PDF- Numerical Reasoning Test Formulas, Numerical -Introduction - Mechanical Aptitude Tests - How to Pass Numerical Reasoning Tests a Step by Step Guide

How to Pass Numerical Reasoning Tests A step-by-step guide to learning key numeracy skills 2nd edition

Heidi Smith

Publisher’s note Every possible effort has been made to ensure that the information contained in this book is accurate at the time of going to press,

and the publishers and authors cannot accept responsibility for any errors or omissions,

- however caused

- or refraining from action,

as a result of the material in this publication can be accepted by the editor,

the publisher or any of the authors

First published in Great Britain and the United States in 2003 by Kogan Page Limited Revised edition 2006 Reprinted 2007,

- 2009 (three times) Second edition 2011 Apart from any fair dealing for the purposes of research or private study,
- or criticism or review,

as permitted under the Copyright,

this publication may only be reproduced,

- stored or transmitted,
- in any form or by any means,

with the prior permission in writing of the publishers,

or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA

Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned addresses: 120 Pentonville Road 1518 Walnut Street,

- koganpage
- 4737/23 Ansari Road Daryaganj New Delhi 110002 India

© Heidi Smith 2003,

- 2011 The right of Heidi Smith to be identified as the author of this work has been asserted by her in accordance with the Copyright,

Designs and Patents Act 1988

- 978 0 7494 6172 0 978 0 7494 6173 7

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library

- 1970– How to pass numerical reasoning tests : a step-by-step guide to learning key numeracy skills / Heidi Smith
- – 2nd ed

Mathematics– Examinations,

- questions,
- Title

- 76–dc22 2010045359 Typeset by Graphicraft Ltd,

This book is dedicated to Dr J V Armitage

- 1 Review the basics 9 Chapter topics 9 Terms used in this chapter 9 Multiplication tables 10 Dividing and multiplying numbers 12 Prime numbers 18 Multiples 19 Working with large numbers 23 Working with signed numbers 26 Averages 29 Answers to Chapter 1 34
- 2 Fractions and decimals 43 Chapter topics 43 Terms used in this chapter 43 What a fraction is 44 Working with fractions 45 Fraction operations 49 Decimal operations 58 Answers to Chapter 2 64
- 3 Rates 69 Chapter topics 69 Terms used in this chapter 69 Converting units 70

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- 4 Percentages 101 Chapter topics 101 Terms used in this chapter 101 Converting between percentages,

fractions and decimals 102 Converter tables 104 Working with percentages 106 Simple interest and compound interest 117 Answers to Chapter 4 120

- 5 Ratios and proportions 133 Chapter topics 133 Terms used in this chapter 133 Working with ratios 134 Ratios and common units of measure 136 Types of ratio 138 Using ratios to find actual quantities 140 Proportions 144 Answers to Chapter 5 149
- 6 Data interpretation 163 Data interpretation questions 164 Answers to Chapter 6 181 Explanations to Chapter 6 questions 183
- 7 Word problems 205 Approaching a word problem 206 Practice test 208 Practice test answers and explanations 209 Glossary 219 Recommendations for further practice 223 Further reading from Kogan Page 227

he success of the First Edition of How to Pass Numerical Reasoning Tests demonstrates that there are many people who,

despite a high level of experience and competence in other areas of their professional life,

are not confident about passing tests that involve numbers

what is not commonly known is that with practice it is possible to improve your numerical reasoning score dramatically

This book was designed with the adult test-taker in mind – someone who is looking for some extra practice material prior to taking a test

percentages – ‘what are the three parts I have to know in order to work out the answer

decimals – ‘what was that quick way to multiply them

workrates – ‘wasn’t there something about adding the total time together and dividing by something

prime numbers and multiples – ‘I remember I can use these to help me,

but can’t quite remember how’

interest rates – ‘what were the two different types and how do I use each of them

This workbook reminds the non-mathematician what these terms are and how to work with them quickly

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The First Edition of How to Pass Numerical Reasoning Tests has helped thousands of applicants prepare for their test

