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Quantitative Methods for Business, fourth edition, employs an accessible ﬁve-part

structure, leading the reader through the subject in a logical sequence.

•

•

•

•

•

Part One introduces the subject, asks why managers use quantitative

methods and reviews essential quantitative tools

Part Two covers data collection and description, focusing in particular on

how to ensure your data is reliable

Part Three shows how quantitative methods can be used for solving

different types of problems

Part Four describes some statistical methods, focusing on probabilities,

sampling and statistical inference

Part Five suggests how statistical ideas may be used in decision analysis,

quality management, inventory control and other areas

Key features

• Worked examples illustrate the principles discussed

• ‘Ideas in practice’ sections show how methods are actually used

• Covers a broad range of materials relevant to managers and students

• Case studies at the end of every chapter consolidate learning objectives

• Extensive pedagogical features, including self-assessment problems, chapter

outlines and summaries, review questions and suggested research projects

QUANTITATIVE METHODS

FOR BUSINESS FOURTH EDITION

All students of management undertake a course in quantitative

methods. These courses come in various guises, including

quantitative analysis, decision analysis, business modelling and

numerical analysis. This book describes a range of quantitative

methods that are widely used in business and which every

student of management will meet somewhere in their course.

Whether studying for an HND, an MBA, a ﬁrst degree or a

professional qualiﬁcation, students will appreciate the author’s

friendly style and practical approach.

Donald Waters

QUANTITATIVE

METHODS

FOR

BUSINESS

FOURTH EDITION

Waters

About the author

Donald Waters is the author of several successful textbooks and is well

known for his clarity of style and his student-friendly texts.

Front cover image:

© Getty Images

an imprint of

9780273694588_COVER.indd 1

Additional student support at

www.pearsoned.co.uk/waters

www.pearson-books.com

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Quantitative Methods

for Business

Visit the Quantitative Methods for Business,

Fourth Edition Companion Website at

www.pearsoned.co.uk/waters to find valuable

student learning material including:

n

n

n

n

n

n

..

Data sets for problems, examples and cases in the

book

Spreadsheet templates for calculations

Additional material to extend the coverage of key

topics

Proofs and derivations of formulae

Answers to problems

Additional worked examples and case studies

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Page ii

We work with leading authors to develop the strongest

educational materials in business, bringing cutting-edge

thinking and best learning practice to a global market.

Under a range of well-known imprints, including Financial

Times Prentice Hall, we craft high quality print and

electronic publications which help readers to understand

and apply their content, whether studying or at work.

To find out more about the complete range of our

publishing please visit us on the World Wide Web at:

www.pearsoned.co.uk

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Quantitative

Methods for

Business

FOURTH EDITION

Donald Waters

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Pearson Education Limited

Edinburgh Gate

Harlow

Essex CM20 2JE

England

and Associated Companies throughout the world

Visit us on the World Wide Web at:

www.pearsoned.co.uk

First published 1993

Second edition published under the Addison-Wesley imprint 1997

Third edition published 2001

Fourth edition published 2008

© Pearson Education Limited 1997, 2001

© Donald Waters 2008

The right of Donald Waters to be identified as author of this work has

been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a

retrieval system, or transmitted in any form or by any means, electronic, mechanical,

photocopying, recording or otherwise, without either the prior written permission of the

publisher or a licence permitting restricted copying in the United Kingdom issued by the

Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.

All trademarks used herein are the property of their respective owners. The use of any

trademark in this text does not vest in the author or publisher any trademark ownership

rights in such trademarks, nor does the use of such trademarks imply any affiliation with or

endorsement of this book by such owners.

The screenshots in this book are reprinted by permission from Microsoft Corporation.

ISBN 978-0-273-69458-8

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

10 9 8 7 6 5 4 3 2 1

11 10 09 08 07

Typeset in 10/12pt Sabon by 35

Printed by Ashford Colour Press Ltd, Gosport

The publisher’s policy is to use paper manufactured from sustainable forests.

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TO CHARLES

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BRIEF CONTENTS

Preface

Part One – Background

1 Managers and numbers

2 Quantitative tools

3 Drawing graphs

Part Two – Collecting and summarising data

4

5

6

7

..

