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Lecture Notes on Engineering Mathematics

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Prelims-H8555.tex 2/8/2007 9: 34 page i Engineering MathematicsPrelims-H8555.tex 2/8/2007 9: 34 page ii In memory of ElizabethPrelims-H8555.tex 2/8/2007 9: 34 page iii Engineering Mathematics Fifth edition John Bird BSc(Hons), CEng, CSci, CMath, FIET, MIEE, FIIE, FIMA, FCollT AMSTERDAM � BOSTON � HEIDELBERG � LONDON � NEW YORK � OXFORD PARIS � SAN DIEGO � SAN FRANCISCO � SINGAPORE � SYDNEY � TOKYO Newnes is an imprint of ElsevierPrelims-H8555.tex 2/8/2007 9: 34 page iv Newnes is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 1989 Second edition 1996 Reprinted 1998 (twice), 1999 Third edition 2001 Fourth edition 2003 Reprinted 2004 Fifth edition 2007 Copyright © 2001, 2003, 2007, John Bird. Published by Elsevier Ltd. All rights reserved The right of John Bird to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissionselsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN: 978-0-75-068555-9 For information on all Newnes publications visit our website at www.books.elsevier.com Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India www.charontec.com Printed and bound in The Netherlands 7891011 1110987654321Prelims-H8555.tex 2/8/2007 9: 34 page v Contents Preface xii 6 Further algebra 48 6.1 Polynominal division 48 6.2 The factor theorem 50 Section 1 Number and Algebra 1 6.3 The remainder theorem 52 1 Revision of fractions, decimals 7 Partial fractions 54 and percentages 3 7.1 Introduction to partial 1.1 Fractions 3 fractions 54 1.2 Ratio and proportion 5 7.2 Worked problems on partial 1.3 Decimals 6 fractions with linear factors 54 1.4 Percentages 9 7.3 Worked problems on partial fractions with repeated linear factors 57 2 Indices, standard form and engineering 7.4 Worked problems on partial notation 11 fractions with quadratic factors 58 2.1 Indices 11 2.2 Worked problems on indices 12 8 Simple equations 60 2.3 Further worked problems on indices 13 8.1 Expressions, equations and 2.4 Standard form 15 identities 60 2.5 Worked problems on standard form 15 8.2 Worked problems on simple 2.6 Further worked problems on equations 60 standard form 16 8.3 Further worked problems on 2.7 Engineering notation and common simple equations 62 prefixes 17 8.4 Practical problems involving simple equations 64 3 Computer numbering systems 19 8.5 Further practical problems 3.1 Binary numbers 19 involving simple equations 65 3.2 Conversion of binary to decimal 19 3.3 Conversion of decimal to binary 20 3.4 Conversion of decimal to Revision Test 2 67 binary via octal 21 3.5 Hexadecimal numbers 23 9 Simultaneous equations 68 4 Calculations and evaluation of formulae 27 9.1 Introduction to simultaneous 4.1 Errors and approximations 27 equations 68 4.2 Use of calculator 29 9.2 Worked problems on 4.3 Conversion tables and charts 31 simultaneous equations 4.4 Evaluation of formulae 32 in two unknowns 68 9.3 Further worked problems on simultaneous equations 70 Revision Test 1 37 9.4 More difficult worked problems on simultaneous 5 Algebra 38 equations 72 5.1 Basic operations 38 9.5 Practical problems involving 5.2 Laws of Indices 40 simultaneous equations 73 5.3 Brackets and factorisation 42 5.4 Fundamental laws and precedence 44 10 Transposition of formulae 77 5.5 Direct and inverse 10.1 Introduction to transposition proportionality 46 of formulae 77Prelims-H8555.tex 2/8/2007 9: 34 page vi vi Contents 10.2 Worked problems on 15.3 Further worked problems on transposition of formulae 77 arithmetic progressions 115 10.3 Further worked problems on 15.4 Geometric progressions 117 transposition of formulae 78 15.5 Worked