Better preparation for psychometric testing leads to increased confidence during the initial assessment and this,

- undoubtedly,

gives you a better chance of getting that job

Introduction Numerical reasoning tests in context Standardized testing is becoming more popular as a means to assess candidates at an early stage of the job application process,

particularly in organizations where the demand for jobs is high

the chances are that you are facing a numerical reasoning test in the very near future

It is likely that you know that you are perfectly capable of doing the job for which you have applied,

and that the only obstacle between you and the next round of interviews is a set of tests that includes a numerical reasoning section

so it is unsurprising that you are not looking forward to the experience

Take heart though

The good news is that the numerical knowledge you need to do well in these tests is the maths you learnt in school

The numerical reasoning section of an aptitude test is the section where many people find they can improve their score,

so it’s worth dedicating a decent amount of test preparation time to this area

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and its purpose is to provide you with the necessary skills to perform well in your numerical reasoning test

There is no magic formula for improving your performance in numerical reasoning tests

- however,

performance is determined by a number of factors aside from basic intelligence

and the secret is to practise as much as you can

it is likely that a quick reminder of the method will help you complete the calculation within the time allowed in the test

Test preparation and good exam technique gives you the confidence to estimate correct answers quickly

practice questions and explanations

What this workbook doesn’t do The content of this workbook is aimed at adult test-takers who want to prepare to take a numerical reasoning test

It is assumed that you don’t necessarily want,

- or have time,
- to become a mathematician,

but you do want to relearn enough maths to help you to do well in the test

This workbook is designed to help you to prepare the required numeracy skills for the standardized aptitude tests currently on the market

This workbook will not explain why mathematical formulae work the way they do

Rather,

it explains how to apply the formula in a practical setting,

in particular in the application of maths in numeracy tests

you would be already immersed in advanced level maths books and wouldn’t need this refresher course

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works through an example and provides you with practice questions to help consolidate your learning

Chapter 1,

- ‘Review the basics’,

guides you through the fundamentals and gives you the opportunity to get started with numeracy questions

- ‘Fractions and decimals’,

explains how to work with parts of whole numbers expressed either as fractions or decimals

- subtract,

multiply and divide fractions and decimals

- ‘Rates’,

reminds you of the formulae to work out speed,

distance and time when you know two (or more) variables

Chapter 4,

- ‘Percentages’,

covers the three variables in a typical percentage question

the whole and the percentage are all explained

Percentage increases and decreases are also explained,

and you will practise questions involving simple and compound interest

- ‘Ratios and proportions’,

examines ‘part to part’ and ‘part to whole’ type ratios and explains how to use ratios to find actual quantities

Chapter 6,

- ‘Data interpretation’,

takes all the skills you have practised and combines them to allow you to practise applying your new skills to data presented in graphs,

- tables and charts

Chapter 7,

- ‘Word problems’,

is brand new to this edition and shows how your analytical skills are under scrutiny just as much as your numerical skills when you are presented with a word problem

The ‘Glossary of terms and formulae’ section lists all the terms and formulae you have used throughout the book and serves as a reference guide

The ‘Recommendations for further practice’ section identifies a number of useful publications and websites for you to use for further practice prior to your test

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an explanation is offered to solve each of the problems

As you work through more and more examples,

you will discover the methods you find easiest to work with,

- so do try other methods,
- either to check your answer,

or to find out whether there is a quicker way to solve the problem

Remember that speed is a key indicator of success

How to use this workbook This workbook is designed to teach the foundation skills relevant to a number of numerical reasoning tests currently on the market

It is progressive and you should work through each of the chapters in order

Concepts explained in the earlier chapters are used and tested again in later chapters

It is good practice for employers and recruiters to send you sample questions prior to the test,

in order that you know what to expect on the day

you can use this workbook to practise the areas most relevant to you if time is short

It is recommended that you work thoroughly through Chapter 1 regardless of the type of test you are taking,

as Chapter 1 provides you with the skills to build a numerical foundation relevant to subsequent chapters

