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Cambridge 0 Level Mathematics

Michael Handbury, John Jeskins, Jean Matthews, Heather West

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We are working with Cambridge International Examinations to gain endorsement for this new, full-colour textbook matched exactly to the syllabus. - Matched exactly to the latest Cambridge O level syllabus - Includes non-calculator questions - Supports revision through a range of past paper questions

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                CAMBRIDGE O LEVEL  MATHEMATICS MICHAEL HANDBURY  JOHN JESKINS JEAN MATTHEWS  HEATHER WEST SERIES EDITOR: BRIAN SEAGER  The question and answers that appear in this book were written by the authors. In an examination, the way marks would be awarded to answers may be different. The Publishers would like to thank the following for permission to reproduce copyright material.  Photo credits p.67 © Getty Images/iStockphoto/Thinkstock/maratr Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser. Hachette UK's policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Email education@bookpoint.co.uk Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: www.hoddereducation.com. © Mike Handbury, John Jeskins, Jean Matthews, Heather West and Brian Seager 2016 First published in 2016 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ www.hoddereducation.com Impression number Year  10 9 8 7 6 5 4 3 2 1  2020 2019 2018 2017 2016  All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any fo; rm or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo © James Thew – Fotolia Illustrations by Integra Software Services Ltd. Typeset in India by Integra Software Services Ltd. Printed in Italy A catalogue record for this title is available from the British Library. ISBN: 9781471859625  Contents INTRODUCTION ................................................................................. xi 1  NUmbeR ....................................................................................................1 types of number .............................................................................................1 Prime factors ...................................................................................................2 Common factors and common multiples ....................................................4  2  SeT laNgUage aND NOTaTION .........................................6 the definition of a set ....................................................................................6 the universal set .............................................................................................6 Venn diagrams ................................................................................................7 the relationship between sets ......................................................................8 the complement of a set ...............................................................................9 subsets .......................................................................................................... 11 Problem solving with Venn diagrams ......................................................... 12 Alternative ways to define a set .................................................................. 12 three-set problems ...................................................................................... 14  3  SqUaReS, SqUaRe ROOTS, CUbeS aND CUbe ROOTS ........................................................ 16 squares and square roots ............................................................................ 16 Cubes and cube roots.................................................................................. 17  4  DIReCTeD NUmbeRS .................................................................... 19 numbers below zero .................................................................................... 19  5  VUlgaR aND DeCImal fRaCTIONS aND peRCeNTageS ......................................................................22 Fractions ........................................................................................................22 Fraction of a quantity ...................................................................................23 equivalent fractions ......................................................................................24 Fractions and decimals ................................................................................25 terminating and recurring decimals...........................................................26 Fractions, decimals and percentages ........................................................27  6  ORDeRINg .............................................................................................29 ordering integers .........................................................................................29 Inequalities ....................................................................................................29 ordering decimals........................................................................................30 ordering fractions ........................................................................................31 ordering fractions, decimals and percentages ........................................32  Photocopying prohibited  Contents  7  STaNDaRD fORm ...........................................................................33 standard form ...............................................................................................33 Calculating with numbers in standard form ..............................................34  8  The fOUR OpeRaTIONS ............................................................36 Calculating with negative numbers ............................................................36 order of operations .....................................................................................37 Multiplying integers .....................................................................................39 Multiplying decimals ....................................................................................40 Dividing integers .......................................................................................... 41 Dividing decimals ......................................................................................... 41 Adding and subtracting fractions...............................................................43 Multiplying fractions ....................................................................................45 Dividing fractions .........................................................................................47  9  eSTImaTION .........................................................................................49 estimating lengths........................................................................................49 Rounding to a given number of decimal places .......................................50 Rounding to a given number of significant figures................................... 51 estimating answers to problems ................................................................53 Working to a sensible degree of accuracy ................................................56  10 lImITS Of aCCURaCy..................................................................58 Bounds of measurement..............................................................................58 sums and differences of measurements....................................................59 Multiplying and dividing measurements ................................................... 61  11 RaTIO, pROpORTION, RaTe ....................................................64 Ratio ..............................................................................................................64 Using ratio to find an unknown quantity ....................................................66 Dividing a quantity in a given ratio .............................................................68 Increasing and decreasing in a given ratio ................................................69 Proportion .....................................................................................................69 Rate ..............................................................................................................72  12 peRCeNTageS.................................................................................... 74 Fractions, decimals and percentages ........................................................ 74 Percentage of a quantity.............................................................................. 74 expressing one quantity as a percentage of another .............................. 76 Percentage change ......................................................................................78 Percentage increase and decrease ............................................................79 Percentage increase and decrease using a multiplier..............................80 Finding the original quantity .......................................................................81  Photocopying prohibited  Contents  13 USe Of aN eleCTRONIC CalCUlaTOR ........................84 order of operations .....................................................................................84 standard form on your calculator ...............................................................85 Checking accuracy .......................................................................................87 Interpreting the calculator display..............................................................89 Calculating with time ...................................................................................90 Fractions on a calculator ..............................................................................92  14 TIme ............................................................................................................94 the 24-hour clock ........................................................................................94 Calculating with time ...................................................................................95 Converting between hours and minutes and hours written as a decimal .........................................................................................97 time zones ....................................................................................................98  15 mONey...................................................................................................100 Value for money ..........................................................................................100 Currency conversion ..................................................................................101  16 peRSONal aND Small bUSINeSS fINaNCe .........104 extracting data from tables and charts ..................................................104 Wages and salaries .....................................................................................105 Discount, profit and loss ............................................................................106 Repeated percentage change ..................................................................107 Compound and simple interest ................................................................109  17 algebRaIC RepReSeNTaTION aND fORmUlae ..... 112 Letters for unknowns .................................................................................. 112 substituting numbers into algebraic expressions .................................. 113 Using harder numbers when substituting................................................ 114 Writing formulae ......................................................................................... 115 Writing equations ....................................................................................... 117 transforming formulae ............................................................................... 118 transforming harder formulae ..................................................................120  18 algebRaIC maNIpUlaTION ................................................ 123 Adding and subtracting with negative numbers .................................... 123 Multiplying and dividing with negative numbers ................................... 124 Combining operations ............................................................................... 125 simplifying algebraic expressions ............................................................ 126 simplifying more complex algebraic expressions .................................. 