problems on 10.4 Harder worked problems on geometric progressions 118 transposition of formulae 80 15.6 Further worked problems on geometric progressions 119 15.7 Combinations and 11 Quadratic equations 83 permutations 120 11.1 Introduction to quadratic equations 83 16 The binomial series 122 11.2 Solution of quadratic 16.1 Pascal’s triangle 122 equations by factorisation 83 16.2 The binomial series 123 11.3 Solution of quadratic 16.3 Worked problems on the equations by ‘completing binomial series 123 the square’ 85 16.4 Further worked problems on 11.4 Solution of quadratic the binomial series 125 equations by formula 87 16.5 Practical problems involving 11.5 Practical problems involving the binomial theorem 127 quadratic equations 88 11.6 The solution of linear and quadratic equations 17 Solving equations by iterative methods 130 simultaneously 90 17.1 Introduction to iterative methods 130 17.2 The Newton–Raphson method 130 17.3 Worked problems on the 12 Inequalities 91 Newton–Raphson method 131 12.1 Introduction in inequalities 91 12.2 Simple inequalities 91 12.3 Inequalities involving a modulus 92 Revision Test 4 133 12.4 Inequalities involving quotients 93 12.5 Inequalities involving square functions 94 Multiple choice questions on 12.6 Quadratic inequalities 95 Chapters 1–17 134 13 Logarithms 97 13.1 Introduction to logarithms 97 Section 2 Mensuration 139 13.2 Laws of logarithms 97 13.3 Indicial equations 100 13.4 Graphs of logarithmic functions 101 18 Areas of plane figures 141 18.1 Mensuration 141 18.2 Properties of quadrilaterals 141 Revision Test 3 102 18.3 Worked problems on areas of plane figures 142 18.4 Further worked problems on 14 Exponential functions 103 areas of plane figures 145 14.1 The exponential function 103 18.5 Worked problems on areas of 14.2 Evaluating exponential functions 103 composite figures 147 x 14.3 The power series for e 104 18.6 Areas of similar shapes 148 14.4 Graphs of exponential functions 106 14.5 Napierian logarithms 108 14.6 Evaluating Napierian logarithms 108 19 The circle and its properties 150 14.7 Laws of growth and decay 110 19.1 Introduction 150 19.2 Properties of circles 150 15 Number sequences 114 19.3 Arc length and area of a sector 152 15.1 Arithmetic progressions 114 19.4 Worked problems on arc 15.2 Worked problems on length and sector of a circle 153 arithmetic progressions 114 19.5 The equation of a circle 155Prelims-H8555.tex 2/8/2007 9: 34 page vii Contents vii 20 Volumes and surface areas of 24.3 Changing from polar into common solids 157 Cartesian co-ordinates 213 20.1 Volumes and surface areas of 24.4 Use of R → P and P → R regular solids 157 functions on calculators 214 20.2 Worked problems on volumes and surface areas of regular solids 157 20.3 Further worked problems on Revision Test 6 215 volumes and surface areas of regular solids 160 20.4 Volumes and surface areas of 25 Triangles and some practical frusta of pyramids and cones 164 applications 216 20.5 The frustum and zone of 25.1 Sine and cosine rules 216 a sphere 167 25.2 Area of any triangle 216 20.6 Prismoidal rule 170 25.3 Worked problems on the solution 20.7 Volumes of similar shapes 172 of triangles and their areas 216 25.4 Further worked problems on 21 Irregular areas and volumes and the solution of triangles and mean values of waveforms 174 their areas 218 21.1 Area of irregular figures 174 25.5 Practical situations involving 21.2 Volumes of irregular solids 176 trigonometry 220 21.3 The mean or average value of 25.6 Further practical situations a waveform 177 involving trigonometry 222 26 Trigonometric identities and equations 225 Revision Test 5 182 26.1 Trigonometric identities 225 26.2 Worked problems on trigonometric