Edexcel International GCSE Maths: Student Book

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CONTENTS 6.4 6.5

Number Chapter 1: Number 1.1 Multiples of whole numbers 1.2 Factors of whole numbers 1.3 Prime numbers 1.4 Square numbers and cube numbers 1.5 Products of prime numbers 1.6 HCF and LCM

6 8 9 10 11 14 15

Chapter 2: Fractions and percentages 2.1 Equivalent fractions 2.2 Fractions and decimals 2.3 Recurring decimals 2.4 Percentages, fractions and decimals 2.5 Calculating a percentage 2.6 Increasing or decreasing quantities by a percentage 2.7 Expressing one quantity as a percentage of another 2.8 Reverse percentage 2.9 Interest and depreciation 2.10 Compound interest problems 2.11 Repeated percentage change

18

Chapter 3: The four rules 3.1 Order of operations 3.2 Choosing the correct operation 3.3 Finding a fraction of a quantity 3.4 Adding and subtracting fractions 3.5 Multiplying and dividing fractions

46

Chapter 4: Directed numbers 4.1 Introduction to directed numbers 4.2 Everyday use of directed numbers 4.3 The number line 4.4 Adding and subtracting directed numbers 4.5 Multiplying and dividing directed numbers

62

Chapter 5: Squares, cubes and roots 5.1 Squares and square roots 5.2 Cubes and cube roots 5.3 Surds

74

Chapter 6: Set language and notation 6.1 Inequalities 6.2 Sets 6.3 Venn diagrams

84

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20 22 24 25 28 31 35 37 40 42 43

48 50 51 54 58

64 65 67 68 72

76 78 79

86 88 90

More notation Practical problems

93 94

Chapter 7: Ratio, proportion and speed 7.1 Ratio 7.2 Speed 7.3 Density and pressure 7.4 Direct proportion 7.5 Proportional variables Chapter 8: Approximation and limits of accuracy 8.1 Rounding whole numbers 8.2 Rounding decimals 8.3 Rounding to significant figures 8.4 Approximation of calculations 8.5 Upper and lower bounds 8.6 Upper and lower bounds for calculations Chapter 9: Standard form 9.1 Standard form 9.2 Calculating with standard form 9.3 Solving problems

96 98 104 108 111 112

114 116 118 119 121 122 125 128 130 132 134

Chapter 10: Applying number and using calculators 10.1 Units of measurement 10.2 Converting between metric units 10.3 Reading scales 10.4 Time 10.5 Currency conversions 10.6 Using a calculator efficiently Examination questions on Number

136 138 139 141 142 145 147 148

Algebra Chapter 11: Algebra and formulae 11.1 The language of algebra 11.2 Substitution into formulae 11.3 Rearranging formulae 11.4 More complicated formulae

156

Chapter 12: Algebraic manipulation 12.1 Simplifying expressions 12.2 Expanding brackets 12.3 Factorisation 12.4 Expanding two brackets 12.5 Multiplying more complex expresions 12.6 Quadratic factorisation 12.7 Factorising ax 2 + bx + c

168

158 161 164 166

170 174 177 179 181 185 187

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12.8 12.9

More than two brackets Algebraic fractions

Chapter 13: Solutions of equations 13.1 Solving linear equations 13.2 Setting up equations 13.3 More complex equations 13.4 Solving quadratic equations by factorisation 13.5 More factorisation in quadratic equations 13.6 Solving quadratic equations by completing the square 13.7 Solving quadratic equations by the quadratic formula 13.8 Simple simultaneous equations 13.9 More complex simultaneous equations 13.10 Linear and non-linear simultaneous equations

188 190 194 196 202 205 206 208 210 213 214 220 222

Chapter 14: Graphs in practical situations 224 14.1 Conversion graphs 226 14.2 Travel graphs 230 14.3 Speed–time graphs 234 Chapter 15: Straight line graphs 15.1 Using coordinates 15.2 Drawing straight line graphs 15.3 More straight line graphs 15.4 The equation y = mx + c 15.5 Finding equations 15.6 Parallel and perpendicular lines 15.7 Graphs and simultaneous equations

238

Chapter 16: Graphs of functions 16.1 Quadratic graphs 16.2 Solving equations with quadratic graphs 16.3 Other graphs 16.4 Estimating gradients 16.5 Graphs of sin x, cos x and tan x 16.6 Transformations of graphs

258

Chapter 17: Integer sequences 17.1 Number sequences 17.2 The nth term of a sequence 17.3 Finding the nth term of an arithmetic sequence 17.4 The sum of an arithmetic sequence