128 expanding a single bracket .......................................................................129 Factorising algebraic expressions ............................................................130 Factorising expressions of the form ax + bx + kay + kby ...................... 131 expanding a pair of brackets ................................................................... 132 expanding more complex brackets .........................................................134  Photocopying prohibited  Contents Factorising expressions of the form a²x² − b²y² ..................................... 135 Factorising expressions of the form x2 + bx + c ......................................136 Factorising expressions of the form a² + 2ab + b ² .................................138 Factorising quadratic expressions of the form ax ² + bx + c .................. 139 simplifying algebraic fractions ................................................................. 142 Adding and subtracting algebraic fractions ...........................................144  19 INDICeS..................................................................................................146 simplifying numbers with indices .............................................................146 Multiplying numbers in index form .......................................................... 147 Dividing numbers in index form ............................................................... 147 simplifying algebraic expressions using indices ....................................148 negative indices .........................................................................................150 Fractional indices........................................................................................ 151 Using the laws of indices with numerical and algebraic expressions ...153  20 SOlUTION Of eqUaTIONS aND INeqUalITIeS ......................................................................155 solving simple linear equations ................................................................155 solving equations with a bracket ..............................................................155 solving equations with the unknown on both sides ...............................156 solving equations involving fractions ...................................................... 157 solving more complex equations involving fractions ............................158 solving inequalities .................................................................................... 159 solving linear simultaneous equations .................................................... 161 solving more complex simultaneous equations .....................................163 solving quadratic equations of the form x 2 + bx + c = 0 by factorisation .... 165 solving quadratic equations of the form ax 2 + bx + c = 0 by factorisation ..166 solving quadratic equations by completing the square ........................ 167 solving quadratic equations using the quadratic formula ..................... 169 solving equations involving algebraic fractions ..................................... 171  21 gRaphICal RepReSeNTaTION Of INeqUalITIeS ................................................................................... 172 showing regions on graphs....................................................................... 172 Representing regions satisfying more than one inequality ................... 175  22 SeqUeNCeS ....................................................................................... 176 number patterns ........................................................................................ 176 Linear sequences ....................................................................................... 178 the n th term ............................................................................................... 179 Finding the formula for the n th term........................................................ 179 the n th term of quadratic, cubic and exponential sequences ............. 181  Photocopying prohibited  Contents  23 VaRIaTION...........................................................................................184 Proportion ...................................................................................................184 Variation as a formula .................................................................................185 other types of variation .............................................................................186  24 gRaphS IN pRaCTICal SITUaTIONS ............................. 191 Conversion graphs ..................................................................................... 191 travel graphs ............................................................................................... 194 Rate of change on a distance–time graph .............................................. 197 Rate of change on a speed–time graph ..................................................200 Area under a speed–time graph ..............................................................203  25 gRaphS Of fUNCTIONS .........................................................207 Quadratic graphs........................................................................................207 Using graphs to solve equations ..............................................................209 Cubic graphs ............................................................................................... 211 Reciprocal graphs ....................................................................................... 213 exponential graphs .................................................................................... 216 estimating the gradient to a curve ........................................................... 217  26 fUNCTION NOTaTION ............................................................... 219 Function notation ....................................................................................... 219 Inverse functions.........................................................................................220  27 COORDINaTe geOmeTRy ......................................................222 the gradient of a straight-line graph .......................................................222 Line segments .............................................................................................223 the length of a line segment ...................................................................223 the general form of the equation of a straight line................................224 Parallel and perpendicular lines................................................................226  28 geOmeTRICal TeRmS ...............................................................229 Dimensions ..................................................................................................229 Angles ..........................................................................................................229 Lines ............................................................................................................230 Bearings .......................................................................................................231 triangles ......................................................................................................232 Quadrilaterals .............................................................................................233 Polygons .....................................................................................................234 solids............................................................................................................235 Congruence and similarity ........................................................................237 Circles ..........................................................................................................240 nets of 3-D shapes ..................................................................................... 241  Photocopying prohibited  Contents  29 geOmeTRICal CONSTRUCTIONS ....................................244 Measuring angles .......................................................................................244 Constructing a triangle using a ruler and protractor ............................. 247 Constructing a geometrical figure using compasses.............................249 Constructing regular polygons in circles .................................................252 Constructing angle bisectors and perpendicular bisectors ..................252 scale drawings and maps ..........................................................................253  30 SImIlaRITy aND CONgRUeNCe .......................................257 Congruence ................................................................................................257 similarity ......................................................................................................260 the areas and volumes of similar shapes.................................................264  31 SymmeTRy ..........................................................................................267 Line symmetry .............................................................................................267 Rotational symmetry ..................................................................................268 symmetry properties of shapes and solids .............................................270 the symmetry properties of the circle .....................................................271  32 aNgleS.................................................................................................. 274 Angles formed by straight lines ................................................................ 274 Angles formed within parallel lines .......................................................... 276 the angles in a quadrilateral .....................................................................280 the angles in a polygon.............................................................................282 the angles in a circle ..................................................................................284  33 lOCI ..........................................................................................................292 simple loci ..................................................................................................292 Problems involving intersection of loci ...................................................295  34 meaSUReS...........................................................................................298 Basic units of length, mass and capacity .................................................298 Area and volume measures .......................................................................299  35 meNSURaTION ................................................................................301 the perimeter of a two-dimensional shape ............................................301 the area of a rectangle ..............................................................................303 the area of a triangle .................................................................................304 the area of a parallelogram ......................................................................306 the area of a trapezium .............................................................................308 the area of shapes made from rectangles and triangles ...................... 310 the circumference of a circle .................................................................... 312 the area of a circle ..................................................................................... 314 Arc length and sector area ........................................................................ 316 the volume of a prism................................................................................ 