Preparation and test technique Being prepared for the test If you don’t know what kind of questions to expect in your test,

there are two things you can do

call the tester or recruiter and ask for a sample practice booklet

It is good practice for recruiters to distribute practice tests to prepare you for the type of tests you will be taking

while you are waiting for the practice questions to arrive,

- practise any sort of maths

Many people find that part of the difficulty they experience with numeracy tests is the lack of familiarity with everyday maths skills

This book aims to help you to overcome

that lack of familiarity by providing you with the opportunity to refresh your memory through plenty of practice drills and questions

Types of question There are a wide range of questions set in a test

multiple choice and data sufficiency answers,

where you are given pieces of information and asked whether you have enough information to answer the questions correctly

The reason for the adoption of this method is to ensure that you can work out correct answers confidently,

without resorting to a ‘multiple guess’ technique

- ‘Data interpretation’,

you are given a range of answers from which to choose the correct one

rather like those the test-writers will set for you

The accompanying explanations will help you to learn about these traps and to help you to avoid them in the test

- calculate it

The question will give you an indication of the level of precision that is expected from you,

- for example,

‘Give the answer to 3 decimal places’

If you are asked for this level of precision,

usually you will have a calculator to assist you

If you are given a range of answers to choose from,

do a quick estimate of the correct answer first

This technique reduces the likelihood of choosing incorrectly under pressure and gives you a narrower range of answers from which to choose the correct answer

Your estimate may be accurate enough to choose the correct answer without completing any additional calculations

This will save you time and allow you to spend more time on difficult questions

- you can

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substitute in possible correct answers to the question and effectively solve the question using the answer as the starting point

Translating the language Part of the difficulty of aptitude tests is in understanding exactly what is being asked of you

be absolutely sure that you understand what you are being asked to do

or whether you need to complete an interim step to provide you with enough information to answer the question

read the labels accompanying the diagrams to make sure you understand whether you are being given percentages or actual values

Read the axis label in case the axes are given in different values

once you have translated the words,

- the maths becomes much easier

so you may as well get into the habit of doing mental arithmetic without it

If you are of the GCSE generation,

maths without a calculator may seem impossible: after all,

calculators are a key tool in maths learning today

all the examples and practice questions in this book have been designed so that you can work out the correct answer without the use of the calculator

the maths can’t be that hard,

? If you set about the practice questions and drills with your calculator,

you will be wasting your test preparation time and practice material

By all means,

check your answers afterwards with your calculator,

but get into the habit of sitting down to take the test without it

do try to stick to the time allocated

One of the skills tested in

aptitude tests is your ability to work quickly and accurately under pressure

sometimes you do not have to finish all the questions in order to do well in a test,

- but of those you do answer,

you must answer those questions correctly

work out which aspect of the problem is taking the extra time

For example,

are you wasting time reworking simple calculations or are you having difficulty in working out exactly what is being asked in the question

? Once you have identified the problem,

you can go back to the relevant section of this book to find suggestions to help overcome the difficulty

What else can you do to prepare

? If you find that you draw a complete blank with some of the mental arithmetic questions,

particularly the timed questions,

find other ways to exercise your grey matter even when you are not in study mode

Add up the bill in your head as you are grocery shopping

if today is Saturday 3 August,

what was the date last Tuesday

? If my birthday is 6 April 1972 and today is Monday,

- on what day was I born

? On the bus on the way to work,

work out roughly how many words there must be on the front page of your neighbour’s newspaper,

based on your estimate of the number of words per line and the number of lines per page

- all levels

In other words,

become proficient at estimating everyday calculations

It will pay off enormously in the test

Getting started This introduction has explained the purpose of this workbook and has recommended a method to use it

Now roll up your sleeves and go straight on to Chapter 1

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Review the basics Chapter topics ●●

Prime numbers

Multiples

Terms used in this chapter Arithmetic mean: The amount obtained by adding two or more numbers and dividing by the number of terms