identities 225 Section 3 Trigonometry 185 26.3 Trigonometric equations 226 26.4 Worked problems (i) on trigonometric equations 227 22 Introduction to trigonometry 187 26.5 Worked problems (ii) on 22.1 Trigonometry 187 trigonometric equations 228 22.2 The theorem of Pythagoras 187 26.6 Worked problems (iii) on 22.3 Trigonometric ratios of acute angles 188 trigonometric equations 229 22.4 Fractional and surd forms of 26.7 Worked problems (iv) on trigonometric ratios 190 trigonometric equations 229 22.5 Solution of right-angled triangles 191 22.6 Angle of elevation and depression 193 27 Compound angles 231 22.7 Evaluating trigonometric 27.1 Compound angle formulae 231 ratios of any angles 195 27.2 Conversion of a sin ωt + b cos ωt 22.8 Trigonometric approximations into R sin(ωt + α) 233 for small angles 197 27.3 Double angles 236 27.4 Changing products of sines 23 Trigonometric waveforms 199 and cosines into sums or 23.1 Graphs of trigonometric functions 199 differences 238 23.2 Angles of any magnitude 199 27.5 Changing sums or differences 23.3 The production of a sine and of sines and cosines into cosine wave 202 products 239 23.4 Sine and cosine curves 202 23.5 Sinusoidal form A sin(ωt ± α) 206 23.6 Waveform harmonics 209 Revision Test 7 241 24 Cartesian and polar co-ordinates 211 24.1 Introduction 211 Multiple choice questions on 24.2 Changing from Cartesian into Chapters 18–27 242 polar co-ordinates 211Prelims-H8555.tex 2/8/2007 9: 34 page viii viii Contents 34.3 Determining resultant Section 4 Graphs 247 phasors by calculation 308 28 Straight line graphs 249 Section 6 Complex Numbers 311 28.1 Introduction to graphs 249 28.2 The straight line graph 249 28.3 Practical problems involving 35 Complex numbers 313 straight line graphs 255 35.1 Cartesian complex numbers 313 35.2 The Argand diagram 314 29 Reduction of non-linear laws to 35.3 Addition and subtraction of linear form 261 complex numbers 314 29.1 Determination of law 261 35.4 Multiplication and division of 29.2 Determination of law complex numbers 315 involving logarithms 264 35.5 Complex equations 317 35.6 The polar form of a complex 30 Graphs with logarithmic scales 269 number 318 30.1 Logarithmic scales 269 35.7 Multiplication and division in n 30.2 Graphs of the form y = ax 269 polar form 320 x 30.3 Graphs of the form y = ab 272 35.8 Applications of complex kx 30.4 Graphs of the form y = ae 273 numbers 321 31 Graphical solution of equations 276 36 De Moivre’s theorem 325 31.1 Graphical solution of 36.1 Introduction 325 simultaneous equations 276 36.2 Powers of complex numbers 325 31.2 Graphical solution of 36.3 Roots of complex numbers 326 quadratic equations 277 31.3 Graphical solution of linear and quadratic equations Revision Test 9 329 simultaneously 281 31.4 Graphical solution of cubic equations 282 Section 7 Statistics 331 32 Functions and their curves 284 32.1 Standard curves 284 37 Presentation of statistical data 333 32.2 Simple transformations 286 37.1 Some statistical terminology 333 32.3 Periodic functions 291 37.2 Presentation of ungrouped data 334 32.4 Continuous and 37.3 Presentation of grouped data 338 discontinuous functions 291 32.5 Even and odd functions 291 38 Measures of central tendency and 32.6 Inverse functions 293 dispersion 345 38.1 Measures of central tendency 345 Revision Test 8 297 38.2 Mean, median and mode for discrete data 345 38.3 Mean, median and mode for grouped data 346 Section 5 Vectors 299 38.4 Standard deviation 348 38.5 Quartiles, deciles and 33 Vectors 301 percentiles 350 33.1 Introduction 301 33.2 Vector addition 301 39 Probability 352 33.3 Resolution of vectors 302 39.1 Introduction to probability 352 33.4 Vector subtraction 305 39.2 Laws