319 the surface area of a prism .......................................................................322  Photocopying prohibited  Contents the volume of a pyramid, a cone and a sphere ......................................324 the surface area of a pyramid, a cone and a sphere ..............................326 the area and volume of compound shapes ............................................330  36 TRIgONOmeTRy .............................................................................334 Bearings .......................................................................................................334 Pythagoras' theorem .................................................................................336 trigonometry ..............................................................................................340 the sine and cosine functions for obtuse angles ..................................349 non-right-angled triangles........................................................................351 Finding lengths and angles in three dimensions ....................................360  37 VeCTORS IN TwO DImeNSIONS .......................................366 Vectors and translations ............................................................................366 Vector notation ...........................................................................................368 Multiplying a vector by a scalar.................................................................373 Addition and subtraction of column vectors...........................................373 Position vectors...........................................................................................375 the magnitude of a vector ........................................................................ 376 Vector geometry .........................................................................................377  38 maTRICeS .............................................................................................381 Displaying information as a matrix ...........................................................381 Multiplying a matrix by a scalar quantity .................................................381 Adding and subtracting matrices .............................................................383 Multiplying two matrices ...........................................................................384 the determinant and the inverse of a matrix ..........................................386 Matrix algebra .............................................................................................388  39 TRaNSfORmaTIONS ...................................................................390 the language of transformations .............................................................390 Reflection ....................................................................................................390 Rotation .......................................................................................................393 translation ...................................................................................................395 enlargement ................................................................................................397 Recognising and describing transformations .........................................403 Combining transformations ......................................................................408 Matrices of transformations ..................................................................... 410  40 pRObabIlITy ..................................................................................... 416 the probability of a single event .............................................................. 416 the probability of an event not occurring ............................................... 418 estimating from a population ................................................................... 419 Relative frequency and probability ..........................................................420 the probability of combined events ........................................................423 Independent events ...................................................................................425 tree diagrams for combined events.........................................................427 the probability of dependent events ......................................................429 Photocopying prohibited  Contents  41 CaTegORICal, NUmeRICal aND gROUpeD DaTa ..............................................................................432 Collecting and grouping data...................................................................432 surveys .........................................................................................................433 Designing a questionnaire ........................................................................435 two-way tables ...........................................................................................435 Averages and range ..................................................................................438 Mean and range..........................................................................................439 Which average to use when comparing data ..........................................441 Working with larger data sets .................................................................. 444 Working with grouped and continuous data ..........................................448  42 STaTISTICal DIagRamS ...........................................................454 Bar charts and pictograms ........................................................................454 Pie charts .....................................................................................................456 Representing grouped and continuous data ..........................................459 scatter diagrams.........................................................................................462 Cumulative frequency diagrams ...............................................................466 Median, quartiles and percentiles ............................................................469 Histograms .................................................................................................. 474  INDEX  ............................................................................................................. 478  Photocopying prohibited  IntRoDUCtIon for the teacher This book is intended to be used by students preparing for Cambridge International Examinations O Level Mathematics (Syllabus D). The structure of the book follows closely the contents of the syllabus, with one chapter devoted to each item of the syllabus. However, in some cases, the full development of a topic is not achieved until a later chapter. For example, further work relevant to Standard Form in Chapter 7 is contained in Chapter 13. It is possible to work through the chapters in order but more variety can be achieved by moving between different areas of the curriculum and the following features will help in planning this. Each chapter starts with a 'By the end of this chapter you will be able to:' box, which repeats the relevant statements from the syllabus. Immediately after this is a 'Check you can:' box, which lists the knowledge and skills required before commencing the chapter. This helps you decide whether topics from earlier in the book need to be covered or revised first. Sometimes, work from a later chapter may be required to complete the topic. If so, that part of the chapter can be delayed until later, or the extra knowledge or technique introduced early. It will then be consolidated in working through the later chapter. Throughout each chapter there are numerous Examples with worked solutions, illustrating each aspect of the topic and helping students understand how to tackle the Exercises which follow. The questions in the exercises give considerable practice to reinforce the learning.* The syllabus requires that some of the work be done without a calculator. This is clearly indicated in the book and questions in the exercises are indicated with There are also regular hints and guidance contained in the Note boxes, helping students avoid common pitfalls and clarifying what is in the text. *Please note that answers to the questions in the exercises will appear on the website which accompanies this book. These are not endorsed by Cambridge International Examinations as a mark scheme but they will provide useful guidance when assessing student progress. Answers can be found at www. hoddereducation.com/Cambridgeextras Brian Seager  Photocopying prohibited  xi  INTRODUCTION  for the student How to use this book The chapters in this book follow the syllabus and each begins with: BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  which lists what you will learn, as detailed in the syllabus.  Following these statements there is:  CHECK YOU CAN: l  which lists what you need to know before you start the chapter. If you are not certain, it is a good idea to revise these first.  Throughout a chapter there are:  Examples Which show you how to apply what you are learning. In each case, a question is followed by a worked solution. These are a model to help you answer the questions in the Exercise which follows.  Another feature of the book is:  Note These contain hints and advice about what you are learning.  xii  Photocopying prohibited  1  nUMBeR  Types of number  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO:  Integers Integers are positive and negative whole numbers: …, –2, –1, 0, 1, 2, …  l  Natural numbers Natural numbers are integers that can be used for counting: 1, 2, 3, 4, 5, …  Rational numbers A rational number is a number that can be written as a fraction a b where a and b are integers and b ≠ 0. Rational numbers include: l l l  identify and use natural numbers, integers (positive, negative and zero), prime numbers, common factors and common multiples, rational and irrational numbers (e.g. π, 2), real numbers.  CHECK YOU CAN:  all integers 581 all terminating decimals, for example 5.81 can be written as 100 all recurring decimals, for example 0.6 = 0.666 666… can be written as 23 .  l  l  Irrational numbers An irrational number is a number that cannot be written as a fraction such as 2 or π.  recognise factors and multiples of a number write a product using index notation, for example 5 × 5 × 5 = 53.  Note Recurring decimals can be shown by placing dots above the digits, e.g. 0.3 = 0.333 333… 1.3 07 = 1.307 307 307…  They give decimals that do not terminate or recur.  Real numbers The set of real numbers is made up of all rational and irrational numbers.  Example 1.1 question Sort the numbers in the list below into rational and irrational numbers. Show how you decide. 0.8652  12  4π  67 19  64  Solution Rational numbers can be written as fractions, so 0.8652, 0.8652 =  8652 10 000  67 19  and 64 are rational.  64 = 8  Irrational numbers cannot be written as fractions, so 12 and 4π are irrational. 12 = 3.464 101…  Photocopying prohibited  4π = 12.566 370…  1  1 NUmbeR  exercise 1.1 1 State which of these numbers are a integers b natural numbers. 8 1534 0 −7 0.6 27  4 5  −12  2 State which of these numbers are rational, showing how you decide. a  17 20  d 5π g  4 25  2 25  b 0.46  c  e 3.141 59  f    −0.234  i  2 3+ 3  h  225  3 State which of these numbers are rational, showing how you decide. a 169 b 0.49 c 5+ 3 d –2.718  e 5π + 2  g  h  27  17  9  f  4π 3π  i  −6 2  4 Write down an irrational number between each pair of numbers. a 3 and 4 b 10 and 11 c 19 and 20  prime factors The factors of a number are all of the numbers that divide exactly into that number. A prime number is a number with only two factors. The factors of 7 are 1 and 7, so 7 is a prime number. The factors of 12 are 1, 2, 3, 4, 6 and 12, so 12 is not a prime number. The only factor of 1 is 1, so 1 is not a prime number. Any number that is not prime can be written as the product of its prime factors. The prime factors of a number can be found either using a factor tree or by dividing repeatedly by prime numbers.  2  Photocopying prohibited  Prime factors  Example 1.2 question Write 60 as the product of its prime factors.  Solution Division method  Factor tree method It doesn't matter how you start the factor tree, the ends of the branches will be the same. 60  15 5  2  60  2  30  3  15  5  5  4 3  2  1  Divide by the prime numbers, starting with 2, until the result is 1.  2  The prime factors of 60 are 2, 3 and 5. Written as a product of its prime factors, 60 = 2 × 2 × 3 × 5. Using index notation, 60 = 22 × 3 × 5.  exercise 1.2 1 Write down all of the factors of each of these numbers. a 8 b 15 c 27 d 54 2 Write down the first ten prime numbers. 3 Write down the prime factors of each of these numbers. a 12 b 20 c 55 d 84 4 Express each of these numbers as the product of its prime factors. a 48 d 350 b 72 c 210 e 75 h 198 f 275 g 120 5 Express each of these numbers as the product of its prime factors. a 495 b 260 c 2700 d 1078 e 420 f 1125 g 112 h 1960 6 a Write each of these square numbers as the product of its prime factors. i 25 ii 36 iii 100 iv 144 b Comment on what you notice about each of the products in a. 7 a Write 96 as the product of its prime factors. b Find the smallest positive integer k such that 96k is a square number. 