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How to Pass Numerical Reasoning Tests

Divisor: The number by which another is divided

Mean: See Arithmetic mean

Median: The middle number in a range of numbers when the set is arranged in ascending or descending order

Test-writers assume that you remember the fundamentals you learnt in school and that you can apply that knowledge and understanding to the problems in the tests

The purpose of this chapter is to remind you of the basics and to provide you with the opportunity to practise them before your test

The skills you will learn in this chapter are the fundamentals you can apply to solving many of the problems in an aptitude test,

so it is worth learning the basics thoroughly

without expending any unnecessary brainpower

- - keep this in reserve for the tricky questions later on

This chapter reviews the basics and includes a number of practice drills to ease you back into numerical shape

- no calculators …

Multiplication tables ‘Rote learning’ as a teaching method has fallen out of favour in recent years

There are good reasons for this in some academic areas but it doesn’t apply to multiplication tables

- but can you recite the

- tables as quickly now

? Recite them to yourself quickly,

- over and over again,
- when you’re out for a run,
- when you’re washing up,

when you’re cleaning your teeth,

when you’re stirring your baked beans

- - any time when you have a spare 10 seconds thinking time

Six times,

seven times and eight times are the easiest to forget,

so drill these more often than the twos and fives

Make sure that you can respond to any multiplication question without pausing even for half a second

you will save yourself valuable seconds in your test and avoid needless mistakes in your calculations

Multiplication tables: practice drill 1 Practise these drills and aim to complete each set within 15 seconds

- (Remember,

the answers are at the end of the chapter

Drill 2

Drill 5

- 3× 7=
- 8× 3=
- 9× 3=

- 7 × 15 =
- 6× 5=
- 11 × 6 =
- 9× 2=

6× 3=

11 × 11 =

- 8× 9=
- 13 × 2 =
- 7× 7=

- 4 × 12 =
- 3× 3=

11 × 13 =

- 12 × 7 =

- 8 × 10 =
- 9 × 12 =
- 13 × 9 =
- 6× 8=

8 × 15 =

- 13 × 4 =
- 2× 4=
- 6 × 14 =
- 6× 7=

- 11 × 2 =
- 8× 5=
- 3 × 15 =
- 13 × 5 =

2× 6=

- 7 × 12 =
- 13 × 3 =
- 9× 8=
- 13 × 4 =

- 9 × 15 =
- 6× 7=
- 4× 5=
- 8× 8=

4 × 12 =

- 9× 9=
- 2× 7=
- 6× 3=
- 7 × 13 =

5 × 14 =

- 3× 8=
- 7 × 12 =
- 12 × 4 =
- 6 × 15 =

3 × 11 =

- 3× 3=
- 2 × 12 =
- 11 × 7 =
- 9× 8=

- 3× 8=

How to Pass Numerical Reasoning Tests

Multiplication tables: practice drill 2 Drill 1

Drill 2

Drill 3

- 8× 9=
- 4× 5=
- 3× 6=
- 6 × 11 =

11 × 11 =

13 × 13 =

- 9× 3=
- 9× 5=
- 5× 3=
- 3× 7=

11 × 14 =

- 13 × 4 =
- 7× 3=
- 5× 6=
- 6× 8=
- 9× 6=
- 6× 8=
- 5× 9=
- 8× 5=
- 12 × 4 =
- 11 × 5 =
- 7× 7=

11 × 15 =

14 × 14 =

- 5× 5=
- 4× 3=

11 × 10 =

- 6× 7=
- 12 × 3 =
- 8 × 14 =
- 7× 8=
- 8× 9=
- 3 × 11 =
- 14 × 4 =
- 9× 7=
- 9× 6=
- 5 × 13 =
- 8× 4=
- 13 × 3 =

12 × 15 =

- 12 × 5 =
- 3 × 14 =
- 9× 6=
- 7× 6=
- 3× 3=

13 × 14 =

- 2 × 12 =
- 12 × 4 =
- 8× 7=
- 4× 2=
- 4× 9=
- 4× 6=
- 8× 2=
- 3× 8=
- 14 × 8 =
- 2× 5=
- 9× 5=
- 3 × 13 =