278

Chapter 18: Indices 18.1 Using indices 18.2 Negative indices

288

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240 242

18.3 18.4

Multiplying and dividing with indices Fractional indices

294 295

Chapter 19: Direct and inverse proportion 300 19.1 Direct proportion 302 19.2 Inverse proportion 307 Chapter 20: Inequalities and regions 20.1 Linear inequalities 20.2 Quadratic inequalities 20.3 Graphical inequalities 20.4 More than one inequality 20.5 More complex inequalities

310

Chapter 21: Functions 21.1 Function notation 21.2 Domain and range 21.3 Inverse functions 21.4 Composite functions 21.5 More about composite functions

326

Chapter 22: Calculus 22.1 The gradient of a curve 22.2 More complex curves 22.3 Turning points 22.4 Motion of a particle

336

Examination questions on Algebra

312 318 319 322 324

328 329 330 332 334

338 341 344 347 352

244 248 251 252 256

260 263 265 268 270 272

280 282 285 286

290 292

Geometry and trigonometry Chapter 23: Angle properties 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 chords 23.8 Setting up equations 23.9 Cyclic quadrilaterals 23.10 Alternate segment theorem 23.11 Intersecting chords Chapter 24: Geometrical terms and relationships 24.1 Measuring and drawing angles 24.2 Bearings 24.3 Congruent shapes 24.4 Similar shapes 24.5 Areas of similar triangles 24.6 Areas and volumes of similar shapes

360 362 365 368 371 374 376 379 381 385 388 391

394 396 398 402 404 407 409

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Chapter 25: Geometrical constructions 25.1 Constructing shapes 25.2 Bisectors 25.3 Scale drawings

414 420 422

Chapter 26: Trigonometry 426 26.1 Pythagoras’ theorem 428 26.2 Trigonometric ratios 432 26.3 Calculating angles 434 26.4 Using sine, cosine and 435 tangent functions 26.5 Which ratio to use 440 26.6 Solving problems using trigonometry 443 26.7 Angles of elevation and depression 446 26.8 Problems in three dimensions 448 26.9 Sine, cosine and tangent of 450 obtuse angles 26.10 The sine rule and the cosine rule 452 26.11 Using sine to find the area of a triangle 460 Chapter 27: Mensuration 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 of a prism 27.8 Volume and surface area of a cylinder 27.9 Arcs and sectors 27.10 Volume and surface area of a cone 27.11 Volume and surface area of a sphere

462

Chapter 28: Symmetry 28.1 Lines of symmetry 28.2 Rotational symmetry 28.3 Symmetry of special two-dimensional shapes

490

Chapter 29: Vectors 29.1 Introduction to vectors 29.2 Using vectors 29.3 The magnitude of a vector

498

Chapter 30: Transformations 30.1 Translations 30.2 Reflections 30.3 Further reflections 30.4 Rotations 30.5 Further rotations 30.6 Enlargements Examination questions on Geometry and Trigonometry

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Statistics and probability

416

464 467 470 471 474 477 480 482 484 486 489

492 494

Chapter 31: Statistical representation 31.1 Frequency tables 31.2 Pictograms 31.3 Bar charts 31.4 Pie charts 31.5 Histograms

536

Chapter 32: Statistical measures 32.1 The mode 32.2 The median 32.3 The mean 32.4 The range 32.5 Which average to use 32.6 Using frequency tables 32.7 Grouped data 32.8 Measuring spread 32.9 Cumulative frequency diagrams

558

Chapter 33: Probability 33.1 The probability scale 33.2 Calculating probabilities 33.3 Probability that an event will not happen 33.4 Addition rule for probabilities 33.5 Probability from data 33.6 Expected frequency 33.7 Combined events 33.8 Tree diagrams

588

Examination questions on Statistics and probability Answers Index

538 541 543 547 551

560 562 565 567 570 572 576 579 582

590 592 595 596 599 602 604 608

616 624 675

495

500 503 508 510 512 514 516 518 521 522

528

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INTRODUCTION Welcome to Collins International GCSE Maths for Edexcel. This page will introduce you to the key features of the book which will help you to succeed in your examinations and to enjoy your maths course.

Why this chapter matters This page is at the start of each chapter.

It tells you why the mathematics in the chapter is important and how it is useful.

Chapter overviews The overview at the start of each chapter shows what you will be studying, the key words you need to know and what you will be expected to know and do in the examination.

Worked examples Worked examples take you through questions step by step and help you understand the topic before you start the practice questions.

Practice questions and answers Every chapter has extensive questions to help you practise the skills you need for the examination. Many of the questions require you to solve problems which is an important part of mathematics.

Colour-coded levels The colour coded panels at the side of the question pages show whether the questions are at Foundation ((blue) or Higher level (yellow). The on some topic headings shows that the content in that topic is at Higher level only.