8 a Write 392 as the product of its prime factors. b Find the smallest positive integer k such that 392k is a cube number.  Photocopying prohibited  3  1 NUmbeR  Common factors and common multiples A common factor of two numbers is a number that is a factor of both of them. 2 is a common factor of 8 and 12 because 8 ÷ 2 = 4 and 12 ÷ 2 = 6. The highest common factor (HCF) of two numbers is the highest number that is a factor of both numbers. The highest common factor of 8 and 12 is 4. A multiple of a number is the product of the number and any integer. A common multiple of two numbers is a number that is a multiple of both of them. 20 is a common multiple of 2 and 5 because 2 × 10 = 20 and 5 × 4 = 20. The lowest common multiple (LCM) of two numbers is the lowest number that is a multiple of both numbers. The lowest common multiple of 2 and 5 is 10.  Example 1.3 question a Find the highest common factor (HCF) of 84 and 180. b Find the lowest common multiple (LCM) of 84 and 180.  Solution First write each number as the product of its prime factors. 84  180  6 2  14 3  2  18 7  3  a  2  6 2  84 = 2 × 2 × 3 × 7 = 22 × 3 × 7  10 5  3  180 = 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5  To find the highest common factor, find all numbers that appear in both lists. 84 = 2 × 2 × 3 × 7 180 = 2 × 2 × 3 × 3 × 5  The highest common factor will be no higher than the smaller of the two numbers.  The highest common factor of 84 and 180 is 2 × 2 × 3 = 12. This means that 12 is the highest number that is a factor of both 84 and 180.  4  ➜  Photocopying prohibited  Common factors and common multiples  b To find the lowest common multiple, find all of the prime numbers that appear in each list and use the higher power of each. The lowest common multiple will be no lower than the larger of the 84 = 22 × 3 × 7 two numbers. 2 2 180 = 2 × 3 × 5 The lowest common multiple of 84 and 180 is 22 × 32 × 5 × 7 = 1260. This means that 1260 is the lowest number that is a multiple of both 84 and 180. It is the lowest number that has both 84 and 180 as factors.  exercise 1.3 1 For each pair of numbers: i express each number as the product of its prime factors ii find the highest common factor (HCF) iii find the lowest common multiple (LCM). a 18 and 24 b 64 and 100 c 50 and 350 d 72 and 126 2 Find the highest common factor (HCF) and lowest common multiple (LCM) of each pair of numbers. a 27 and 63 b 20 and 50 c 48 and 84 d 50 and 64 e 42 and 49 3 For each pair of numbers: i express each number as the product of its prime factors ii find the highest common factor (HCF) iii find the lowest common multiple (LCM). a 260 and 300 b 340 and 425 c 756 and 2100 d 1980 and 2376 4 Find the HCF and LCM of each pair of numbers. a 5544 and 2268 b 2016 and 10 584 5 a Find the highest common factor of 45, 60 and 75. b Find the lowest common multiple of 45, 60 and 75. 6 A rectangle measures 240 mm by 204 mm. It is split up into identical squares. Find the largest possible side length for the squares. 7 There are two lighthouses near a port. The light in the first lighthouse flashes every 22 seconds. The light in the second lighthouse flashes every 16 seconds. At 10 p.m. one evening both lights are switched on. What is the next time that the lights flash at the same time? 8 Buses to Shenley leave the bus station every 40 minutes. Buses to Winley leave every 15 minutes. At 8.15 a.m. buses to both Shenley and Winley leave the bus station. When is the next time that buses to both places leave at the same time?  Photocopying prohibited  5  2  set LAngUAge AnD notAtIon  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  use language, notation and Venn diagrams to describe sets and represent relationships between sets.  CHECK YOU CAN: l  recall the meaning of the terms integer, prime, multiple and factor.  The definition of a set A set is a collection of numbers, shapes, letters, points or other objects. They form a set because they fulfil certain conditions. The notation for a set is a pair of curly brackets: {…}. The individual members of a set are called elements. A set can be defined by giving a rule which satisfies all the elements or by giving a list of the elements. For example, A = {integers from 1 to 10}  or  A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}  exercise 2.1 List the elements of the following sets. 1 2 3 4 5  {integers from 11 to 18} {the first five prime numbers} {the factors of 12} {the multiples of 8 less than 50} {vowels}  Note If you are giving a rule for a set then it must be precise. In A above, do not just use 'integers'.  The universal set All the sets in Exercise 2.1 are finite sets. They have a fixed number of elements. Sets can also have an infinite number of elements. For example, {multiples of 8}  or  {prime numbers}  Most of the sets you will be dealing with will be finite sets. Sometimes we make sure we are dealing with a finite set by defining a universal set. A universal set is a set from which – for a particular situation – all other sets will be taken. For example, if you define the universal set as positive integers less than 50, then the set in Exercise 2.1 question 4 could simply be defined as {multiples of 8} since you are only considering integers less than 50. Similarly, if you define the universal set as positive integers less than 12, then the set in question 2 could have been defined as {prime numbers}.  6  Photocopying prohibited  Venn diagrams  Notation The symbol for a universal set is e The symbol ∈ means 'is an element of'. The symbol ∉ means 'is not an element of'. So  3 ∈ {prime numbers}  and  4 ∉ {prime numbers}.  Venn diagrams Venn diagrams are a way of showing sets and the relationships between sets. Venn diagrams were introduced in 1880 by John Venn. In a Venn diagram the universal set is shown by a rectangle. Other sets are drawn as circles or ovals within the rectangle. The diagrams here show three typical Venn diagrams. Q  P  This Venn diagram shows two sets, P and Q, where there are some elements that are in both sets.  P  Q  This Venn diagram shows two sets, P and Q, where there are no elements that are in both sets. Sets P and Q are disjoint.  P Q  This Venn diagram shows two sets, P and Q, where all the elements of set Q are also in set P. Set Q is a subset of set P.  Venn diagrams are not restricted to two sets. You may be asked to draw a Venn diagram for three or more sets. You may be asked to draw a Venn diagram and fill in the elements of the sets.  Photocopying prohibited  7  2 SeT laNgUage aND NOTaTION  Example 2.1 e = {integers from 1 to 20}  P = {factors of 12}  Q = {prime numbers}  question a  Draw a Venn diagram to show the elements of the universal set and its subsets P and Q.  b List the elements that are in both set P and set Q.  Solution a  Q  P 1 4  6 12  5 2 3  Note  7 11 13  17 19  Do not forget to fill in the elements of e that are in neither P nor Q.  8 9 10 14 15 16 18 20  b 2, 3  We do not put an element in a Venn diagram more than once. So, for example, if A = {letters in the word label} the set is {l, a, b, e} and you put those four letters in the Venn diagram. You would not put the letter l in twice.  exercise 2.2 Draw a Venn diagram to show the elements of the universal set and its subsets P and Q. 1 e = {integers from 1 to 20} P = {factors of 18} Q = {odd numbers}  2 e = {integers less than 25} P = {multiples of 4} Q = {prime numbers}  3 e = {first 15 letters of the alphabet} 4 e = {positive integers less than 21} P = {letters in the word golf} P = {even numbers} Q = {letters in the word beam} Q = {multiples of 4} 5 e = {days of the week} P = {days with six letters} Q = {days beginning with S}  6 e = {multiples of 3 less than 50} P = {factors of 36} Q = {odd numbers}  The relationship between sets The intersection of two sets, P and Q, is all the elements that are in both set P and set Q. You can write this as P ∩ Q. In Example 2.1, P ∩ Q = {2, 3}. 8  Photocopying prohibited  The complement of a set  The union of two sets, P and Q, is all the elements that are in set P or set Q or both. You can write this as P ∪ Q. In Example 2.1, P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 17, 19} If a set has no elements, it is called the empty set. The symbol for the empty set is Ø. It may seem a trivial idea but it is quite important when dealing with some sets. For example, if P = {triangles}  e = {polygons}  Q = {quadrilaterals}  then the Venn diagram looks like this. P  Q  The sets are disjoint and so the intersection, P ∩ Q = Ø.  exercise 2.3 For each of the questions in Exercise 2.2, find a P∩Q  b P ∪ Q.  The complement of a set A′ means those elements of the universal set which are not in set A. It is called the complement of A. For example, if  e = {positive integers less than 13} and A = {factors of 12} = {1, 2, 3, 4, 6, 12} then A′ = {5, 7, 8, 9, 10, 11}  Photocopying prohibited  9  2 SeT laNgUage aND NOTaTION  Example 2.2 question On separate copies of the diagram shade these sets. a  P′  Q  P  b (P ∩ Q)′ c  (P ∪ Q)′  Solution a  b P  Q  c P  Q  Q  P  exercise 2.4 For each of the questions in Exercise 2.2, list the elements of these sets. a Q′ b (P ∪ Q)′  exercise 2.5 1 On separate copies of the diagram shade these sets. a P′ b (P ∩ Q)′ c (P ∪ Q)′  2 On separate copies of the diagram shade these sets. a P∩Q b P∪Q c P′  P  Q  P Q  3 e = {polygons} A = {quadrilaterals} B = {regular polygons} a Draw a Venn diagram to represent these sets. b Describe in words the set A ∩ B.  10  Photocopying prohibited  Subsets  4 a On separate copies of the diagram shade these sets. i (P ∩ Q)′ ii P′ ∪ Q′ b What do you notice?  P  5 a On separate copies of the diagram shade these sets. i (P ∪ Q)′ ii P′ ∩ Q′ c What do you notice?  P  Q  Q  Subsets Example 2.3 question Find all the subsets of P = {a, b, c, d}.  Solution Subsets with one element:  {a}, {b}, {c}, {d}  Subsets with two elements:  {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}  Subsets with three elements:  {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}  In addition to these 14 subsets, the set P itself is regarded as a subset:  {a, b, c, d}  the empty set also is regarded as a subset:  Ø  This makes 16 subsets in total.  A set with 4 elements has 16 or 24 subsets. In general, a set with n elements has 2n subsets. In Example 2.3, the first 14 subsets of set P and the empty set are known as proper subsets. The proper subsets do not include the set itself. A set with n elements has 2n − 1 proper subsets  Notation The symbol ⊆ means 'is a subset of'. So A ⊆ B means set A is a subset of set B. It includes the possibility that set A could be the set B itself or the empty set. The symbol ⊆ means 'is not a subset of'. So A ⊆ B means set A is not a subset of set B.  Photocopying prohibited  11  2 SeT laNgUage aND NOTaTION  The symbol ⊂ means 'is a proper subset of'. So A ⊂ B means set A is a proper subset of set B. It excludes the possibility that set A could be the set B itself. The symbol ⊄ means 'is not a proper subset of'. So A ⊄ B means set A is not a proper subset of set B.  problem solving with Venn diagrams If A is a set then n(A) means the number of elements in set A. For example, if A = {a, b, c, d}, then n(A) = 4. When you are using a Venn diagram to solve a problem you can write the number of elements in the subsets rather than filling in all the elements.  Example 2.4 question There are 32 students in a class. They can choose to study history (H) or geography (G) or both or neither:  a  l  18 study history  l  20 study geography  l  8 study both history and geography.  Draw a Venn diagram to show this information.  b Find the number of students who study neither history nor geography.  Solution a  Since 8 students study both, 10 students study history but not geography  12 students study geography but not history. b The number of students who study history or geography or both is 10 + 8 + 12 = 30. So the number of students who study neither history nor geography is  G  H 10  8  12  2  32 − 30 = 2.  alternative ways to define a set If a set is finite, you can define it by listing its elements. For example, P = {1, 4, 9, 16, 25}. However, if a set is infinite, you cannot list all the elements so you define it by giving the rule used to form it.  12  Photocopying prohibited  Alternative ways to define a set  There are various ways to do this. For example, A = {x : x is a natural number}  Note  B = {(x, y) : y = 2x + 1}  In definitions like this the colon ( : ) is read as 'where'.  