11 × 12 =

- 6 × 10 =

Dividing and multiplying numbers Long multiplication Rapid multiplication of multiple numbers is easy if you know the multiplication tables inside-out and back-to-front

you break the problem down into a number of simple calculations by dividing the multiplier up into units of tens,

- hundreds,
- thousands and so on

In Chapter 2 you will work through practice drills involving division and multiplication of decimals

Worked example Q

? To multiply a number by 237,

break the problem down into a number of simpler calculations

- tens and units

Review the basics

to multiply by 237 you multiply by: 7 (units) 3 (tens) 2 (hundreds) (It doesn’t matter in which order you complete the calculation

) 2,348 237 × 16,436 70,440 469,600 556,476

- = = = = =

! This exercise is intended to help you to speed up your mental arithmetic

Set a stopwatch and aim to complete this drill in five minutes

Long multiplication: practice drill 2 Set a stopwatch and aim to complete this practice drill in five minutes

like long multiplication calculations,

can be completed quickly and easily without a calculator if you know the multiplication tables well

There are four steps in a long division calculation,

and as long as you follow these in order,

you will arrive at the right answer

but recognize which number you are dividing into

This is called the dividend

- - this will become very important when you divide very large or very small numbers

Divide (D) Multiply (M) Subtract (S) Bring down (B)

Do-Mind-Slippery-Bananas

- 12 divides into 15 once,

so write ‘1’ on top of the division bar

1 12 156

Step 2: Multiply Multiply the result of step 1 (1) by the divisor (12): 1 × 12 = 12

Write the number 12 directly under the dividend (156)

- 1 12 156 12

1 12 156

1 12 156

- 12 36

Step 1: Divide 12 into 36 and write the result (3) on top of the long division sign

13 12 156

Write the number 36 directly below the new dividend (36)

13 12 156

-12 36 36

13 12 156

Step 4: There aren’t any more digits to Bring down,

so the calculation is complete

- 156 ÷ 12 = 13 You will learn about long division with remainders and decimals in Chapter 2

Long division: practice drill 1 Set a stopwatch and aim to complete these calculations in four minutes

You may check your answers with a calculator only once you have finished all the questions in the drill

- 99 ÷ 11
- 91 ÷ 13
- 117 ÷ 13
- 182 ÷ 14
- 696 ÷ 58
- 3,024 ÷ 27
- 2,890 ÷ 34
- 636 ÷ 53
- 1,456 ÷ 13

Q10 2,496 ÷ 78

Long division: practice drill 2 Set a stopwatch and aim to complete the following practice drill within five minutes

- 1,288 ÷ 56
- 1,035 ÷ 45
- 6,328 ÷ 56
- 5,625 ÷ 125
- 2,142 ÷ 17
- 7,952 ÷ 142
- 10,626 ÷ 231
- 11,908 ÷ 458
- 81,685 ÷ 961

Prime numbers An integer greater than 1 is a prime number if its only positive divisors are itself and 1

All prime numbers apart from 2 are odd numbers

Even numbers are divisible by 2 and cannot be prime by definition

- 1 is not a prime number,

because it is divisible by one number only,

The following is a list of all the prime numbers below 100

you don’t waste time trying to find other numbers to divide into them

- ! 0-10

2 3 5 7

- 11 13 17 19

23 29

31 37

- 41 43 47

53 59

61 67

- 71 73 79

83 89

How many prime numbers are there between 1 and 100

Between 1 and 100,

there are five prime numbers ending in 1

What is the result of the product of the first three prime numbers minus the sum of the second three prime numbers

Q10 What is the sum of the second 12 prime numbers minus the sum of the first 12 prime numbers