Exam practice Each of the four main sections in the book ends with sample exam questions from past examinations. These will show you the types of questions you will meet in the exams. Mark schemes are available in the teacher pack.

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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 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?

Note: ≈ means ‘approximately equal to’.

52 = 5 50 2 =

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

500 2 =

×5=

0 = 25

500 ×

4096 = (4 + 0 ) 81 = (8 + 1)2

Some number patterns have special names.

Can you pair up these patterns and their 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

= 100 10 × 10 00 10 = 10 × 0 1 10 × 10 000 × 10 = 0 1 × 10 × 10

10, 5, 2, 1

32, 16, 8, 4, 2, 1

0

42 = 16 3342 = 111 556

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 1.2 ‘Factors of whole numbers’ will help you to work this out!)

25, 5, 1

250 00

342 = 1156

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

18, 9, 6, 3, 2, 1

00

500 =

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

96

25

50 × 5

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

1×1= 2×2= 3×3= 4×4=

1

1+3

1+3+

1+3+

5

5+7

6

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Chapter p

Number Topics

Level

1 Multiples of whole

Key words

FOUNDATION

multiple, common multiple, even, odd

FOUNDATION

factor, factor pair, common factor

3 Prime numbers

FOUNDATION

prime number

4 Square numbers and

FOUNDATION

square, square number, cube, cube number

FOUNDATION

product

FOUNDATION

highest common factor, lowest common multiple

numbers

2 Factors of whole numbers

cube numbers

5 Products of prime numbers

6 HCF and LCM

What you need to be able to do in the examinations: FOUNDATION ● ● ● ●

Use the terms odd, even and prime numbers, factors and multiples.

Identify prime factors, common factors and common multiples.

Express integers as the product of powers of prime factors.

Find Highest Common Factors (HCF) and Lowest Common Multiples (LCM).

7

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1.1

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. Multiples of 2 are even 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 …

35 is a common multiple of 5 and 7. Other common multiples of 5 and 7 are 70, 105, 140 and so on.

FOUNDATION

EXERCISE 1A 1

Write out the first five multiples of: a

3

b

7

c

9

d

11

e

16

Remember: the first multiple is the number itself.

2

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

3

multiples of 4

b

multiples of 7

c

multiples of 6.

72

135

102

161

197

132

78

91

216

514

Odd numbers cannot be multiples of even numbers. Whole numbers are either even or odd.

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

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.

How many numbers between 1 and 100 inclusive are multiples of both 6 and 9?

List the numbers.

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1.2

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

The common factors of 12 and 20 are 1, 2 and 4. They are factors of both numbers.

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.

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 it is counted only once.

So, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

EXERCISE 1B

2

What are the factors of each of these numbers? a

10

b

28

c

18

d

17

e

25

f

40

g

30

h

45

i

24

j

16

d

117

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

110

b

201

e

130

f

240

c

145

FOUNDATION

1

9

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FOUNDATION

CHAPTER 1: Number

3

4

1.3

Find the common factors of each of the following pairs 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

55 and 77

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.

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.

So, a prime number is a whole number that has only two factors: itself and 1.

Note: 1 is not a prime number, since it has only one factor – itself.

The prime numbers up to 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

FOUNDATION

EXERCISE 1C 1

Write down the prime numbers between 20 and 30.

2

Write down the only prime number between 90 and 100.

3

Decide which of these numbers are not prime numbers.

462

4

108

848

365

711

When three different prime numbers are multiplied together the answer is 105.

What are the three prime numbers?

5

A shopkeeper has 31 identical soap bars.

He is trying to arrange the bars on a shelf in rows, each with the same number of bars.

Is it possible?

Explain your answer.

10

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Square numbers and cube numbers

1.4

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, …

If dots are arranged in three dimensional cubes we get cube numbers.

1×1×1=1

2×2×2=8

3 × 3 × 3 = 27

4 × 4 × 4 = 64

We can write these as 13, 23, 33, 43, … and we read them as ‘one cubed’, ‘two cubed’ and so on.

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CHAPTER 1: Number

FOUNDATION

EXERCISE 1D 1

The square number pattern starts: 1

4

9

16

25

…

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

…

…

…

a

b 5

Work out each of the following. You may use your calculator.

32 + 42

and

52

52 + 122

and

132

72 + 242

and

252

92 + 402

and

412

Describe what you notice about your answers to part a.

Find: a

53

b

63

c

103

6

Show that 1331 is a cube number.

7

Which is larger, 103 or 302? Find the difference between them.

8

a

Show that (1 + 2 + 3)2 = 13 + 23 + 33.

b

Is it true that (1 + 2 + 3 + 4)2 = 13 + 23 + 33 + 43?

9 10

Look for the connection with the square numbers on the top line.