C = {x : 2  x  5}  So A = {1, 2, 3, 4, 5, …} and is an infinite set.  exercise 2.6 1 Find all the subsets of {p, q, r}. 2 How many subsets does the set {5, 6, 7, 8, 9, 10} have? 3 Muna has 35 books in her electronic book reader 20 are crime stories (C) 12 are books of short stories (S) 7 are books of short crime stories. a Copy and complete this Venn diagram to show the number of books of each type in Muna's e-reader. b Find i n(C ∩ S′) ii n(C ∪ S)′. 4 In a class of 30 students 22 study physics (P) 19 study chemistry (C) 6 study neither physics nor chemistry. a Find the number of students who study both physics and chemistry. b Show the information in a Venn diagram. 5 In a sports club with 130 members 50 play tennis but not football 40 play football but not tennis 15 play neither football nor tennis. a Draw a Venn diagram to show this information. b Find the number of members who play both tennis and football. 6 For two sets A and B, n(A) = 20, n(B) = 30 and n(A ∪ B) = 38. By drawing a Venn diagram or otherwise, find n(A ∩ B). 7 For two sets A and B, n(A) = 20, n(B) = 30 and n(A ∩ B) = 11. By drawing a Venn diagram or otherwise, find n(A ∪ B). 8 Two sets P and Q are such that n(P ∩ Q) = 0. Show the sets P and Q on a Venn diagram. 9 Two sets P and Q are such that n(P ∩ Q) = n(P). Show the sets P and Q on a Venn diagram. 10 Two sets P and Q are such that n(P ∪ Q) = n(P). Show the sets P and Q on a Venn diagram. 11 e = {x : x is an integer} and P = (x : −2  x < 4}. List the elements of P.  Photocopying prohibited  B = {(x, y) : y = 2x + 1} is read as 'set B consists of the points (x, y) where y = 2x + 1'.  C  S  13  2 SeT laNgUage aND NOTaTION  Three-set problems Example 2.5 question In a sixth form of 200 students, three of the subjects students can study are mathematics (M), technology (T) and psychology (P). 110 students study mathematics, 85 study technology and 70 study psychology 45 study mathematics and technology 35 study mathematics and psychology 19 study technology and psychology 9 study all three subjects. a Copy and complete this Venn diagram. b Find i  the number of students who study none of the three subjects  ii  n(M ∩ T ∩ P′)  M  T 36 9  iii n[M ∩ (T ∪ P)]. P  Solution a  b i M  T 36  39 26  ii  200 − (39 + 36 + 9 + 26 + 30 + 10 + 25) = 200 − 175 = 25 36  iii 36 + 9 + 26 = 71  30  9 10  25 P  25  exercise 2.7 1  14  e = {integers from 2 to 15 inclusive} A = {prime numbers} B = {multiples of 4} C = {multiples of 2} a Draw a Venn diagram to represent the sets e, A, B and C. b Find i n(B ∪ C)′ ii n(A ∪ B) ∩ C′.  Photocopying prohibited  Three-set problems  2 Copy the diagram. Insert a, b, c and d in the correct subsets in the Venn diagram given the following information. i a∈P∩Q∩R ii b ∈ (P ∪ Q ∪ R)′ iii c ∈ (P ∪ Q)′ ∩ R iv d ∈ P ∩ Q ∩ R′  P  Q  R  3 Use set notation to describe the sets represented by the shaded area in these Venn diagrams. a b A  C  P  B  Q  R  4 e = {integers} E = {multiples of 2} T = {multiples of 3} F = {multiples of 4} a Use set notation to express M = {multiples of 12}, as simply as possible, in terms of E, T and F. b Simplify E ∪ F. 5 The Venn diagram shows the number of elements in each subset. A a Find n(B ∩ C). 2x x b You are given that n(A ∪ B) = n(C). 7 Find x. 5 8  10 C  6 The sets P, Q and R are subsets of the universal set e. Q⊂P Q∩R=Ø P∩R≠Ø Draw a Venn diagram to show the sets P, Q, R and e. 7 Copy and complete these statements about sets P, Q and R. a P∩…=Ø b R…Q c R∪…=Q d n(R ∩ Q) = n(…) e n(P) + n(…) = n(P ∪ R)  Photocopying prohibited  B  4  8  P  Q R  15  sQUARes, sQUARe Roots, CUBes AnD CUBe Roots  3  You should already have met the topics in this chapter. This chapter is a quick reminder.  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  calculate squares, square roots, cubes and cube roots of numbers.  Squares and square roots As you can see in the diagram, the square with side 3 has an area of 3 × 3 = 9 squares and the square with side 4 has an area of 4 × 4 = 16 squares.  CHECK YOU CAN: l  l  Note  multiply numbers, with and without a calculator find the area of a square.  3  You can say that the square of 3 is 9 or 3 squared is 9 and write it as 32 = 9.  4 3 4  The integers 1, 4, 9, 16, 25, … are the squares of the integers 1, 2, 3, 4, 5, … . Because 16 = 42, the positive square root of 16 is 4. It is written as 16 = 4. Similarly 36 = 6 and 81 = 9. You can use your calculator to find squares and square roots.  Example 3.1  Note  question  Some calculators operate slightly differently. Learn which keys you need to use to do these operations on your calculator.  Work out these using a calculator. a  472  b  729  Solution a  472 = 2209  b  729 = 27  On your calculator you need to press  4 7 x2 =  On your calculator you need to press √  7 2 9 =  You also need to be able to find squares and square roots without a calculator. Here is a list of the squares of the integers 1 to 15. You should learn these square numbers. You will also need them for finding square roots without a calculator. Integer  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  Square number  1  4  9  16  25  36  49  64  81  100  121  144  169  196  225  16  Photocopying prohibited  Cubes and cube roots  Example 3.2  Solution  question  Work out the squares first. 82 = 8 × 8 = 64  Work out 82 − 42.  42 = 4 × 4 = 16 So 82 − 42 = 64 − 16 = 48  exercise 3.1 1 Write down the square of these numbers. a 7 b 12 c 5 d 10 f 8 g 11 h 3 i 6 2 Write down the positive square root of these numbers. a 49 b 121 c 81 d 36 f 169 g 144 h 225 i 100 3 Work out these. a 132 b 112 c 142 d 62 4 Work out these.  e 9 j 4 e 25 j 196 e 92  a b 81 5 Work out these.  144  c  16  d  100  e  64  a b 529 6 Work out these. a 202 b 7 Work out these. a 62 − 52 e 42 + 52 8 Work out these. a 212 − 92  256  c  324  d  841  e  784  252  c 132  b 22 + 32 f 62 − 32  d 242 c 72 − 42 g 52 − 42 − 32  e 332 d 32 − 22 h 132 − 52 − 92  b 242 + 72 − 102 c 172 − 152 + 112 d 202 + 212 + 222  Cubes and cube roots The cube in the diagram has a volume of 2 × 2 × 2 = 8.  2  Note  The cube of a number is the number multiplied by itself, and then by itself again.  2  The integers 1, 8, 27, 64, 125, 216, … are the cubes of the integers 1, 2, 3, 4, 5, 6, … . Because 8 = 23 the cube root of 8 is 2. It is written  You can say that the cube of 2 is 8 or 2 cubed is 8 and write it as 23 = 2 × 2 × 2 = 8.  2  3  as 8 = 2. Similarly 3 27 = 3 and 3 64 = 4.  Photocopying prohibited  17  3 SqUaReS, SqUaRe ROOTS, CUbeS aND CUbe ROOTS  You need to be able to find cubes and cube roots without a calculator. Here is a list of the cubes of the integers 1 to 10. You should learn these cube numbers. Integer  1  2  3  4  5  6  7  8  9  10  Cube number  1  8  27  64  125  216  343  512  729  1000  Example 3.3 question Work out 3 216 without a calculator.  Solution 3  216 = 6  You know that 63 = 216, so you also know that the cube root of 216 is 6.  You can also use your calculator to find cubes and cube roots.  Example 3.4  Note  question  Some calculators operate slightly differently.  Work out these using a calculator. a  153  b  3  4913  Learn which keys you need to use to do these operations on your calculator.  Solution a  153 = 3375  On your calculator press  1  5 x3 =  b  3  On your calculator press  3  4 9 1  4913 = 17  3 =  exercise 3.2 1 Write down the cube of these numbers. a 4 b 5 c 3 d 10 e 8 2 Write down the cube root of these numbers. a 1 b 64 c 1000 3 Work out these. a 73 b 93 c 203 d 253 e 1.53 f 2.73 g 5.43 4 Work out these. 3 d 1000 000 a 3 343 b 3 729 c 3 1331 e 3 216 f 3 1728 g 3 512 5 Work these out. Give your answers to 2 decimal places. a 3 56 b 3 230 c 3 529 d 3 1100 e 3 7526 3 6 Find the length of a cube whose volume is 45 cm . Give your answer to 2 decimal places.  18  Photocopying prohibited  4  DIReCteD nUMBeRs  Numbers below zero Some numbers are less than zero. These are called negative numbers. They are written as ordinary numbers with a negative sign in front.  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  Negative numbers are used in many situations.  4 3 2 1 0 -1 -2 -3 -4  Money earned by a company Thousand dollars  Thermometer measuring temperature  Depth of water in a river  30 20 10 0 –10 –20 –30  m  Profit Loss  2 1 0 –1 –2  CHECK YOU CAN: l  l  add and subtract whole numbers read whole numbers from a scale.  Note  Buttons in a lift 3 2 1 0 –1  use directed numbers in practical situations.  Remember that zero is neither positive nor negative.  Car park  Ground floor Basement  Example 4.1 question Suravi measured the daytime and night-time temperatures in her garden for 2 days. Here are her results. Day Temperature (°C)  a  Monday daytime 7  Monday night-time −2  Tuesday daytime 3  Tuesday night-time −5  How much did the temperature change between each reading?  b The daytime temperature on Wednesday was 4 °C warmer than the Tuesday night-time temperature. What was the Wednesday daytime temperature? Use the temperature scale to help you.  Solution a  From 7 °C to −2 °C you go down 9 °C. From −2 °C to 3 °C you go up 5 °C. From 3 °C to −5 °C you go down 8 °C.  Photocopying prohibited  b Find −5 °C on the scale and move 4 °C up. You get to −1 °C.  8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6  °C  Monday daytime  Tuesday daytime  Monday night-time Tuesday night-time  19  4 DIReCTeD NUmbeRS  Example 4.2 question Abal's bank account is overdrawn by $75. How much must he put in this account for it to be $160 in credit?  Solution Abal's bank account is overdrawn by $75. This means Abal owes the bank $75. It can be shown as −$75. $160 in credit means that there is $160 in the account. It can be shown as +$160. From −75 to +160 is 235, so he must put $235 into the account.  exercise 4.1 1 Copy and complete the table. Start temperature/°C  move/°C  a  4  Up 3  b  −2  Down 4  end temperature/°C  c  10  Down 14  d  −5  Down 3  e  −10  −2  f  10  −9  g  −4  2  h  Up 7  10  i  Down 6  −9  j  Up 2  −8  2 The floors in a shopping centre are numbered −3, −2, −1, 0, 1, 2, 3, 4, 5. a Ali parks his car on floor −2. He takes the lift and goes up 6 floors. Which floor is he now on? b While shopping, Ubah goes down 3 floors ending on floor −1. Which floor did she start from? 3 What is the difference in temperature between each of these? a 17 °C and −1 °C b −8 °C and 12 °C c −19 °C and −5 °C d 30 °C and −18 °C e 13 °C and −5 °C f −10 °C and 15 °C g −20 °C and −2 °C h 25 °C and −25 °C 4 To the nearest degree, the hottest temperature ever recorded on Earth was 58 °C in 1922 and the coldest ever recorded was −89 °C in 1983. What is the difference between these temperatures? 5 Geta's bank account is $221 in credit. She puts $155 into the account. Later she spends $97 on clothes and $445 on some electronic goods. What will Geta's bank account show now? 20  Photocopying prohibited  Numbers below zero  6 The table shows the heights above sea level, in metres, of seven places. place  height/m  Mount Everest Bottom of Lake Baikal  8 863 −1 484  Bottom of Dead Sea  −792  Ben Nevis  1 344  Mariana Trench  −11 022  Mont Blanc World's deepest cave  4 807 −1 602  What is the difference in height between the highest and lowest places? 7 The highest temperature ever recorded in England was 38.5 °C in 2006 and the lowest ever recorded was −26.1 °C in 1982. What is the difference between these temperatures? 8 The table shows the coldest temperatures ever recorded on each continent. Continent  lowest temperature  Africa  −23.9 °C  Antarctica  −89.2 °C  Asia  −71.2 °C  Australia  −23.0 °C  North America  −60.0 °C  South America  −32.8 °C  Europe  −58.1 °C  What is the difference between the highest and lowest of these temperatures? 9 This is an extract from a tide table for Dungeness, Washington State, USA. Day  high/low  Tide time  1  High  2.03 a.m.  7.2  Low  9.28 a.m.  −1.8  High  2.27 p.m.  7.5  Low  9.53 p.m.  5.6  2  height/feet above normal  High  2.48 a.m.  7.2  Low  10.09 a.m.  −2.2  High  3.00 p.m.  7.7  Low  10.42 p.m.  5.4  What is the difference in height between the highest high tide and the lowest low tide?  Photocopying prohibited  21  VULgAR AnD DeCIMAL FRACtIons AnD PeRCentAges  5  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  l  use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts recognise equivalence and convert between these forms.  CHECK YOU CAN: l  l  l  identify a fraction of a shape understand and use decimal notation find the highest common factor of two numbers.  