For example,

- 54 is a multiple of 9 and 72 is a multiple of 8

if the last digit is 0 or is an even number 3,

if the sum of its digits are a multiple of 3 4,

if the last two digits are a multiple of 4 5,

if the last digit is 0 or 5 6,

if it is divisible by 2 and 3 9,

if its digits sum to a multiple of 9 There is no consistent rule to find multiples of 7 or 8

because 8 is divisible by an even number

Is 91,542 divisible by 3

because 9+1+5+4+2 = 21 and 21 is a multiple of 3

- because 16 is a multiple of 4

- because the last digit is 5

- because the last digit,

is divisible by an even number and the sum of all the digits,

- is a multiple of 3

Multiples: practice drill Set a stopwatch and aim to complete the following 10-question drill in five minutes

The following numbers are multiples of which of the following integers: 2,

- ? Drill 1

989,136

240,702

161,174

Lowest common multiple The lowest common multiple is the least quantity that is a multiple of two or more given values

you can simply multiply them together,

but this will not necessarily give you the lowest common multiple of both integers

To find the lowest common multiple,

you will work with the prime numbers

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Review the basics

Step 2: Line up common prime factors

? Step 1: Express each of the integers as the product of its prime factors To find the prime factors of an integer,

divide that number by the prime numbers,

- starting with 2

The product of the prime factors of an integer is called the prime factorization

Now divide the remainder,

by the next prime factor after 2: 1 33

So the prime factors of 6 are 2 and 3

(Remember that 1 is not a prime number

) The product of the prime factors of an integer is called the prime factorization,

so the prime factorization of 6 = 2 × 3

Divide 9 by the first prime number that divides without a remainder: 3 39

Now divide the result by the first prime number that divides without a remainder

How to Pass Numerical Reasoning Tests

Step 2: Line up common prime factors Line up the prime factors of each of the given integers below each other: 6 = 2 × 3 9 = 3 × 3 Notice that 6 and 9 have a common prime factor (3)

Step 3: Find the product of the prime factors Multiply all the prime factors together

- count this only once
- 6 = 2 × 3 9 = 3 × 3 Prime factorization =
- 3 × 3 = 18

The lowest common multiple of 6 and 9 = 18

Lowest common multiple: practice drill 1 Set a stopwatch and aim to complete the following drill in four minutes

- 12 and 9

Q3 3 and 5 Q4

- 12 and 15

Q5 8 and 14 Q6 9 and 18 Q7 4 and 7 Q8

- 13 and 7
- 12 and 26

Lowest common multiple: practice drill 2 Set a stopwatch and aim to complete the following drill in four minutes

Find the lowest common multiple of the following sets of numbers: Q1

- 2 and 5
- 4 and 5
- 5 and 9
- 6 and 5
- 6 and 7
- 2 and 5 and 6
- 3 and 6 and 7
- 3 and 7 and 8
- 3 and 6 and 11

Working with large numbers Test-writers sometimes set questions that ask you to perform an operation on very large or very small numbers

as it is easy to be confused by a large number of decimal places or zeros

billions and trillions The meaning of notations such as millions,

billions and trillions is ambiguous

The terms vary and the UK definition of these terms is different from the US definition

make sure you know the difference

US definition Million: 1,000,000 or ‘a thousand thousand’ (same as UK definition) Billion: 1,000,000,000 or ‘a thousand million’ Trillion: 1,000,000,000,000 or ‘a thousand billion’

UK definition Million: 1,000,000 or ‘a thousand thousand’ (same as US definition) Billion: 1,000,000,000,000 or ‘a million million’ Trillion: 1,000,000,000,000,000,000 or ‘a million million million’ The US definitions are more commonly used now

- assume the US definition

Multiplying large numbers To multiply large numbers containing enough zeros to make you go cross-eyed,

follow these three steps: Step 1: Multiply the digits greater than 0 together

Step 2: Count up the number of zeros in each number

Step 3: Add that number of zeros to the result of Step 1

? Step 1: Multiply the digits greater than 0 together 2×2=4 Step 2: Count up the number of zeros in each number 2,000,000 × 2,000 = nine zeros