Collecting data

Diagrams for presenting data

Using numbers to describe data

Describing changes with index numbers

xvii

1

3

18

43

63

65

90

120

148

Part Three – Solving management problems

167

8

9

10

11

12

13

169

200

232

265

289

319

Finance and performance

Regression and curve fitting

Forecasting

Simultaneous equations and matrices

Planning with linear programming

Rates of change and calculus

Part Four – Introducing statistics

341

14

15

16

17

343

366

397

419

Uncertainty and probabilities

Probability distributions

Using samples

Testing hypotheses

Part Five – Management problems with uncertainty

447

18

19

20

21

22

449

478

504

528

555

Making decisions

Quality management

Inventory management

Project networks

Queues and simulation

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Brief contents

Glossary

Appendices

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Index

575

Solutions to review questions

Probabilities for the binomial distribution

Probabilities for the Poisson distribution

Probabilities for the Normal distribution

Probabilities for the t-distribution

Critical values for the χ2 distribution

587

601

606

610

611

612

614

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CONTENTS

Preface

Part One – Background

1 Managers and numbers

Chapter outline

Why use numbers?

Solving problems

Useful software

Chapter review

Case study – Hamerson and Partners

Problems

Research projects

Sources of information

2 Quantitative tools

Chapter outline

Working with numbers

Changing numbers to letters

Powers and roots

Chapter review

Case study – The Crown and Anchor

Problems

Research projects

Sources of information

3 Drawing graphs

Chapter outline

Graphs on Cartesian co-ordinates

Quadratic equations

Drawing other graphs

Chapter review

Case study – McFarlane & Sons

Problems

Research projects

Sources of information

Part Two – Collecting and summarising data

4 Collecting data

Chapter outline

Data and information

..

xvii

1

3

3

3

7

11

14

14

15

15

16

18

18

19

25

31

39

39

40

41

41

43

43

43

51

55

59

60

61

62

62

63

65

65

65

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Contents

Types of data

Using samples to collect data

Organising data collection

Chapter review

Case study – Natural Wholemeal Biscuits

Problems

Research projects

Sources of information

5 Diagrams for presenting data

Chapter outline

Data reduction and presentation

Tables of numerical data

Diagrams of data

Continuous data

Chapter review

Case study – High Acclaim Trading

Problems

Research projects

Sources of information

6 Using numbers to describe data

Chapter outline

Measuring data

Measures of location

Measures of spread

Other measures of data

Chapter review

Case study – Consumer Advice Office

Problems

Research projects

Sources of information

7 Describing changes with index numbers

Chapter outline

Measuring change

Changing the base period

Indices for more than one variable

Chapter review

Case study – Heinz Muller Engineering

Problems

Research projects

Sources of information

Part Three – Solving management problems

8 Finance and performance

Chapter outline

Measures of performance

Break-even point

69

72

79

86

87

88

89

89

90

90

90

93

98

109

115

116

117

119

119

120

120

120

122

133

141

145

145

146

146

147

148

148

148

154

156

161

162

163

164

165

167

169

169

169

174

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Contents

Value of money over time

Discounting to present value

Mortgages, annuities and sinking funds

Chapter review

Case study – OnlineInkCartridges.com

Problems

Research projects

Sources of information

9 Regression and curve fitting

Chapter outline

Measuring relationships

Linear relationships

Measuring the strength of a relationship

Multiple regression

Curve fitting

Chapter review

Case study – Western General Hospital

Problems

Research projects

Sources of information

10 Forecasting

Chapter outline

Forecasting in organisations

Judgemental forecasts

Projective forecasts

Forecasts with seasonality and trend

Chapter review

Case study – Workload planning

Problems

Research projects

Sources of information

11 Simultaneous equations and matrices

Chapter outline

Simultaneous equations

Matrix notation

Matrix arithmetic

Using matrices to solve simultaneous equations

Chapter review

Case study – Northern Feedstuffs

Problems

Research projects

Sources of information

12 Planning with linear programming

Chapter outline

Constrained optimisation

Formulation

..