fractions A fraction is a number written in the form a , where a and b are integers. b  The value on the top of the fraction is known as the numerator. The value on the bottom of the fraction is known as the denominator. The fraction 74 is a proper fraction because the numerator is smaller than the denominator. The fraction 74 is an improper fraction because the numerator is larger than the denominator. The fraction 1 43 is a mixed number because it is formed from an integer and a proper fraction. An improper fraction can be written as a mixed number. 7 4 3 3 = + =1 4 4 4 4  Example 5.1 question a  Write  17 6  b Write 3 4 as an improper fraction.  as a mixed number.  5  Solution a  Divide the numerator by the denominator and write the remainder as a fraction over the denominator.  Note The denominator of the mixed number is the same as the denominator of the improper fraction.  17 ÷ 6 = 2 remainder 5 so  17 5 =2 6 6  b Multiply the integer by the denominator of the fraction and add the numerator. 3 × 5 + 4 = 19 so  4 5  3 =  19 5  exercise 5.1 1 State whether each of these is a proper fraction, an improper fraction or a mixed number. a 19 b 1 23 c 86 d 8 65 18 2 Write each of these improper fractions as a mixed number. a f 22  11 8 10 3  b  11 5  c  g  19 8  h  9 4 23 4  e  15 16  d  7 2  e  15 7  i  33 10  j  37 9  Photocopying prohibited  Fraction of a quantity  3 Write each of these mixed numbers as an improper fraction. a 11  b 25  8 22 5  f  g  8 2 3 3  c 33 h  4 1 2 10  d 5 12 2 83  i  e 32 j  9 46 7  fraction of a quantity A fraction can be used to describe a share of a quantity. The denominator shows how many parts the quantity is divided into. The numerator shows how many of those parts are required.  Example 5.2 question a  Work out  1 8  b Work out  of 56.  5 8  of 56.  Solution a  Finding So 5 8  b  1 8  of 56 is the same as dividing 56 into 8 parts.  1 8  of 56 = 56 ÷ 8 = 7  5 8  of 56 = 5 × 7 = 35  means 5 × 81 .  So  To find a fraction of a quantity, divide by the denominator and multiply by the numerator.  exercise 5.2 1 Work out these. a  3 4  of 64  b  2 3  of 96  c  5 7  of 35  d  9 10  of 160  e  11 of 180 12  2 In a school of 858 students, 6 are boys. 11 How many boys are there? 3 In an election, the winning party got 95 of the votes. There were 28 134 votes in total. How many votes did the winning party get? 4 A shop offers 14 off everything as a special offer. A mobile phone normally costs $168. How much does it cost with the special offer? 7 share of $120 or a 78 share of $104? 5 Which is larger, a 10 Show how you decide. 6 Which is larger, a 83 share of $192 or a 2 share of $180? 5 Show how you decide.  Photocopying prohibited  23  5 VUlgaR aND DeCImal fRaCTIONS aND peRCeNTageS  equivalent fractions These squares can be divided into equal parts in different ways.  The fraction represented by the shaded parts is  1 4  or  2 8  4 or 16 .  These three fractions are equal in value and are equivalent. 1 4  2 8  =  4 = 16 but  1 4  is in its simplest form or in its lowest terms.  Example 5.3 question Write each fraction in its lowest terms. a  6 10  b  18 24  Solution a  The highest common factor of 6 and 10 is 2, so divide the numerator and denominator by 2. 6 3 = 10 5  Note  in its lowest terms.  18 and 24 also have common factors of both 9 2 and 3, so 12 and 68 are also equivalent to 18 , 24 3 but 4 is the simplest form of this fraction.  b The highest common factor of 18 and 24 is 6, so divide the numerator and denominator by 6. 18 3 = 24 4  in its lowest terms.  exercise 5.3 1 Fill in the missing numbers in each set of equivalent fractions. a  1 5 = = = 4 8 12  b  1 7 = = = 5 10 20  c  2 4 12 = = = 5 25  d  2 4 6 = = = 9 36  e  2 1 = = 7 35  f  4 16 = = 9 72  2 Express each fraction in its lowest terms. a  8 10  b  2 12  c  15 21  d  e  14 21  f  25 30  g  20 40  h  i  16 24  j  k  20 120  l  500 1000  m  56 70  n  150 300 64 72  o  60 84  p  120 180  12 16 18 30  3 A bag contains 96 balls. 36 of the balls are red. What fraction of the balls are red? Give your answer in its lowest terms. 24  Photocopying prohibited  Fractions and decimals  4 Akbar drove 64 km of a 120 km journey on motorways. What fraction of the journey did he drive on motorways? Give your answer in its lowest terms. 5 The table gives information about the members of a club. male  female  adult  90  60  Child  55  45  Answer these questions, giving each answer as a fraction in its lowest terms. a What fraction of the members are adult males? b What fraction of the members are female? c What fraction of the members are children?  fractions and decimals We can use place value in a decimal to convert the decimal to a fraction.  Example 5.4 question Convert the decimal 0.245 to a fraction in its lowest terms.  Note  Solution Units  .  Tenths  hundredths  Thousandths  0  .  2  4  5  0.245 =  5 2 4 + + 10 100 1000  You can use the place value of the final digit to write the decimal directly as a single fraction. The final digit here, 5, represents 245 thousandths so 0.245 is equivalent to 1000 .  5 40 200 = 1000 + 1000 + 1000 Convert each fraction to its equivalent with a denominator of 1000. 245 = 1000  =  The HCF of 245 and 1000 is 5, so divide the numerator and the denominator by 5 to simplify the fraction.  49 200  0.245 is equivalent to  49 200  exercise 5.4 Convert each of these decimals to a fraction in its lowest terms. 1 0.7 6 0.056  2 0.29 7 0.008  Photocopying prohibited  3 0.85 8 0.02  4 0.07 9 0.545  5 0.312 10 0.1345  25  5 VUlgaR aND DeCImal fRaCTIONS aND peRCeNTageS  Terminating and recurring decimals In Chapter 1, you learnt that both terminating and recurring decimals were rational numbers. So both terminating and recurring decimals can be written as fractions. Also all fractions can be written as a terminating or a recurring decimal. You can convert the fraction 5 8  = 5 ÷ 8 = 0.625  5 8  to a decimal using division.  This is a terminating decimal because it finishes at the digit 5. You can convert the fraction 1 = 1 ÷ 6 = 0.166 666...  1 6  to a decimal using division.  6  This is a recurring decimal because the digit 6 repeats indefinitely.  Dot notation for recurring decimals Dot notation can be used when writing recurring decimals. Dots are placed over the digits that recur. For example, 1  3 = 0.333 333… is written using dot notation as 0.3. 1 0.16 . 6 = 0.166 666… is written using dot notation as 124   999 = 0.124 124 124… is written using dot notation as 0.124.  Example 5.5 question Convert  13 25  to a decimal.  b Convert  7 11  to a decimal.  a c  Convert 0.6̇ to a fraction using  1 3  = 0.3̇.  Solution a b  13 25 7 11 7 11  c  26  = 13 ÷ 25 = 0.52 = 7 ÷ 11 = 0.636 363… In this case the digits 6 and 3 recur so you can write the answer using dot notation.  = 0.63  0.6̇ = 2 × 0.3̇ = 2 ×  1 3  =  2 3  Photocopying prohibited  Fractions, decimals and percentages  exercise 5.5 1 Convert each of these fractions to a decimal. 5 3 a 83 b 16 c 11 d 79 e 80 40 250 2 Convert each of these fractions to a recurring decimal. Write your answers using dot notation. 1 a 23 b 65 c 12 d 4 15 3 Given that 0.1 = 1 , write each of these as a fraction.  a 0.2 4 Given that of these. a  2 27  9  1 27  b 0.3   and = 0.037 b  e  16 33  c 0.5 find the decimal equivalent of each  1  , = 0.09 11  5 27  c  10 27  d  2 11  e  6 11  fractions, decimals and percentages The term per cent means 'out of 100'. 75  For example 75% means 75 out of every 100 or 100 . 75 100  can be written in decimal form as 0.75. 75  So 75% is equivalent to 100 and 0.75. You can find fraction and decimal equivalents of all percentages. There are some fraction, decimal and percentage equivalents that are useful to remember. fraction  Decimal  percentage  1 2  0.5  50%  1 4  0.25  25%  3 4  0.75  75%  1 10  0.1  10%  1 5  0.2  20%  Photocopying prohibited  27  5 VUlgaR aND DeCImal fRaCTIONS aND peRCeNTageS  Example 5.6 question a  Convert  3 8  b Convert 65% to a fraction in its lowest terms.  to a percentage.  Solution a  3 8  = 3 ÷ 8 = 0.375  Convert to a decimal by dividing.  Note  0.375 × 100 = 37.5 Multiply by 100 for percentage. So  3 8  You can convert directly from a fraction to a percentage using multiplication.  = 37.5%  b 65% = 65% =  65 100 13 20  The HCF of 65 and 100 is 5, so divide the numerator and the denominator by 5 to simplify the fraction.  exercise 5.6 1 Convert each of these percentages to a fraction. Write your answers in their lowest terms. a 35% b 65% c 8% d 120% 2 Convert each of these percentages to a decimal. a 16% b 27% c 83% d 7% e 31% f 4% g 17% h 2% i 150% j 250% k 9% l 12.5% 3 Convert each of these fractions to a decimal. a  1 100  b  17 100  c  2 50  d  8 5 5 16  e  1 8  5 g 3 h 17 i f 8 40 20 4 Convert each of the decimals you found in question 3 to a percentage.  5 Convert each of these fractions to a percentage. Give your answers correct to 1 decimal place. 1 a 1 b 65 c 12 d 6 6 Write three fractions that are equivalent to 40%.  5 12  e  3 70  7 Write three fractions that are equivalent to 0.125. 8 a Convert 160% to a decimal. b Convert 160% to a mixed number in its lowest terms. 9 The winning party in an election gained 7 of the votes. 12 What percentage is this? Give your answer correct to the nearest 1%. 10 Imran saves 92 of his earnings. What percentage is this? Give your answer correct to the nearest 1%. 11 In a survey about the colour of cars, 22% of the people said they 3 preferred red cars, 20 of the people said they preferred silver cars 6 and said they preferred black cars. 25  Which colour car was most popular? Show how you decide. 12 In class P, 73 of the students are boys. In class Q, 45% of the students are boys. Which class has the higher proportion of boys? Show how you decide. 28  Photocopying prohibited  6  oRDeRIng  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, , .  Ordering integers Example 6.1 question Put these masses in order, smallest first. 1.2 kg  1500 g  175 g  2 kg  0.8 kg  Solution  CHECK YOU CAN: l  l l  l  l  use the symbols = and ≠ correctly order whole numbers work with negative numbers convert between metric units: l m, cm and mm l kg and g l litres (l), cl and ml convert between fractions, decimals and percentages.  Write each value with the same units first. (It is usually easier to use the smallest unit.) 1200 g  1500 g  175 g  2000 g  800 g  Then order them, writing your answer using the original units. So order is 175 g  0.8 kg  1.2 kg  1500 g  2 kg.  exercise 6.1 1 Write each set of temperatures in order, lowest first. a −2 °C 7 °C 0 °C −5 °C 3 °C b −2 °C 5 °C 1 °C 2 °C −1 °C c 7 °C −7 °C 4 °C −9 °C −3 °C d 9 °C 4 °C −2 °C 7 °C −8 °C e −4 °C 5 °C −2 °C 3 °C −7 °C 2 Write each set of lengths in order of size, smallest first. a 2.42 m 1600 mm 284 cm 9m 31 cm b 423 cm 6100 mm 804 cm 3.2 m 105 mm 3 Write each set of masses in order of size, smallest first. a 4000 g 52 000 g 9.4 kg 874 g 1.7 kg b 4123 g 2104 g 3.4 kg 0.174 kg 2.79 kg 4 Write each set of capacities in order of size, smallest first. a 2.4 litres 1600 ml 80 cl 9 litres 51 cl b 3.1 litres 1500 ml 180 cl 1 litre 51.5 ml  Inequalities a < b means 'a is less than b'. a  b means 'a is less than or equal to b'. a > b means 'a is greater than b'. a  b means 'a is greater than or equal to b'.  Photocopying prohibited  29  6 ORDeRINg  exercise 6.2 1 Rewrite these: insert > or < in each part as appropriate. a 7 °C is … than 2 °C. b −10 °C is … c −3 °C is … than 0 °C. d −7 °C is … e 10 °C is … than 0 °C. f −2 °C is … g 4 °C is … than −1 °C. h −2 °C is …  than 5 °C. than −12 °C. than −5 °C. than 2 °C.  Ordering decimals Example 6.2 question Put these decimals in order of size, smallest first. 0.412  0.0059  0.325  0.046  0.012  Solution Add zeros to make the decimals all the same length. 0.4120  0.0059  0.3250  0.0460  0.0120  Now, remove the decimal points and any zeros in front of the digits. 4120  59  3250  460  120  Note As you get used to this, you can omit the second step.  The order in size of these values is the order in size of the decimals. The order is 0.0059  0.012  0.046  0.325  0.412  exercise 6.3 1 Put these numbers in order of size, smallest first. a 462, 321, 197, 358, 426, 411 b 89 125, 39 171, 4621, 59 042, 6317, 9981 c 124, 1792, 75, 631, 12, 415 d 9425, 4257, 7034, 5218, 6641, 1611 e 1 050 403, 1 030 504, 1 020 504, 1 040 501, 1 060 504, 1 010 701 2 Put these decimals in order of size, smallest first. a 0.123, 0.456, 0.231, 0.201, 0.102 b 0.01, 0.003, 0.1, 0.056, 0.066 c 0.0404, 0.404, 0.004 04, 0.044, 0.0044 d 0.71, 0.51, 0.112, 0.149, 0.2 e 0.913, 0.0946, 0.009 16, 0.090 11, 0.091 3 Put these numbers in order of size, smallest first. a 3.12, 3.21, 3.001, 3.102, 3.201 b 1.21, 2.12, 12.1, 121, 0.12 c 7.023, 7.69, 7.015, 7.105, 7.41 d 5.321, 5.001, 5.0102, 5.0201, 5.02 e 0.01, 12.02, 0.0121, 1.201, 0.0012 f 8.097, 8.79, 8.01, 8.1, 8.04  30  Photocopying prohibited  Ordering fractions  Ordering fractions To put fractions in order, convert them to equivalent fractions all with the same denominator, and order them by the numerator.  