How many cube numbers are there between 2000 and 4000?

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

Find two different square numbers with a sum of 85.

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CHAPTER 1: Number

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

EXERCISE 1E Write out the first three numbers that are multiples of both of the numbers shown. a 2

3 and 4

b

4 and 5

c

3 and 5

d

6 and 9

e

FOUNDATION

1

5 and 7

Here are four numbers.

14

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

One dog barks every 8 seconds and another dog barks every 12 seconds. If both dogs bark together, how many seconds will it be before they both bark 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.

7

a

a multiple of 3 and a multiple of 4

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

a number with 5 factors exactly

g

a multiple of 5 and a factor of 20

h

a prime number that is one more than a square number

i

a cube number

j

a number which is a quarter of a cube number

13

12

21

8

15 17

9

18 10

14

6 16

Arrange these four cards to make a cube number.

1

2

7

9 13

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1.5

Products of prime numbers

Every positive integer can be written as a product of prime numbers.

For example, 5472 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 19.

We write this more concisely as 5472 = 25 × 32 × 19.

25 means 2 × 2 × 2 × 2 × 2 and 32 means 3 × 3.

EXAMPLE 2

Write 702 as a product of prime numbers.

Keep dividing by prime numbers, starting with the lowest.

702 is even so it is divisible by 2: 2)702 351 351 is not divisible by 2 but it is divisible by 3.

3)351 117 Divide by 3 again: 3)117 39

Remember the prime numbers are 2, 3, 5, 7, 11, 13, 17 …

Divide by 3 again: 3) 39 13 13 is prime so we stop there.

These calculations can be written more concisely like this: 2)702 3)351 3)117 3) 39 13 Now just write down all the prime numbers shown: 702 = 2 × 3 × 3 × 3 × 13 (Check that this is correct with a calculator)

FOUNDATION

We can write this more concisely as 702 = 2 × 33 × 13.

EXERCISE 1F 1

Calculate the following: a

24 × 3

b

33 × 72

c

25 × 55

d

35 × 5

e

24 × 54

f

210 × 34

24 × 3 = 2 × 2 × 2 × 2 × 3

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CHAPTER 1: Number

3

Write each of these as a product of prime numbers: a

72

b

100

c

252

d

560

e

285

f

729

g

444

h

896

i

675

j

1323

a

Choose any 3 digit number.

Multiply it by 7, multiply the answer by 11 and then multiply that answer by 13.

FOUNDATION

2

What happens? b

1.6

Does what happened in part a happen with any three digit number? Why?

HCF and LCM

A common factor is a factor common to two or more numbers.

The numbers 60 and 72 have a number of common factors, including 2 and 3: 60 = 22 × 3 × 5 72 = 23 × 32 The prime factors common to both are 22 and 3.

2 × 2 × 3 ×5 2× 2 × 2 × 3 ×3 Multiply these together to find the highest common factor (HCF).

The HCF of 60 and 72 is 22 × 3 = 12 This is the highest number that is a factor of 60 and 72.

Multiples of 60 are 60, 120, 180, …

Multiples of 72 are 72, 144, 216, …

They will have a number of common multiples.

We can use prime factors to find the lowest common multiple (LCM): 60 = 22 × 3 × 5 72 = 23 × 32

Choose the highest power of each number in either list, e.g. 23 not 22

Any common multiple must contain all the factors of both numbers.

It must contain 23 and 32 and 5.

The LCM of 60 and 72 = 23 × 32 × 5 = 360

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CHAPTER 1: Number

FOUNDATION

EXERCISE 1G 1

2

3

4

5

a

Show that 2 is a common factor of 10 and of 20.

b

Is it the highest common factor?

Find the highest common factor (HCF) of each of these pairs of numbers. You should be able to spot these without writing out a list of prime factors. a

8 and 12

b

9 and 12

c

4 and 20

d

15 and 24

e

20 and 50

f

100 and 150

Find the highest common factor (HCF) of each of these pairs of numbers. a

24 and 30

b

36 and 48

c

72 and 96

d

60 and 84

e

108 and 63

f

66 and 78

g

84 and 140

h

165 and 385

a

Show that 60 is a common multiple of 2 and of 3.

b

Is it the lowest common multiple?

Write each number as a product of prime factors.

Find the lowest common multiple (LCM) of these pairs of numbers. a

2 and 5

b

2 and 7

c

3 and 5

d

3 and 7

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CHAPTER 1: Number

Write each of these pairs of numbers as a product of prime factors. Hence find the LCM. a

12 and 15

b

16 and 24

c

12 and 14

d

25 and 40

e

18 and 21

f

60 and 80

g

32 and 48

h

70 and 55

FOUNDATION

6

17

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