of probability 353 34 Combination of waveforms 307 39.3 Worked problems on 34.1 Combination of two periodic probability 353 functions 307 39.4 Further worked problems on 34.2 Plotting periodic functions 307 probability 355Prelims-H8555.tex 2/8/2007 9: 34 page ix Contents ix 39.5 Permutations and 44.5 Tangents and normals 411 combinations 357 44.6 Small changes 412 Revision Test 10 359 Revision Test 12 415 40 The binomial and Poisson distribution 360 45 Differentiation of parametric 40.1 The binomial distribution 360 equations 416 40.2 The Poisson distribution 363 45.1 Introduction to parametric equations 416 41 The normal distribution 366 45.2 Some common parametric 41.1 Introduction to the normal equations 416 distribution 366 45.3 Differentiation in parameters 417 41.2 Testing for a normal 45.4 Further worked problems on distribution 371 differentiation of parametric equations 418 Revision Test 11 374 46 Differentiation of implicit functions 421 46.1 Implicit functions 421 46.2 Differentiating implicit functions 421 Multiple choice questions on 46.3 Differentiating implicit Chapters 28–41 375 functions containing products and quotients 422 46.4 Further implicit Section 8 Differential Calculus 381 differentiation 423 42 Introduction to differentiation 383 47 Logarithmic differentiation 426 42.1 Introduction to calculus 383 47.1 Introduction to logarithmic 42.2 Functional notation 383 differentiation 426 42.3 The gradient of a curve 384 47.2 Laws of logarithms 426 42.4 Differentiation from first 47.3 Differentiation of logarithmic principles 385 functions 426 n x 42.5 Differentiation of y = ax by 47.4 Differentiation of f (x) 429 the general rule 387 42.6 Differentiation of sine and cosine functions 388 Revision Test 13 431 ax 42.7 Differentiation of e and ln ax 390 Section 9 Integral Calculus 433 43 Methods of differentiation 392 43.1 Differentiation of common 48 Standard integration 435 functions 392 48.1 The process of integration 435 43.2 Differentiation of a product 394 48.2 The general solution of 43.3 Differentiation of a quotient 395 n integrals of the form ax 435 43.4 Function of a function 397 48.3 Standard integrals 436 43.5 Successive differentiation 398 48.4 Definite integrals 439 44 Some applications of differentiation 400 49 Integration using algebraic 44.1 Rates of change 400 substitutions 442 44.2 Velocity and acceleration 401 49.1 Introduction 442 44.3 Turning points 404 49.2 Algebraic substitutions 442 44.4 Practical problems involving 49.3 Worked problems on maximum and minimum integration using algebraic values 408 substitutions 442Prelims-H8555.tex 2/8/2007 9: 34 page x x Contents 49.4 Further worked problems on 54.3 The mid-ordinate rule 471 integration using algebraic 54.4 Simpson’s rule 473 substitutions 444 49.5 Change of limits 444 Revision Test 15 477 50 Integration using trigonometric substitutions 447 55 Areas under and between curves 478 50.1 Introduction 447 55.1 Area under a curve 478 50.2 Worked problems on 2 2 55.2 Worked problems on the area integration of sin x, cos x, 2 2 under a curve 479 tan x and cot x 447 55.3 Further worked problems on 50.3 Worked problems on powers the area under a curve 482 of sines and cosines 449 55.4 The area between curves 484 50.4 Worked problems on integration of products of sines and cosines 450 50.5 Worked problems on integration 56 Mean and root mean square values 487 using the sin θ substitution 451 56.1 Mean or average values 487 50.6 Worked problems on integration 56.2 Root mean square values 489 using the tan θ substitution 453 57 Volumes of solids of revolution 491 57.1 Introduction 491 Revision Test 14 454 57.2 Worked problems on volumes of solids of revolution 492 51 