..

xi

181

184

192

195

195

196

198

198

200

200

200

205

212

218

223

227

227

228

230

231

232

232

232

235

237

251

261

261

262

263

263

265

265

265

271

273

282

285

286

286

287

288

289

289

290

290

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Contents

Using graphs to solve linear programmes

Sensitivity of solutions to changes

Solving real problems

Chapter review

Case study – Elemental Electronics

Problems

Research projects

Sources of information

13 Rates of change and calculus

Chapter outline

Differentiation

Finding the maximum and minimum

Marginal analyses

Integration

Chapter review

Case study – Lundquist Transport

Problems

Research projects

Sources of information

296

302

304

313

313

314

317

318

319

319

319

324

330

333

337

337

338

339

339

Part Four – Introducing statistics

341

14 Uncertainty and probabilities

343

343

343

347

353

361

362

363

364

364

Chapter outline

Measuring uncertainty

Calculations with probabilities

Conditional probabilities

Chapter review

Case study – The Gamblers’ Press

Problems

Research projects

Sources of information

15 Probability distributions

Chapter outline

Frequency distributions

Combinations and permutations

Binomial distribution

Poisson distribution

Normal distribution

Chapter review

Case study – Machined components

Problems

Research projects

Sources of information

16 Using samples

Chapter outline

Purpose of sampling

366

366

366

368

371

376

382

393

394

394

396

396

397

397

397

..

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Contents

Sampling distribution of the mean

Confidence intervals

One-sided confidence intervals

Using small samples

Chapter review

Case study – Kings Fruit Farm

Problems

Research projects

Sources of information

17 Testing hypotheses

Chapter outline

Aim of hypothesis testing

Significance levels

Tests with small samples

Testing other hypotheses

Chi-squared test for goodness of fit

Tests of association

Chapter review

Case study – Willingham Consumer Protection Department

Problems

Research projects

Sources of information

419

419

420

423

428

429

434

439

441

442

443

445

445

447

18 Making decisions

449

19 Quality management

Chapter outline

Measuring quality

Quality control

Tools for quality control

Acceptance sampling

Process control

Chapter review

Case study – Bremen Engineering

Problems

..

399

403

409

411

415

415

416

417

418

Part Five – Management problems with uncertainty

Chapter outline

Giving structure to decisions

Decision making with certainty

Decision making with uncertainty

Decision making with risk

Sequential decisions

Chapter review

Case study – The Newisham Reservoir

Problems

Research projects

Sources of information

..

xiii

449

449

452

453

458

465

472

473

473

476

477

478

478

478

484

486

490

495

498

499

500

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Contents

Research projects

Sources of information

20 Inventory management

Chapter outline

Background to stock control

The economic order quantity

Stock control for production

Variable demand

Periodic review

ABC analysis of stock

Chapter review

Case study – Templar Manufacturing

Problems

Research projects

Sources of information

21 Project networks

Chapter outline

Network analysis

Drawing project networks

Timing of projects

Project evaluation and review technique

Chapter review

Case study – Westin Contractors

Problems

Research projects

Sources of information

22 Queues and simulation

Chapter outline

Features of queues

Single-server queues

Simulation models

Monte Carlo simulation

Chapter review

Case study – The Palmer Centre for Alternative Therapy

Problems

Research projects

Sources of information

501

502

504

504

504

508

513

515

518

521

523

524

525

526

527

528

528

528

530

535

545

549

550

550

552

554

555

555

555

557

560

565

572

572

573

574

574

Glossary

575

Appendix A

Solutions to review questions

587

Appendix B

Probabilities for the binomial distribution

601

Appendix C

Probabilities for the Poisson distribution

606

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Contents

xv

Appendix D

Probabilities for the Normal distribution

610

Appendix E

Probabilities for the t-distribution

611

Appendix F

Critical values for the χ2 distribution

612

Index

614

Supporting resources

Visit www.pearsoned.co.uk/waters to find valuable online resources

Companion Website for students

n Data sets for problems, examples and cases in the book

n Spreadsheet templates for calculations

n Additional material to extend the coverage of key topics

n Proofs and derivations of formulae

n Answers to problems

n Additional worked examples and case studies

For instructors

n Complete, downloadable Instructor’s Manual

n PowerPoint slides that can be downloaded and used for

presentations

n Review of key aims and points of each chapter

n Worked solutions to problems

n Comments on case studies

n Copies of figures and artwork from the book

n Additional worked examples and case studies

Also: The Companion Website provides the following features:

n

n

n

Search tool to help locate specific items of content

E-mail results and profile tools to send results of quizzes to

instructors

Online help and support to assist with website usage and

troubleshooting

For more information please contact your local Pearson Education

sales representative or visit www.pearsoned.co.uk/waters

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P R E FA C E

Managers are the people who run their organisations. They need many skills

for this – including problem-solving, leadership, communications, analysis,

reasoning, experience, judgement, and so on. They also have to understand

quantitative methods, which give essential tools for making rational decisions

in complex circumstances. This does not mean that managers have to be

professional mathematicians, but they do need to understand quantitative

reasoning and interpret numerical results.

All students of management do a course in quantitative methods. These

courses come in various guises, including quantitative analysis, decision analysis,

business modelling, numerical analysis, and so on. This book describes a range

of quantitative methods that are widely used in business, and which every

student of management will meet somewhere in their course. It gives an

introduction to quantitative methods that is appropriate for the early years of

an HND, an undergraduate business course, an MBA, or many professional

courses. It is aimed at anyone who wants to see how quantitative ideas are

used in business.

Management students come from different backgrounds, so we cannot

assume much common knowledge or interests. This book starts with the

assumption that you have no previous knowledge of management or quantitative methods. It works from basic principles and develops ideas in a logical

sequence, moving from core ideas through to real applications.

Management students often find quantitative ideas difficult. Typically,

you are not interested in mathematical abstraction, proofs and derivations,

but are more concerned with how useful a result is, and how you can apply

it. This is why the book has a practical rather than a theoretical approach.

We have made a deliberate decision to avoid proofs, derivations and rigorous

(often tedious) mathematics. Some formal procedures are included, but

these are kept to a minimum. We assume that computers – particularly

spreadsheets – do the routine arithmetic, with Microsoft Excel used to illustrate many of the calculations (but you can get equivalent results from any

spreadsheet). There is additional material on the Companion Website at

www.pearsoned.co.uk/waters.

Contents

Managers can use almost any kind of quantitative methods in some circumstances, so there is an almost unlimited amount of material that we could put

into the book. To keep it to a reasonable length we have concentrated on

the most widely used topics, taking a balanced view without emphasising

some topics at the expense of others. And the book takes a deliberately broad

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Preface

approach, describing many topics rather than concentrating on the details

of a few.

For convenience the book is divided into five parts that develop the subject

in a logical sequence.

n

n

n

n

n

Part One gives an introduction to quantitative methods for managers.

These first three chapters lay the foundations for the rest of the book, saying

why managers use quantitative methods, and giving a review of essential

quantitative tools.

Part Two describes data collection and description. All quantitative methods

need reliable data, so these chapters show how to collect this, summarise

it, and present it in appropriate forms.

Part Three shows how to use these quantitative ideas for solving different

types of problems, including finance, performance, regression, forecasting,

simultaneous equations, matrices, linear programming and calculus.

Part Four describes some statistical methods, focusing on probabilities,

probability distributions, sampling and statistical inference.

Part Five shows how to use these statistical ideas for problems with uncertainty, including decision analysis, quality management, inventory control,

project networks, queues and simulation.

Many people find probabilistic ideas more difficult than deterministic ones,

so we have drawn a clear separation between the two. The first three parts

describe deterministic methods, and the last two parts cover problems with

uncertainty. The whole book gives a solid foundation for understanding

quantitative methods and their use in business.

Format

Each chapter uses a consistent format which includes:

n

n

n

n

n

n

n

n

n

n

n

n

a list of chapter contents

an outline of material covered and a list of things you should be able to do

after finishing the chapter

the main material of the chapter divided into coherent sections

worked examples to illustrate methods

‘ideas in practice’ to show how the methods are actually used

short review questions throughout the text to make sure you understand

the material (with solutions in Appendix A)

key terms highlighted in the chapter, with a glossary at the end of the

book

a chapter review listing the material that has been covered

a case study based on material in the chapter

problems (with solutions given on the Companion Website at

www.pearsoned.co.uk/waters)

research projects, which allow you to look deeper into a topic

sources of information, including references, suggestions for further reading

and useful websites.