Example 6.3 Note  question Which is the bigger,  3 4  or  Multiplying the two denominators together will always work to find a common denominator, but the lowest common multiple is sometimes smaller.  5 ? 6  Solution First find a common denominator. 24 is an obvious one, as 4 × 6 = 24, but a smaller one is 12. 3 = 9 5 = 10 10 12  is bigger than  9 , 12  so  4 12 6 12 5 is bigger than 34 . 6  Alternatively, you can convert each fraction to a decimal and compare the decimals as before.  Example 6.4 question Put these fractions in order, smallest first. 3 10  1 4  9 20  2 5  1 2  Solution Use division to convert the fractions to decimals (3 ÷ 10, 1 ÷ 4, etc.). 0.3  0.25  0.45  0.4  0.5  Now make the number of decimal places the same. 0.30  0.25  So order is  0.45  1 4  3 10  0.40 2 5  9 20  0.50 1 . 2  exercise 6.4 1 Write each pair of fractions, inserting > or < as appropriate. a  2 7 … 3 9  b  5 7 … 6 8  c  3 7 … 8 20  d  3 5 … 4 8  e  7 5 … 9 6  f  3 4 … 10 15  2 Write each of these sets of fractions in order, smallest first. a  7 10  3 11 3 4 20 5  d  13 5 3 7 1 16 8 4 16 2  Photocopying prohibited  b  7 3 7 5 12 4 8 6  c  13 2 3 2 1 15 3 10 5 2  e  2 1 9 17 3 5 2 20 40 8  f  11 7 3 17 16 8 4 32  Note Try both methods shown in the examples. The first is useful when calculators are not allowed.  31  6 ORDeRINg  Ordering fractions, decimals and percentages Example 6.5 question Put these numbers in order, smallest first. 1 4  3%  11 40  0.41  0.35  A common error is to write 3% as 0.3, rather than 0.03.  Solution Convert them all to decimals. 0.25,  0.03,  Note  0.41,  0.275,  0.35  0.275  0.35  0.41.  So the order is 0.03  0.25  Write the numbers in their original form. 3%  11 40  1 4  0.35  0.41  exercise 6.5 1 Put these numbers in order, smallest first. 7 20 10  4 , 5  88%, 0.83, 17 ,  3 , 8  35%, 0.45, 2 ,  3 , 5  30%, 0.7, 3 ,  2 Put these numbers in order, smallest first. 5 5 12  3 Put these numbers in order, smallest first. 2 4 3  4 Soccer teams United, City and Rovers have all played the same number of games. United have won 3 of the games they have played, City have won 35% 8 and Rovers have won 0.4. Put the teams in order of the number of matches they have won. 5 A group of boys were asked to name their favourite sport. 2 chose football, 0.27 chose rugby and 28% chose gymnastics. 7 List the sports in the order of their popularity, the most popular first.  32  Photocopying prohibited  7  stAnDARD FoRM  Standard form  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO:  Standard form is a way of making very large numbers and very small numbers easy to deal with.  l  In standard form, numbers are written as a number between 1 and 10 multiplied by a power of 10, for example 6.3 × 105.  use the standard form A × 10n where n is a positive or negative integer, and 1  A < 10.  Large numbers CHECK YOU CAN:  Example 7.1  l  Write these numbers in standard form.  question  Solution  a  a  500 000  500 000 = 5 × 100 000 = 5 × 105  b 6 300 000  b 6 300 000 = 6.3 × 1 000 000 = 6.3 × 106  c  c  45 600  understand and use index notation.  Note You can write down the answer without any intermediate steps.  45 600 = 4.56 × 10 000 = 4.56 × 104  Move the decimal point until the number is between 1 and 10. Count the number of places the point has moved: that is the power of 10.  Small numbers Example 7.2  Note  question  You can write down the answer without any intermediate steps.  Write these numbers in standard form. a  0.000 003 b 0.000 056 c 0.000 726  Move the decimal point until the number is between 1 and 10.  Solution a  0.000 003 =  b 0.000 056 = c  0.000 726 =  1 3 = 3 × 1000000 1000000 1 5.6 = 5.6 × 100000 100000 7.26 10000  Photocopying prohibited  = 7.26 ×  1 10000  =3×  1 106  = 5.6 ×  = 3 × 10 −6 1  105 1  = 7.26 ×  = 5.6 × 10 −5  104  = 7.26 × 10−4  Count the number of places the point has moved, put a minus sign in front and that is the power of 10.  33  7 STaNDaRD fORm  exercise 7.1 1 Write these numbers in standard form. a 7000 b 84 000 c 563 d 6 500 000 e 723 000 f 27 g 53 400 h 693 i 4390 j 412 300 000 k 8 million l 39.2 million 2 Write these numbers in standard form. a 0.003 b 0.056 c 0.0008 d 0.000 006 3 e 0.000 082 f 0.0060 g 0.000 000 38 h 0.78 i 0.003 69 j 0.000 658 k 0.000 000 000 56 l 0.000 007 23 3 These numbers are in standard form. Write them as ordinary numbers. a 5 × 104 b 3.7 × 105 c 7 × 10−4 d 6.9 × 106 −3 4 7 e 6.1 × 10 f 4.73 × 10 g 2.79 × 10 h 4.83 × 10 −5 i 1.03 × 10 −2 j 9.89 × 108 k 2.61 × 10 –6 l 3.7 × 102 3 −4 −7 m 3.69 × 10 n 6.07 × 10 o 5.48 × 10 p 1.98 × 109 4 A billion is a thousand million. In 2015 the population of the world was approximately 7.2 billion. Write the population of the world in standard form.  Calculating with numbers in standard form When you need to multiply or divide numbers in standard form you can use your knowledge of the laws of indices.  Example 7.3  Note  question Work out these. Give your answers in standard form. a  (7 × 103) × (4 × 104)  b (3 × 108) ÷ (5 × 103)  Solution a  (7 × 103) × (4 × 104) = 7 × 4 × 103 × 104 = 28 × 107 = 2.8 × 108  b (3 × 108) ÷ (5 × 103) =  3 × 108 5 × 103  103 × 104 = 1000 × 10 000 = 10 000 000 = 107  Note 3 5 108 103  = 0.6 =  and  100000 000 1000  = 100 000 = 105  = 0.6 × 105 = 6 × 104  34  Photocopying prohibited  Calculating with numbers in standard form  When you need to add or subtract numbers in standard form it is much safer to change to ordinary numbers first.  Example 7.4 question Work out these. Give your answers in standard form. a  (7 × 103) + (1.4 × 104)  b (7.2 × 105) + (2.5 × 104) c  (5.3 × 10–3) – (4.9 × 10–4)  Solution a  7 000 + 14 000 21 000 = 2.1 × 104  b  720 000 + 25 000 745 000 = 7.45 × 105  c  0.005 30 – 0.000 49 0.004 81 = 4.81 × 10 −3  exercise 7.2 1 Work out these. Give your answers in standard form. a (4 × 103) × (2 × 104) b (6 × 107) × (2 × 103) 3 2 c (7 × 10 ) × (8 × 10 ) d (9 × 107) ÷ (3 × 104) 3 4 e (4 × 10 ) × (1.3 × 10 ) f (4.8 × 103) ÷ (1.2 × 10 –2) 6 –2 g (8 × 10 ) × (9 × 10 ) h (4 × 108) ÷ (8 × 102) –4 –3 i (7 × 10 ) × (8 × 10 ) j (5 × 10 –5) ÷ (2 × 104) 3 4 k (4 × 10 ) + (6 × 10 ) l (7 × 106) – (3 × 103) 5 4 m (6.2 × 10 ) – (3.7 × 10 ) n (4.2 × 109) + (3.6 × 108) 6 4 o (7.2 × 10 ) – (4.2 × 10 ) p (7.8 × 10 –5) + (6.1 × 10 –4) 2 Work out these. Give your answers in standard form. a (6.2 × 105) × (3.8 × 107) b (6.3 × 107) ÷ (4.2 × 102) 8 –3 c (6.67 × 10 ) ÷ (4.6 × 10 ) d (3.7 × 10 –4) × (2.9 × 10 –3) 8 3 e (1.69 × 10 ) ÷ (5.2 × 10 ) f (5.8 × 105) × (3.5 × 103) 6 2 g (5.2 × 10 ) h (3.1 × 10 –4)2 6 4 i (3.72 × 10 ) – (2.8 × 10 ) j (7.63 × 105) + (3.89 × 104) –3 –4 k (5.63 × 10 ) – (4.28 × 10 ) l (6.72 × 10 –3) + (2.84 × 10 –5) –5 –3 m (4.32 × 10 ) – (4.28 × 10 ) n (7.28 × 108) + (3.64 × 106)  Photocopying prohibited  35  8  tHe FoUR oPeRAtIons Calculating with negative numbers  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  use the four operations for calculations with whole numbers, decimals and vulgar (and mixed) fractions, including correct ordering of operations and use of brackets.  You should know the rules for working with negative numbers. These are covered in detail in Chapter 18 but summarised here. Adding a negative number is the same as subtracting a positive number. 5 + −7 is the same as 5 − 7 so 5 + −7 = −2 Subtracting a negative number is the same as adding a positive number. 5 − −7 is the same as 5 + 7 so 5 − − 7 = 12 When you multiply or divide two negative numbers, the result is positive.  CHECK YOU CAN: l  l  l  add, subtract, multiply and divide integers without a calculator add and subtract decimals without a calculator find equivalent fractions.  −2 × −7 = 14 −15 ÷ −3 = 5 When you multiply or divide one positive number and one negative number, the result is negative. 2 × −7 = −14 −15 ÷ 3 = −5 You can extend the rules for multiplying and dividing to calculations with more than two numbers. If the number of negative signs is even, the result is positive. If the number of negative signs is odd, the result is negative.  Example 8.1 question Work out these. a  4−5+3−6  b −7 × −7  c  −3 × −6 ÷ −2  Solution a  4−5+3−6 =4+3−5−6  First collect the numbers to add and the numbers to subtract.  = 7 − 11  Find separate totals for each.  = −4  Finally do the subtraction, making sure that the sign is correct.  b −7 × −7 =7×7 = 49  36  There are two negative signs, so the answer will be positive. This calculation is equivalent to the line above.  ➜ Photocopying prohibited  Order of operations  c  −3 × −6 ÷ −2  There are three negative signs, so the answer will be negative.  = 18 ÷ −2  This calculation is equivalent to the line above.  = −9  The answer to part b is an important result. It shows that the square root of 49 can be negative as well as positive. All positive numbers have two square roots.  exercise 8.1 Work out these. 1 4 7 10 13 16 19  4+3−2−1 9−2−3+2 2−3+4−7 −4 − 3 + 2 + 7 − 5 − 1 24 ÷ −6 6 × 10 ÷ −5 6 × −4 × 3 ÷ −2  2 6−3−5+4 5 −4 + 2 − 2 + 5 8 2−3−5−6+4+1 11 −5 × 4 14 −32 ÷ 4 17 −3 × −4 ÷ −6 20 −1 × −2 × −3 × −4 × −5  3 4+3+2−5 6 4−1−3−4 9 7+3+2−5−4+3 12 −6 × −5 15 −45 ÷ −5 18 −84 ÷ −12 × −3  Order of operations If you are asked to work out the answer to a calculation involving more than one operation, it is important that you use the correct order of operations. For example, if you want to work out 2 + 3 × 4, should you carry out the addition or the multiplication first? This is the correct order of operations when carrying out any calculation: l  first work out anything in brackets  l  then work out any powers (such as squares or square roots)  l  then do any multiplication or division  l  finally do any addition or subtraction.  So to work out 2 + 3 × 4, you should first do the multiplication, then the addition. 2 + 3 × 4 = 2 + 12 = 14 If you want the addition to be done first, then brackets are needed in the calculation. (2 + 3) × 4 = 5 × 4 = 20  Some calculations are written like fractions, for example 6 × 4 . 5+3 In this case the fraction line works in the same way as brackets. First evaluate the numerator, then the denominator and then do the division. 6 × 4 = 24 = 3 5+3 8 This calculation could also be written as 6 × 4 ÷ (5 + 3). Photocopying prohibited  37  8 The fOUR OpeRaTIONS  Example 8.2 question Work out these. a  7+4÷2−5  b (6 − 2)2 + 5 × −3  Solution a  7+4÷2−5 =7+2−5  First work out the division.  =4  The addition and subtraction can be done in a single step.  b (6 − 2)2 + 5 × −3 = 42 + 5 × −3  First work out the brackets.  = 16 + 5 × −3  Next work out the power.  = 16 + −15  Then the multiplication.  =1  And finally the addition.  exercise 8.2 Work out these. 1 2×5+4 4 2 + 52 7 (13 − 2) × 5  2 2 × (5 + 4) 5 5 × 42 8 2 × 33  10 (5 − 3) × (8 − 3)  11 6 +  12 4×3 16 20 5+3 19 (20 − 2 × 6)2  14  12 4  ×3  17  20 5  +3  13  22 3 × 23  8 2  20 12 − 6 × 3 23 −7 × −12 −8 + 4  9×4 26 4 × 2 − 5 × −4 −3 + −6 28 Hassan says that 12 ÷ 2 + 4 = 2. a Explain what he has done wrong. b Work out the correct answer. 29 Aisha says that 3 × 22 = 36. a Explain what she has done wrong. b Work out the correct answer. 25  38  3 2+5×4 6 13 − 2 × 5 9 (2 × 5)2 12 6 + 8 2 6 + 4 −2 15 5 18 3 × 5 − 6 × 4 8 21 4 × 3 − 5 × 4 24 8 × −6 −4 + −8 27 4 + 5 × −2 − 6  Photocopying prohibited  Multiplying integers  30 Write out each of the following calculations, with brackets if necessary, to give the answers stated. a 3 + 6 × 5 − 1 to give i 44 ii 32 iii 27 b 6 + 42 − 16 ÷ 2 to give i 6 ii 3 iii 92 c 12 − 8 ÷ 4 + 4 to give i 11 ii 5 iii 14 d 18 + 12 ÷ 6 − 3 to give i 2 ii 17 iii 22 iv 10  multiplying integers There are a number of methods that can be used for multiplying three-digit by two-digit integers. These methods can be adapted for use with integers of any size.  Example 8.3 question Work out 352 × 47.  Solution Method 1: Long multiplication 3  5  ×  2  4  7  1  4 2  0 4  8 6  0 4  1  6  5  4  4  (352 × 40) (352 × 7)  Method 2: Grid method ×  300  50  2  40  12 000  2 000  80  7  2 100  350  14  = 14 080 = 2 464 16 544  exercise 8.3 Work out these. 1 138 × 13 2 581 × 23 3 614 × 14 4 705 × 32 5 146 × 79 6 615 × 46 7 254 × 82 8 422 × 65 9 428 × 64 10 624 × 75 11 A theatre has 42 rows of seats. There are 28 seats in each row. How many seats are there altogether? 12 A packet contains 36 pens. A container holds 175 of these packets. How many pens are in the container? Photocopying prohibited  39  8 The fOUR OpeRaTIONS  multiplying decimals Use this method for multiplying two decimal numbers: l  count the number of decimal places in the numbers you are multiplying  l  ignore the decimal points and do the multiplication  l  put a decimal point in your result so that there are the same number of decimal places in your answer as in the original calculation.  