Integration using partial fractions 455 57.3 Further worked problems on 51.1 Introduction 455 volumes of solids of 51.2 Worked problems on revolution 493 integration using partial fractions with linear factors 455 58 Centroids of simple shapes 496 51.3 Worked problems on integration 58.1 Centroids 496 using partial fractions with 58.2 The first moment of area 496 repeated linear factors 456 58.3 Centroid of area between a 51.4 Worked problems on integration curve and the x-axis 496 using partial fractions with 58.4 Centroid of area between a quadratic factors 457 curve and the y-axis 497 58.5 Worked problems on θ centroids of simple shapes 497 52 The t = tan substitution 460 2 58.6 Further worked problems 52.1 Introduction 460 on centroids of simple shapes 498 52.2 Worked problems on the 58.7 Theorem of Pappus 501 θ t = tan substitution 460 2 52.3 Further worked problems on 59 Second moments of area 505 θ 59.1 Second moments of area and the t = tan substitution 462 2 radius of gyration 505 59.2 Second moment of area of 53 Integration by parts 464 regular sections 505 53.1 Introduction 464 59.3 Parallel axis theorem 506 53.2 Worked problems on 59.4 Perpendicular axis theorem 506 integration by parts 464 59.5 Summary of derived results 506 53.3 Further worked problems on 59.6 Worked problems on second integration by parts 466 moments of area of regular sections 507 54 Numerical integration 469 59.7 Worked problems on second 54.1 Introduction 469 moments of area of 54.2 The trapezoidal rule 469 composite areas 510Prelims-H8555.tex 2/8/2007 9: 34 page xi 62.2 Solution of simultaneous Revision Test 16 512 equations by determinants 548 62.3 Solution of simultaneous equations using Cramers rule 552 Section 10 Further Number and Algebra 513 Revision Test 17 553 60 Boolean algebra and logic circuits 515 60.1 Boolean algebra and switching circuits 515 Section 11 Differential Equations 555 60.2 Simplifying Boolean expressions 520 60.3 Laws and rules of Boolean 63 Introduction to differential algebra 520 equations 557 60.4 De Morgan’s laws 522 63.1 Family of curves 557 60.5 Karnaugh maps 523 63.2 Differential equations 558 60.6 Logic circuits 528 63.3 The solution of equations of dy 60.7 Universal logic gates 532 the form = f (x) 558 dx 63.4 The solution of equations of 61 The theory of matrices and dy determinants 536 the form = f (y) 560 dx 61.1 Matrix notation 536 63.5 The solution of equations of 61.2 Addition, subtraction and dy the form = f (x) · f (y) 562 multiplication of matrices 536 dx 61.3 The unit matrix 540 61.4 The determinant ofa2by2 matrix 540 Revision Test 18 565 61.5 The inverse or reciprocal of a 2 by 2 matrix 541 61.6 The determinant ofa3by3 matrix 542 Multiple choice questions on 61.7 The inverse or reciprocal of a Chapters 42–63 566 3 by 3 matrix 544 62 The solution of simultaneous Answers to multiple choice questions 570 equations by matrices and determinants 546 62.1 Solution of simultaneous Index 571 equations by matrices 546Prelims-H8555.tex 2/8/2007 9: 34 page xii Preface Engineering Mathematics 5th Edition covers a wide statistics, differential calculus, integral calculus, further range of syllabus requirements. In particular, the book number and algebra and differential equations. is most suitable for the latest National Certificate and Diploma courses and City & Guilds syllabuses in This new edition covers, in particular, the following Engineering. syllabuses: This text will provide a foundation in mathemat- ical principles, which will enable students to solve (i) Mathematics for Technicians, the core unit for mathematical, scientific and associated engineering National Certificate/Diploma courses in Engi- principles. In