more about cubes (and cube roots) in Chapter 5.

EXERCISE 1C The cube number pattern starts: 1

8

27

64

CORE

1

…

Copy and continue the pattern above until you have written down the first 12 cube numbers. You may use your calculator for this.

Chapter 1: Number

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Chapter 1 . Topic 2 1 CORE

2

Work out the answer to each of these number sentences. 1+8= 1 + 8 + 27 = 1 + 8 + 27 + 64 = Look carefully at the pattern of the three number sentences. What kind of numbers are these? Now write down the next three number sentences in the pattern and work them out.

3

4

Find the next three numbers in each of these number patterns. (They are all based on cube numbers.) You may use your calculator. 1

8

27

64

a 2

9

28

65

…

…

…

b 0

7

26

63

…

…

…

c 2

16

54

128

…

…

…

d 1000

729

512

343

…

…

…

a Work out the values of these expressions. 13 + 53 + 33 33 + 73 + 03 33 + 73 + 13 b Describe what you notice about your answers to part a.

5

Work out the values of these expressions: 123 + 13 and 93 + 103. Your answer is Bender’s (a character in Futurama) serial number. It is sometimes called the Hardy–Ramanujan number after the Indian mathematician Ramanujan who noticed that this is the smallest number that can be expressed as the sum of two cubes in two different ways.

6

Work out the values of these expressions: 692 and 693 on your calculator. What do you notice about the digits in your answers?

1.2 Multiples of whole numbers When you multiply any whole number by another whole number, the answer is called a multiple of either of those numbers. For example, 5 × 7 = 35, which means that 35 is a multiple of 5 and it is also a multiple of 7. Here are some other multiples of 5 and 7: multiples of 5 are:

5

10

15 20

25

30

35

multiples of 7 are:

7

14

21 28

35

42

…

12

…

1.2 Multiples of whole numbers

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1.3

Chapter 1 . Topic 3

EXERCISE 1D Write out the first five multiples of: a 3

b 7

c 9

d 11

Remember: the first multiple is the number itself. 2

72 3

135

b multiples of 7

102

161

197

132

c multiples of 6. 78

91

e 16 Advice and Tips

Use your calculator to see which of the numbers below are: a multiples of 4

CORE

1

216

514

There is no point testing odd numbers for multiples of even numbers such as 4 and 6.

Find the biggest number that is smaller than 100 and that is: a a multiple of 2

b a multiple of 3

c a multiple of 4

d a multiple of 5

e a multiple of 7

f

a multiple of 6.

4

A party of 20 people are getting into taxis. Each taxi holds the same number of passengers. If all the taxis fill up, how many people could be in each taxi? Give two possible answers.

5

Here is a list of numbers. 6

8

12

15

18

28

a From the list, write down a multiple of 9. b From the list, write down a multiple of 7. c From the list, write down a multiple of both 3 and 5. 6

How many numbers between 1 and 100 are multiples of both 6 and 9? List the numbers.

1.3 Factors of whole numbers A factor of a whole number is any whole number that divides into it exactly. So: the factors of 20 are

1

2

4

5

10

20

the factors of 12 are

1

2

3

4

6

12

Factor facts Remember these facts. • 1 is always a factor and so is the number itself. • When you have found one factor, there is always another factor that goes with it – unless the factor is multiplied by itself to give the number. For example, look at the number 20: 1 × 20 = 20

so 1 and 20 are both factors of 20

2 × 10 = 20

so 2 and 10 are both factors of 20

4 × 5 = 20

so 4 and 5 are both factors of 20.

These are called factor pairs.

Chapter 1: Number

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You may need to use your calculator to find the factors of large numbers.

Example 1 Find the factors of 36. Look for the factor pairs of 36. These are: 1 × 36 = 36

2 × 18 = 36

3 × 12 = 36

4 × 9 = 36

6 × 6 = 36

6 is a repeated factor so is counted only once. So, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

EXERCISE 1E CORE

1

2

3

What are the factors of each of these numbers? a 10

b 28

c 18

d 17

e 25

f

g 30

h 45

i

j

24

16

What is the biggest factor that is less than 100 for each of these numbers? a 110

b 201

c 145

d 117

e 130

f

240

Find the largest common factor for each pair of numbers. a 2 and 4

b 6 and 10

c 9 and 12

d 15 and 25

e 9 and 15

f

12 and 21

g 14 and 21

h 25 and 30

i

30 and 50

j 4

40

55 and 77

Advice and Tips Look for the largest number that has both numbers in its multiplication table.

Find the highest odd number that is a factor of 40 and a factor of 60.

Lowest common multiple The lowest common multiple (LCM) of two numbers is the smallest number that appears in the multiplication tables of both numbers. For example, the LCM of 3 and 5 is 15, the LCM of 2 and 7 is 14 and the LCM of 6 and 9 is 18.

Example 2 Find the LCM of 18 and 24. Write out the 18 times table: 18, 36, 54, 72 , 90, 108, … . Write out the 24 times table: 24, 48, 72 , 96, 120, … You can see that 72 is the smallest (least) number in both (common) tables (multiples).

14

1.3 Factors of whole numbers

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1.4

Chapter 1 . Topic 4

Highest common factor The highest common factor (HCF) of two numbers is the biggest number that divides exactly into both of them. For example, the HCF of 24 and 18 is 6, the HCF of 45 and 36 is 9 and the HCF of 15 and 22 is 1.

Example 3 Find the HCF of 28 and 16. Write out the factors of 28: 1, 2, 4 , 7, 14, 28 Write out the factors of 16: 1, 2, 4 , 8, 16 You can see that 4 is the biggest (highest) number in both (common) lists (factors).

EXERCISE 1F

2

3

Find the LCM of each pair of numbers. a 24 and 56

b 21 and 35

c 12 and 28

d 28 and 42

e 12 and 32

f

g 15 and 25

h 16 and 36

CORE

1

18 and 27

Find the HCF of each pair of numbers. a 24 and 56

b 21 and 35

c 12 and 28

d 28 and 42

e 12 and 32

f

g 15 and 25

h 16 and 36

i

42 and 27

j

48 and 64

k 25 and 35

l

36 and 54

18 and 27

The HCF of two numbers is 6. The LCM of the same two numbers is 72. What are the numbers? Explain how you reached you answer.

1.4 Prime numbers What are the factors of 2, 3, 5, 7, 11 and 13? Notice that each of these numbers has only two factors: itself and 1. They are all examples of prime numbers.

Chapter 1: Number

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