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Preface

xix

To summarise

This is a book on quantitative methods for business and management. The book:

n

n

n

n

n

n

n

n

n

is an introductory text that assumes no previous knowledge of business,

management or quantitative methods

takes a broad view and is useful for students doing a wide range of courses,

or people studying by themselves

covers a lot of material, concentrating on the most widely-used methods

develops the contents in a logical order

presents ideas in a straightforward, reader-friendly style

avoids abstract discussion, mathematical proofs and derivations

illustrates principles by examples from a wide range of applications

uses spreadsheets and other software to illustrate calculations

includes a range of learning features to help you understand the material.

Companion Website

The Companion Website for the book is www.pearsoned.co.uk/waters. This

contains valuable teaching and learning information including:

For students:

Study material designed to help your understanding

n Data sets for problems, examples and cases in the book

n Spreadsheet templates for calculations

n Additional material to extend the coverage of key topics

n Proofs and derivations of formulae

n Answers to problems

n Additional worked examples and case studies

n

For lecturers adopting the book for courses:

n A secure password-protected site with teaching material

n A review of key aims and points for each chapter

n Worked solutions to problems

n Comments on case studies

n Copies of figures and artwork from the book

n Additional worked examples and case studies.

Acknowledgements and trademarks

Excel, Microsoft Project, PowerPoint and Visio are trademarks of Microsoft

Corporation; Microsoft Excel screenshots are reprinted with permission from

Microsoft Corporation.

A lot of software is available for quantitative methods. The following list

gives packages that are mentioned in the book, with their developers. You

can find more information about products from company websites.

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Preface

Ability Office Spreadsheet is a trademark of Ability Software; ConceptDraw

and ConceptDraw Project are trademarks of Computer Systems Odessa

Corporation; CorelDraw and Quattro Pro are trademarks of Corel Corporation; GAMS is a trademark of GAMS Development Corporation; GPSS and

SLX are trademarks of Wolverine Software Corporation; Harvard graphics

and DrawPlus are trademarks of Serif Corporation; ILOG is a trademark

of ILOG OPL Development Studio; Jmp and SAS are trademarks of SAS

Institute, Inc.; LINDO and What’sBest are trademarks of Lindo Systems,

Inc.; Lotus 1-2-3 and Freelance Graphics are trademarks of Lotus Development Corporation; Minitab is a trademark of Minitab, Inc.; OpenOffice

Calc is a trademark of OpenOffice.Org; Oracle Projects is a trademark of

Oracle Corporation; PowerProject is a trademark of ASTA Development,

Inc.; Primavera Project Planner is a trademark of Primavera Systems, Inc.;

SmartDraw is a trademark of SmartDraw.com; SOPT is a trademark of

SAIOTECH, Inc.; S-plus is a trademark of Mathsoft, Inc.; SPSS is a trademark of SPSS, Inc.; StarOffice is a trademark of Sun Microsystems, Inc.;

SuperProject is a trademark of Computer Associates International; Systat

and Sigmaplot are trademarks of Systat Software Group; TurboProject is a

trademark of IMSI; and XPRESS is a trademark of Frontline Systems, Inc.

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PA R T O N E

Background

Managers are the people who run their organisations. They need many skills,

with key ones being the ability to analyse problems and make the best decisions

to solve them. Each problem is in some way distinct, but they all share common

features – and, in particular, they generally have some quantitative features.

This book describes the quantitative methods that managers use most often

to analyse and solve their problems.

The book is divided into five parts, each of which covers a different aspect

of quantitative methods. This first part describes the underlying concepts of

quantitative methods, setting the context for the rest of the book. The second

part shows how to collect and summarise data, and the third part uses this data

to solve some common management problems. The fourth part introduces some

statistics, and the fifth part uses these to solve problems with uncertainty.

There are three chapters in this first part. Chapter 1 shows that managers

constantly use numbers, and they must understand a range of quantitative

ideas. The rest of the book describes key methods. Before you look at them in

detail, you have to be familiar with some basic quantitative tools. Chapters 2

and 3 review these tools – with Chapter 2 describing numerical skills and

algebra, and Chapter 3 showing how to draw graphs.