addition, the material will provide neering, to include all or part of the following engineering applications and mathematical principles chapters: necessary for advancement onto a range of Incorporated 1. Algebraic methods: 2, 5, 11, 13, 14, 28, 30 Engineer degree profiles. It is widely recognised that a (1, 4, 8, 9 and 10 for revision) students’ ability to use mathematics is a key element in 2. Trigonometric methods and areas and vol- determining subsequent success. First year undergrad- umes: 18–20, 22–25, 33, 34 uates who need some remedial mathematics will also 3. Statistical methods: 37, 38 find this book meets their needs. 4. Elementary calculus: 42, 48, 55 In Engineering Mathematics 5th Edition,new (ii) Further Mathematics for Technicians, the material is included on inequalities, differentiation of optional unit for National Certificate/Diploma parametric equations, implicit and logarithmic func- courses in Engineering, to include all or part of tions and an introduction to differential equations. the following chapters: Because of restraints on extent, chapters on lin- 1. Advanced graphical techniques: 29–31 ear correlation, linear regression and sampling and 2. Algebraic techniques: 15, 35, 38 estimation theories have been removed. However, 2. Trigonometry: 23, 26, 27, 34 these three chapters are available to all via the 3. Calculus: 42–44, 48, 55–56 internet. A new feature of this fifth edition is that a free Inter- (iii) The mathematical contents of Electrical and net download is available of a sample of solutions Electronic Principles units of the City & Guilds (some 1250) of the 1750 further problems contained in Level 3 Certificate in Engineering (2800). the book – see below. (iv) Any introductory/access/foundation course Another new feature is a free Internet download involving Engineering Mathematics at (available for lecturers only) of all 500 illustrations University, Colleges of Further and Higher contained in the text – see below. education and in schools. Throughout the text theory is introduced in each chapter by a simple outline of essential definitions, Each topic considered in the text is presented in a way formulae, laws and procedures. The theory is kept to that assumes in the reader little previous knowledge of a minimum, for problem solving is extensively used that topic. to establish and exemplify the theory. It is intended Engineering Mathematics 5th Edition provides that readers will gain real understanding through see- a follow-up to Basic Engineering Mathematics and ing problems solved and then through solving similar a lead into Higher Engineering Mathematics 5th problems themselves. Edition. For clarity, the text is divided into eleven topic This textbook contains over 1000 worked problems, areas, these being: number and algebra, mensura- followed by some 1750 further problems (all with tion, trigonometry, graphs, vectors, complex numbers,Prelims-H8555.tex 2/8/2007 9: 34 page xiii answers). The further problems are contained within structure. Lecturers’ may obtain a complimentary set some 220 Exercises; each Exercise follows on directly of solutions of the Revision Tests in an Instructor’s from the relevant section of work, every two or three Manual available from the publishers via the internet – pages. In addition, the text contains 238 multiple- see below. choice questions. Where at all possible, the problems A list of Essential Formulae is included in mirror practical situations found in engineering and sci- the Instructor’s Manual for convenience of reference. ence. 500 line diagrams enhance the understanding of Learning by Example is at the heart of Engineering the theory. Mathematics 5th Edition. At