Example 8.4 question Work out these. a  0.4 × 0.05  b 4.36 × 0.52  Solution a  0.4 × 0.05  There are three decimal places in the calculation, so there will be three in the answer.  4 × 5 = 20  Multiply the figures, ignoring the decimal points.  0.4 × 0.05 = 0.020  Insert a decimal point so that there are three decimal places in the answer. In this case we need to add an extra zero between the decimal point and the 2.  0.4 × 0.05 = 0.02 b 4.36 × 0.52  There are four decimal places in the calculation, so there will be four in the answer.  436 × 52 = 22 672  Multiply the figures, ignoring the decimal points. Use whichever method you prefer.  4.36 × 0.52 = 2.267 2  Insert a decimal point so that there are four decimal places in the answer.  exercise 8.4 1 Given that 63 × 231 = 14 553, write down the answers to these. a 6.3 × 2.31 b 63 × 23.1 c 0.63 × 23.1 d 63 × 0.231 e 6.3 × 23 100 2 Given that 12.4 × 8.5 = 105.4, write down the answers to these. a 124 × 8.5 b 12.4 × 0.85 c 0.124 × 8.5 d 1.24 × 8.5 e 0.124 × 850 3 Work out these. a 0.3 × 5 b 0.6 × 0.8 c 0.2 × 0.4 d 0.02 × 0.7 e 0.006 × 5 f 0.07 × 0.09 4 Work out these. a 42 × 1.5 b 5.9 × 6.1 c 10.9 × 2.4 d 2.34 × 0.8 e 5.46 × 0.7 f 6.23 × 1.6 5 Work out these. a 0.5 + 0.2 × 0.3 b 0.1 × 0.8 − 0.03 c 0.2 + 0.42 d 0.9 × 0.8 − 0.2 × 0.7  40  Photocopying prohibited  Dividing decimals  Dividing integers There are a number of methods that can be used for dividing three-digit by two-digit integers. These methods can be adapted for use with integers of any size.  Example 8.5 question Work out 816 ÷ 34.  Solution Method 1: Long division  Method 2: Chunking  816 ÷ 34 = 24  816 ÷ 34 = 24  24 34 816 – 680  (34 × 20)  136 – 136  816 – 340  (34 × 10)  476 – 340  (34 × 10)  136 136 0  (34 × 4)  0 The answer is 24.  (34 × 4)  The answer is 10 + 10 + 4 = 24.  exercise 8.5 Work out these. 1 987 ÷ 21 2 684 ÷ 18 3 864 ÷ 16 4 924 ÷ 28 6 352 ÷ 22 7 855 ÷ 45 8 992 ÷ 31 9 918 ÷ 27 11 Eggs are packed in boxes of 12. How many boxes are needed for 828 eggs? 12 A group of 640 people are going on a bus trip. Each bus can carry 54 people. How many buses are needed?  5 544 ÷ 32 10 576 ÷ 18  Dividing decimals Use this method for dividing by a decimal: l l  l  l  first write the division as a fraction make the denominator an integer by multiplying the numerator and the denominator by the same power of 10 cancel the fraction to its lowest terms if possible; this gives you easier numbers to divide divide to find the answer.  Photocopying prohibited  41  8 The fOUR OpeRaTIONS  Example 8.6 question Work out these. a  b 25.7 ÷ 0.08  0.9 ÷ 1.5  Solution a  0.9 ÷ 1.5 =  0.9 1.5  Write the division as a fraction.  =  9 15  Make the denominator an integer by multiplying both the numerator and the denominator by 10.  =  3 5  3 5  =5  Simplify. 0.6 3.0  Divide, adding an extra zero after the decimal point to complete the division.  0.9 ÷ 1.5 = 0.6 b 25.7 ÷ 0.08 =  25.7 0.08  Write the division as a fraction.  =  2570 8  Make the denominator an integer by multiplying both the numerator and the denominator by 100.  =  1285 4  Simplify.  1285 4  =  321.25 4 1285.00  Use whichever method you prefer for the division. Divide, adding extra zeros after the decimal point to complete the division.  25.7 ÷ 0.08 = 321.25  exercise 8.6 1 Given that 852 ÷ 16 = 53.25, write down the answers to these. a 852 ÷ 1.6 b 8.52 ÷ 16 c 85.2 ÷ 1.6 d 85.2 ÷ 0.16 e 852 ÷ 0.016 2 Given that 482 ÷ 25 = 19.28, write down the answers to these. a 4.82 ÷ 2.5 b 0.482 ÷ 2.5 c 48.2 ÷ 0.25 d 4.82 ÷ 250 e 4.82 ÷ 0.025 3 Work out these. a 8 ÷ 0.2 b 1.2 ÷ 0.3 c 5.6 ÷ 0.7 d 9 ÷ 0.3 e 15 ÷ 0.03 f 6.5 ÷ 1.3 4 Work out these. a 1.55 ÷ 0.05 b 85.8 ÷ 0.11 c 5.55 ÷ 1.5 d 0.68 ÷ 1.6 e 87.6 ÷ 0.24 f 1.35 ÷ 1.8 5 Work out these. a 0.6 + 0.2 ÷ 0.1 b 0.5 + 0.7 0.3 × 0.2 1.8 c d 2.6 × 2 0.5 0.32 0.1  42  Photocopying prohibited  Adding and subtracting fractions  adding and subtracting fractions In the diagram each rectangle is divided into 20 small squares.  +  The diagram shows  =  1 4  + 2. 5  The result of the addition has 13 squares shaded or  13 . 20  Use this method to add two fractions:  1 4 1 4  l  change the fractions to equivalent fractions with the same denominator  l  add the numerators.  = +  5 =and = 2= 8 20 5 20 2 5 8 = = 13 + 5 20 20 20  The method is the same for subtracting two fractions. Change them to equivalent fractions with the same denominator then subtract the numerators. If the fractions are mixed numbers, change them to improper fractions first.  Example 8.7 question Work out these. a  2 3  +  5 6  b  3 4  –  1 3  c  2 3 + 11 5  3  d  3 1 – 15 4  6  Solution a  b  2 3  3 4  +  –  5 6  1 3  =  4 6  =  9 6  =  3 2  =  9 12  =  5 12  +  5 6  Write the fractions with a common denominator. The lowest common multiple of 3 and 6 is 6, so use 6 as the common denominator. The result is an improper fraction and not in its lowest terms.  = 121 −  4 12  Photocopying prohibited  Simplify the fraction and write it as a mixed number. The lowest common multiple of 4 and 3 is 12, so use 12 as the common denominator.  ➜  43  8 The fOUR OpeRaTIONS  c  2 3 + 11 =  13 5  =  39 15  =  59 15  5  3  +  4 3  First change the mixed numbers to improper fractions.  +  20 15  The lowest common multiple of 5 and 3 is 15.  =  3 14 15  d 3 1 − 15 = 13 − 11 4  6  4  The result is an improper fraction, so write it as a mixed number. First change the mixed numbers to improper fractions.  6  =  39 12  −  =  17 12  = 112  22 12 5  The lowest common multiple of 4 and 6 is 12. The result is an improper fraction, so write it as a mixed number.  Note You should always give your final answer as a fraction in its lowest terms.  exercise 8.7 1 Add these fractions. 4 7  +  1 6  c  2 3  +  1 4  d 5+4 e 8+ 2 Subtract these fractions.  1 5  f  3 4  +  2 5  1 3  c  2 3  −  1 4  a  2 7  +  2 7  3  3  1  a  1 3  b  −  11  1 7  5 6  b  2  −  5  1  7  5  d 12 − 3 e 8−3 f 9 − 12 3 Add these fractions. Write your answers as simply as possible. a  1 3 b 110 + 5  3 + 2 15 10  c  21 +1 1 5 10  1 d 42  +2 3 10  3 e 110 +  2 5  4 Subtract these fractions. Write your answers as simply as possible. a  51  10  –  7 10  b 15 − 6  2 3  c  11  c  6  c  72  5  −  7 10  d 11 2  −  3 4  e  2  7 4 −1 10 5  5 Work out these. a  31 + 21 2  5  b 4 7 – 13 8  4  1 5 –3 12 3  d 43 + 25  e  5 5 – 11  13  d 52 – 31  e  4  4  8  6  4  1 – 12  31  6 Work out these. a  44  47  9  +  25 6  b 47  13  –  41 2  5  –  4  7  2  Photocopying prohibited  4  Multiplying fractions  multiplying fractions Multiplying a fraction by a whole number In this diagram,  1 8  is red.  In this diagram, five times as much is red, so 1 × 5 = 5 . 8  8  This shows that to multiply a fraction by an integer, you multiply the numerator by the integer. Then simplify by cancelling and changing to a mixed number if possible.  Multiplying a fraction by a fraction In this diagram,  1 4  is red.  In this diagram, 1 of the red is dotted. This is So  1 12 1 3  3  of the original rectangle. ×  1 1 = . 4 12  To multiply fractions, multiply the numerators and multiply the denominators, then simplify if possible.  Photocopying prohibited  45  8 The fOUR OpeRaTIONS  Example 8.8 question Work out these. a  7 12  ×2  3 4  b  ×  6 7  2 2 × 15  c  3  6  Solution a  7 12  × 2= =  b  c  3 42  ×  14 12 7 6  63 7  2 2 × 15 = 3  6  Multiply the numerator by 2. Simplify the fraction and write it as a mixed number.  = 11 6  =  3 2  =  9 14  8 3  ×  =  84 3  =  4 3  =  44 9  ×  ×  3 7  Note  The result is a proper fraction in its lowest terms.  11 6  ×  Cancel the common factor of 2 in the numerator and the denominator.  Cancelling common factors before multiplying makes the arithmetic simpler. If you multiply first, you may have to cancel to give a fraction in its lowest terms.  First change the mixed numbers to improper fractions. 11 63  Cancel the common factor of 2 in the numerator and the denominator.  Multiply the numerators and multiply the denominators.  11 3  = 48 9  The result is an improper fraction, so write it as a mixed number.  exercise 8.8 Work out these. Write each answer as an integer, a proper fraction or a mixed number in its lowest terms. 1  1 2  ×4  2 7× 5  6 24 × 12 11  1 4  16  3 × 7 7 9  ×  1  2 3  1  21 4 2 × 2 6  46  1 2  7  2 3  ×4  12  2 3  ×  17  1 3  4  3 4  × 12  5  2 5  ×5  ×2  9  4 5  ×3  10  1 5  ×3  3 9× 8  4 9  3 5  13  4 × 1 9 2  14  1 × 2 3 3  15  5 × 3 6 5  1 × 5 2 6  18  3 × 5 10 11  19  2 × 5 3 8  20  3 × 5 5 12  1  2  22 1 2 × 3 3  1  2  23 4 5 × 1 3  1  2  24 3 3 × 2 5  1  2  25 3 5 × 1 3  Photocopying prohibited  Dividing fractions  Dividing fractions When you work out 6 ÷ 3, you are finding how many 3s there are in 6. When you work out 6 ÷ In this diagram,  1 3  1 3  , you are finding out how many 1 s there are in 6.  of the rectangle is red.  3  This diagram shows 6 of these rectangles. You can see that 18 of the red squares will fit into this diagram. So 6 ÷  1 3  = 6 × 3 = 18  You can see that dividing by 1 3  1 3  is the same as multiplying by 3.  is known as the reciprocal of 3.  This can be extended to division by a non-unit fraction. For example, when you work out 4 ÷ 2 , you are finding out how many 2 s 3 3 there are in 4. In this diagram,  2 3  of the rectangle is red.  This diagram shows 4 of these rectangles. You can see that 6 of these 2-square shapes will fit into this diagram. You can think of the calculation in two steps. 4÷ So  1 3  Or  = 4 × 3 = 12 (there are 12 small squares in the 4 rectangles). 4÷ 4÷  2 3 2 3  = 12 ÷ 2 = 6 =4×  3 2  =  12 2  = 6.  You can see that dividing by 2 3  2 3  is the same as multiplying by 3 .  is known as the reciprocal of 3 .  2  2  To find the reciprocal of any fraction, turn it upside down. The reciprocal of  a b  is b . a  Use this method for dividing by a fraction: l  find the reciprocal of the fraction you are dividing by  l  multiply by the reciprocal.  Photocopying prohibited  47  8 The fOUR OpeRaTIONS  Example 8.9 question Work out these. a  8 9  b  ÷2  9 10  3 4  ÷  c  2 3 ÷ 15 4  8  Solution a  8 9  ÷2  =  8 9  ÷  2 1  2 is the same as  =  8 9  ×  1 2  The reciprocal of  =  84 9  = = b  9 10  ÷  3 4  4 9 4 9  = =  c  3  1  ×  2 1 2 1  is 21 , so multiply by this reciprocal.  Cancel the common factor of 2 in the numerator and the denominator.  21  ×1  9 10  ×  93 510  4 3  ×  The reciprocal of 42 31  2 1  =  3 5  ×  =  6 5  = 15  1  3 4  Remember that when you have a fraction calculation involving mixed numbers, you should first change them to improper fractions.  Cancel the common factors of 2 and 3 in the numerator and the denominator. The result is an improper fraction, so write it as a mixed number.  24 ÷ 18 =  11 4  ÷  13 8  First change the mixed numbers to improper fractions.  =  11 4  ×  8 13  The reciprocal of  =  11 41  ×  =  22 13  = 1 13  5  Note  is 34 , so multiply by this reciprocal.  82 13  13 8  is  8 13  Always check that your final answer is given as a mixed number in its lowest terms.  , so multiply by this reciprocal.  Cancel the common factor of 4 in the numerator and the denominator. The result is an improper fraction, so write it as a mixed number.  9  exercise 8.9 Work out these. Write each answer as an integer, a proper fraction or a mixed number in its lowest terms. 1 9÷  2  4  1 3 1÷ 3 3 4 2 1 ÷3 3  5  4 ÷ 3 5 10 2 ÷ 7 5 10 3 ÷ 11 18 4 1 5 ÷ 33 3 5  11  7 10 13 16 19 48  8  +  21 3  3 4 4 9 5 6 1 4  1 2  3  ÷2  6  ÷ 10  9  ÷  ÷  3 8  12  2 3 4 5 7 9 2 5  ÷3 ÷4 ÷ ÷  1 9 1 5  14 2 1 ÷ 1 1  15 2 2 ÷ 1 1  17  18 3 1 + 2 4 × 2 1  20  3 3 21 ÷ 31 4 2 4 2 + 31 5 4  5  2  ÷  32 3  −  2 5  2  23 4 Photocopying prohibited  9  estIMAtIon  estimating lengths To estimate lengths, compare with measures you know.  BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO: l  Example 9.1 question  l  Estimate the height of the lamp post.  Solution 7.5 m  l  Any answer from 6.5 to 8.5 m would be acceptable. The lamp post is about four times the height of the man.  make estimates of numbers, quantities and lengths give approximations to specified numbers of significant figures and decimal places round off answers to reasonable accuracy in the context of a given problem.  Note The height to an adult person's waist is about 1 m (100 cm) and their full height is about 1.7 m (170 cm).  CHECK YOU CAN: l  exercise 9.1 1 Estimate a the height of the fence  l  b the length of the fence.  l  l  l  Note 2 Estimate the height of the tre

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