Chapters in the book follow a logical path through the material, so it is

best to take each one in turn. However, you can be flexible, as the map overleaf shows the relationships between chapters.

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Map 1 Map of chapters – Part One

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Page 3

CHAPTER

1

Managers and numbers

Contents

Chapter outline

Why use numbers?

Solving problems

Useful software

Chapter review

Case study – Hamerson and Partners

Problems

Research projects

Sources of information

3

3

7

11

14

14

15

15

16

Chapter outline

Managers analyse problems and make the best decisions to solve them.

Their problems invariably have some numerical features, so managers must

understand and use a range of quantitative methods. This chapter introduces

the underlying ideas of quantitative methods. It discusses the importance of

numerical information, the general approach of quantitative methods, and

the way that quantitative models are used to solve problems.

After finishing this chapter you should be able to:

n

n

n

n

n

Appreciate the importance and benefits of numbers

Say why quantitative methods are particularly useful for managers

Understand the use of models

Describe a general approach to solving problems

Use computers for calculations.

Why use numbers?

On an ordinary day, you might notice that the temperature is 17°C, petrol costs

92 pence per litre, 1.3 million people are unemployed, house prices rose by

12% last year, some people want a pay rise of £1.50 an hour, a football team

has won its last seven games, 78% of people want shops to open longer hours,

your telephone bill is £95, and a candidate won 32,487 votes in an election.

These numbers give essential information. They have the benefit of giving a

clear, precise and objective measure. When the temperature is 30 degrees,

you know exactly how hot it is; when a bar contains 450 grams of chocolate,

you know exactly how big it is; and your bank manager can say exactly how

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Managers and numbers

much money is in your account. On the other hand, when you cannot measure

something it is much more difficult to describe and understand. When you

get a pain in your stomach it is very difficult to describe the kind of pain,

how bad it is, or how it makes you feel. When you read a book it is difficult

to say how good the book is or to describe the pleasure it gave you.

A second benefit of numbers is that you can use them in calculations. If

you buy three bars of chocolate that cost 30 pence each, you know the total

cost is 90 pence; if you pay for these with a £5 note you expect £4.10 in

change. If you start a 120 km journey at 12:00 and travel at 60 km an hour,

you expect to arrive at 14:00.

n

n

Any reasoning that uses numbers is quantitative.

Any reasoning that does not use numbers, but is based on judgement and opinions, is

qualitative.

WORKED EXAMPLE 1.1

An automatic ticket machine only accepts pound

coins. The numbers of tickets it gives are:

£1 – 1 ticket, £2 – 3 tickets, £3 – 4 tickets,

£4 – 5 tickets, £5 – 7 tickets

How can you get the cheapest tickets?

Solution

You can do a simple calculation to find the best

value for money. You know that:

£1 gives 1 ticket, so each ticket costs £1/1 = £1

£2 gives 3 tickets, so each ticket costs £2/3 = £0.67

n £3 gives 4 tickets, so each ticket costs £3/4 = £0.75

n £4 gives 5 tickets, so each ticket costs £4/5 = £0.80

n £5 gives 7 tickets, so each ticket costs £5/7 = £0.71

n

n

Buying three tickets for £2 clearly gives the lowest

cost per ticket.

Numbers increase our understanding of a situation – and it is impossible to

lead a normal life without them. This does not mean that we all have to be

mathematical whiz-kids – but it does mean that we have to understand some

numerical reasoning and know how to work with numbers.

Often we do not need precise answers, but are happy with rough estimates.

If you can read a page a minute, you know that you can finish a 57-page report

in about an hour. If you see a car for sale, you do not know exactly how much

it costs to run, but a rough calculation shows whether you can afford it; if you

get a bill from a builder you can quickly check that it seems reasonable; before

you go into a restaurant you can get an idea of how much a meal will cost.

Numbers and management

Managers have to understand quantitative reasoning, as their decisions are

almost invariably based on calculations. When they want to increase profits, they

measure the current profits and set numerical targets for improvement. And they

continually measure performance, including return on investment, turnover,

share price, capacity, output, productivity, sales, market share, number of

customers, costs, and so on. Annual accounts review overall performance,

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