regular intervals throughout the text are some 18 Revision tests to check understanding. For example, JOHN BIRD Revision test 1 covers material contained in Chapters Royal Naval School of Marine Engineering, 1 to 4, Revision test 2 covers the material in Chapters HMS Sultan, 5 to 8, and so on. These Revision Tests do not have formerly University of Portsmouth and answers given since it is envisaged that lecturers could Highbury College, set the tests for students to attempt as part of their course PortsmouthPrelims-H8555.tex 2/8/2007 9: 34 page xiv Free web downloads Additional material on statistics Chapters on Linear correlation, Linear regression and Sampling and estimation theories are available for free to students and lecturers at http://books.elsevier.com/ companions/9780750685559 In addition, a suite of support material is available to lecturers only from Elsevier’s textbook website. Solutions manual Within the text are some 1750 further problems arranged within 220 Exercises. A sample of over 1250 worked solutions has been prepared for lecturers. Instructor’s manual This manual provides full worked solutions and mark scheme for all 18 Revision Tests in this book. Illustrations Lecturers can also download electronic files for all illustrations in this fifth edition. To access the lecturer support material, please go to http://textbooks.elsevier.com and search for the book. On the book web page, you will see a link to the Instruc- tor Manual on the right. If you do not have an account for the textbook website already, you will need to reg- ister and request access to the book’s subject area. If you already have an account but do not have access to the right subject area, please follow the ’Request Access to this Subject Area’ link at the top of the subject area homepage.Ch01-H8555.tex 1/8/2007 18: 7 page 1 Section 1 Number and AlgebraThis page intentionally left blank Ch01-H8555.tex 1/8/2007 18: 7 page 3 Chapter 1 Revision of fractions, decimals and percentages Alternatively: 1.1 Fractions Step (2) Step (3) 2 When 2 is divided by 3, it may be written as or 2/3 or 3 ↓↓ 2 2/3. is called a fraction. The number above the line, (7 × 1) + (3 × 2) 3 1 2 + = i.e. 2, is called the numerator and the number below 3 7 21 the line, i.e. 3, is called the denominator. ↑ When the value of the numerator is less than the Step (1) value of the denominator, the fraction is called a proper 2 fraction; thus is a proper fraction. When the value 3 Step 1: the LCM of the two denominators; of the numerator is greater than the denominator, the 1 7 Step 2: for the fraction , 3 into 21 goes 7 times, fraction is called an improper fraction. Thus is 3 3 7 × the numerator is 7 × 1; an improper fraction and can also be expressed as a 2 mixed number, that is, an integer and a proper frac- Step 3: for the fraction , 7 into 21 goes 3 times, 7 7 tion. Thus the improper fraction is equal to the mixed 3 × the numerator is 3 × 2. 3 1 number 2 . 3 1 2 7 + 6 13 When a fraction is simplified by dividing the numer- Thus + = = as obtained previously. 3 7 21 21 ator and denominator by the same number, the pro- cess is called cancelling. Cancelling by 0 is not 2 1 permissible. Problem 2. Find the value of 3 − 2 3 6 1 2 Problem 1. Simplify + One method is to split the mixed numbers into integers 3 7 and their fractional parts. Then     The lowest common multiple (i.e. LCM) of the two 2 1 2 1 3 − 2 = 3 + − 2 + denominators is 3 × 7, i.e. 21. 