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Review the basics

Step 3: Add that number of zeros to the right of the result of Step 1 Result of Step 1 = 4 nine zeros = 000,000,000 Answer = 4,000,000,000 (4 billion,

US definition)

Multiplying large numbers: practice drill Use the US definition of billion and trillion to complete this practice drill

- 9 hundred × 2 thousand
- 2 million × 3 million
- 3 billion × 1 million
- 12 thousand × 4 million
- 24 million × 2 billion
- 18 thousand × 2 million
- 2 thousand × 13 million
- 3 hundred thousand × 22 thousand
- 28 million × 12 thousand

Q10 14 billion × 6 thousand

but cancel out equivalent zeros before you start

000 ÷ 2,

Now you are left with an easier calculation: 4,000 ÷ 2 Answer = 2,000

Dividing large numbers: practice drill Set a stopwatch and aim to complete the following drill in four minutes

240,000

6,720,000

475,000

19,500,000

23,800,000

- 149,500,000

9,890,000

15,540,000

Working with signed numbers Multiplication of signed numbers There are a few simple rules to remember when multiplying signed numbers

Positive × positive = positive Negative × negative = positive Negative × positive = negative Positive × negative = negative

The product of an odd number of negatives = negative N×N×N=N The product of an even number of negatives = positive N×N×N×N=P

Worked example P × P = P 2 × 2 = 4 N × N = P

- -2 = 4 N × P = N

-2 × 2 =

- -4 P × N = N 2 ×
- -4 N × N × N = N
- -8 N × N × N × N = P

-2 = 16

Division of signed numbers Positive ÷ positive = positive Negative ÷ negative = positive Negative ÷ positive = negative Positive ÷ negative = negative

Worked example P ÷ P = P 2 ÷ 2 = 1 N ÷ N = P

- -2 = 1 P ÷ N = N 2 ÷
- -1 N ÷ P = N

-2 ÷ 2 =

How to Pass Numerical Reasoning Tests

Multiplication and division of signed numbers: practice drill I Set a stopwatch and aim to complete each drill within five minutes

Multiplication and division of signed numbers: practice drill 2 Q1

Averages One way to compare sets of numbers presented in tables,

graphs or charts is by working out the average

This is a technique used in statistical analysis to analyse data and to draw conclusions about the content of the data set

- the mode and the median

Arithmetic mean The arithmetic mean (also known simply as the average) is a term you are probably familiar with

To find the mean,

simply add up all the numbers in the set and divide by the number of terms

Emma scores 77,

- 81 and 82 in each section

- ? Arithmetic mean =
- (77 + 81 + 82) = 240 3

- ? Arithmetic mean =
- (0 + 6 + 12 + 18) = 36 4

The arithmetic mean of the set is 36 ÷ 4 = 9

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Tip: remember to include the zero as a value in your sum of the number of values

- 9 and q = 5

? Step 1: Rearrange the formula to help you to find the sum of values Sum of values = arithmetic mean × number of values Step 2: Now plug in the numbers Sum of values

= arithmetic mean × number of values

(Don’t forget to count the fourth value q in the number of values

) Sum of values = 22 Step 3: Subtract the sum of known values from the sum of values Sum of values

- - sum of known values = q 22
- 18 (3 + 6 + 9)

The mode The mode is the number (or numbers) that appear(s) the most frequently in a set of numbers

? 21, 22, 23, 22, 22, 25, 25, 22, 21 21 appears twice

- 22 appears four times
- 23 appears once
- 25 appears twice

- 1 appears once
- 01 appears three times
- 1 appears twice
- 01 appears three times

So the mode numbers are 0

when the numbers are put in ascending or descending order

- 37 First order the numbers in ascending (or descending) order

As there are nine numbers in the set,

the fifth number in the series is the median

- 37 is the median value

- 1 First put the numbers of the set in order (either ascending or descending)
- 10 This time there is an even number of values in the set