3 6 3 6 Expressing each fraction so that their denominators 2 1 are 21, gives: = 3 + − 2 − 3 6 1 2 1 7 2 3 7 6 4 1 3 1 + = × + × = + = 1 + − = 1 = 1 3 7 3 7 7 3 21 21 6 6 6 2 7 + 6 13 Another method is to express the mixed numbers as = = 21 21 improper fractions.Ch01-H8555.tex 1/8/2007 18: 7 page 4 4 Engineering Mathematics 1 8 9 2 9 2 11  8 7 24 8 × 1 × 8   Since 3 = , then 3 = + = = × × = 3 3 3 3 3 5 3 7 5 × 1 × 1 1 1   1 12 1 13 64 4 Similarly, 2 = + = = = 12 6 6 6 6 5 5 2 1 11 13 22 13 9 1 Thus 3 − 2 = − = − = = 1 3 6 3 6 6 6 6 2 3 12 Problem 6. Simplify ÷ as obtained previously. 7 21 Problem 3. Determine the value of 3 3 12 7 5 1 2 ÷ = 4 − 3 + 1 12 7 21 8 4 5 21 Multiplying both numerator and denominator by the   reciprocal of the denominator gives: 5 1 2 5 1 2 4 − 3 + 1 = (4 − 3 + 1) + − + 8 4 5 8 4 5 1 3  3 21 3   3 × 5 × 5 − 10 × 1 + 8 × 2  3 7 12 7 1  4 4  = 2 + = = = 1 1 40 12 12  21  1 4   × 25 − 10 + 16 21 21  12   1  1 = 2 + 40 This method can be remembered by the rule: invert the 31 31 = 2 + = 2 second fraction and change the operation from division 40 40 to multiplication. Thus: 1 3 3 12 3 21  3   3 14 ÷ = × = as obtained previously. Problem 4. Find the value of × 7 21 12  4 7  1 4 7 15 3 1 Dividing numerator and denominator by 3 gives: Problem 7. Find the value of 5 ÷ 7 5 3 1 3 14 1 14 1 × 14  × = × =  7 15 7 5 7 × 5  5 The mixed numbers must be expressed as improper fractions. Thus, Dividing numerator and denominator by 7 gives: 2 1 × 14  1 × 2 2  14  3 1 28 22 28 3 42  = = 5 ÷ 7 = ÷ = × = 7 × 5 1 × 5 5 1  5 3 5 3 5 22 55  11 This process of dividing both the numerator and denom- inator of a fraction by the same factor(s) is called Problem 8. Simplify cancelling.     1 2 1 3 1 − + ÷ × 3 1 3 3 5 4 8 3 Problem 5. Evaluate 1 × 2 × 3 5 3 7 The order of precedence of operations for problems Mixed numbers must be expressed as improper frac- containing fractions is the same as that for integers, tions before multiplication can be performed. Thus, i.e. remembered by BODMAS (Brackets, Of, Division, Multiplication, Addition and Subtraction). Thus, 3 1 3 1 × 2 × 3 5 3 7           5 3 6 1 21 3 1 2 1 3 1 = + × + × + − + ÷ × 5 5 3 3 7 7 3 5 4 8 3 Section 1Ch01-H8555.tex 1/8/2007 18: 7 page 5 Revision of fractions, decimals and percentages 5 1 1 4 × 2 + 5 × 1 3  = − ÷ (B) 2 3 2 1 2 3 20 24  2. (a) + (b) − +  8 7 11 9 7 3 2   1 13 8  43 47 = − × (D) (a) (b) 3 20  1  5 77 63 1 26 3 2 1 4 5 = − (M) 3. (a) 10 − 8 (b) 3 − 4 + 1 3 5 7 3 4 5 6 (5 × 1) − (3 × 26)   = (S) 16 17 15 (a) 1 (b) 21 60 −73 13 = =−4 3 5 17 15 15 15 4. (a) × (b) × 4 9 35 119   Problem 9. Determine the value of 5 3 (a) (b)   12 49 7 1 1 1 3 1 of 3 − 2 + 5 ÷ − 3 7 2 13 7 4 6 2 4 8 16 2 5. (a) × × 1 (b) × 4 × 3 5 9 7 17 11 39     7 1 1 1 3 1 3 of 3 − 2 + 5 ÷ − (a) (b) 11 6 2 4 8 16 2 5 7 1 41 3 1 3 45 1 5 = of 1 + ÷ − (B) 6. (a) ÷ (b) 1 ÷ 2 6 4 8 16 2 8 64 3 9   8 12 7 5 41 3 1 (a) (b) = × + ÷ − (O) 15 23 6 4 8 16 2   2  1 3 8 1 7 7 5 41 16 1  7. + ÷ − 1 = × + × − (D) 2 5 15 3 24 6 4 8 3 2 1       35 82 1 7 5 3 15 4 = + − (M) 8. of 15 × + ÷ 5 24 3 2 15 7 4 16 5   35 + 656 1 1 2 1 3 2 13 = − (A) 9. × − ÷ + − 24 2 4 3 3 5 7 126       691 1 2 1 2 1 3 28 = − (A) 10. × 1 ÷ + + 1 2 24 2 3 4 3 4 5 55 691 − 12 = (S) 11. If a storage tank is holding 450 litres when it is 24 three-quarters full, how much will it contain 679 7 when it is two-thirds full? = = 28 24 24 400 litres 12. Three people, P, Q and R contribute to a fund. P provides 3/5 of the total, Q provides 2/3 of Now try the following exercise the remainder, and R provides £8. Determine (a) the total of the fund, (b) the contributions Exercise 1 Further problems on fractions of P and Q. (a) £60 (b) £36, £16 Evaluate the following: 1 2 7 1 1.2 Ratio and proportion 1. (a) + (b) − 2 5 16 4   9 3 The ratio of one quantity to another is a fraction, and (a) (b) 10 16 is the number of times one quantity is contained in another quantity of the same kind. If one quantity is Section 1