Draw a line in the middle of the set: 1 0

The median of the series is the average of the two numbers on either side of the dividing line

the median number in the series is the arithmetic mean of 2 and 4: (2 + 4) ÷ 2 = 3

Review the basics

What is the arithmetic mean of the following sets of numbers

What is the median of the following sets of numbers

- 39 What is the mode in the following sets of numbers
- ? Q11 21,

-2 Q13 0,

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How to Pass Numerical Reasoning Tests

Answers to Chapter 1 Multiplication tables: practice drill 1 Drill 1

Multiplication tables: practice drill 2 Drill 1

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Q10 4,174,236

Long multiplication: practice drill 2 Q1 162 Q2 209 Q3 252 Q4 437 Q5 494 Q6 3,587 Q7 4,121 Q8 4,576 Q9

Q10 73,225

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Long division practice: drill 1 Q1 9 Q2 7 Q3 9 Q4 13 Q5 12 Q6

Q6 56 Q7 46 Q8 26 Q9 85 Q10 133

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Prime numbers: practice drill Q1

Q10 569

Drill 2

Drill 4

2,3,4,6,9

- 2,3,4,6 [8]

2,3,4,6

2,3,4,6

- 2,3,4,6,9 [8]

2,4 [8]

2,3,6 [7]

2,4 [8]

2,3,4,6

2,4 [8]

- 2,3,4,6 [8]

2,4 [8]

2,4 [8]

2,4 [8]

this is indicated in the answer table with [7] or [8]

Lowest common multiple: practice drill 1 Q1

- 8 and 6 Answer = 24 8 6

- 12 and 9 Answer = 36 12 9

Prime factorization = 2 × 2 × 3 × 3 Q3

- 3 and 5 Answer = 15 3 5

- 12 and 15 Answer = 60 12 15

- 8 and 14 Answer = 56 8 14

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- 9 and 18 Answer = 18 9 18

Prime factorization = 2 × 3 × 3 Q7

- 4 and 7 Answer = 28 4 = 2 × 2 7 = 1 × 7 Prime factorization = (1 ×) 2 × 2 × 7
- 13 and 7 Answer = 91 13 7

- 12 and 26 Answer = 156 12 26

Prime factorization = 2 × 2 × 3 × 13 Q10 7 and 15 Answer = 105 7 = 1 × 7 15 = 3 × 5 Prime factorization = (1 ×) 3 × 5 × 7

Lowest common multiple: practice drill 2 Q1 10 Q2 20 Q3 45 Q4 30 Q5 42 Q6 30 Q7 42 Q8

Q9 66 Q10 84

1 million,

- 8 hundred thousand
- (1,800,000)

6 trillion

- (6,000,000,000,000)
- 3 thousand trillion
- (3,000,000,000,000,000)

48 billion

- (48,000,000,000)
- 48 thousand trillion
- (48,000,000,000,000,000)

36 billion

- (36,000,000,000)

26 billion

- (26,000,000,000)

6 billion,

- 6 hundred million
- (6,600,000,000)
- 336 billion
- (336,000,000,000)
- 84 trillion
- (84,000,000,000,000)

Review the basics

170,000

How to Pass Numerical Reasoning Tests

Multiplication and division of signed numbers: practice drill 2 Q1

Averages: practice drill Arithmetic mean:

Median:

1 4

1 and 0

CHAPTER 2

Fractions and decimals Chapter topics ●●

What a fraction is

Fraction operations

Terms used in this chapter Denominator: The number below the line in a vulgar fraction

How to Pass Numerical Reasoning Tests

Improper fraction: A fraction in which the numerator is greater than or equal to the denominator

Numerator: The number above the line in a vulgar fraction

Proper fraction: A fraction less than one,

with the numerator less than the denominator

- rather than decimally

In Chapter 1,

you practised operations with whole numbers

In this chapter you will practise number operations on parts of numbers

there are a few extra tricks you can learn to complete these puzzles quickly and accurately

What a fraction is Proper and improper fractions A fraction is a part of a whole number,

or a value expressed as one number divided by another

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