MATHS Quest. General Mathematics HSC COURSE. Robert Rowland I T I O N S E C O N D E D

Transcription

1 MATHS Quest General Mathematics HSC COURSE S E C O N D E D I T I O N Robert Rowland

2 Second edition published 007 by John Wiley & Sons Australia, Ltd McDougall Street, Milton, Qld 06 First edition published 00 Typeset in 0.5/.5 pt Times John Wiley & Sons Australia, Ltd 00, 007 The moral rights of the author have been asserted. National Library of Australia Cataloguing-in-Publication data Rowland, Robert, 96. Maths quest: general mathematics HSC. 50 nd ed. Includes index. For secondary school students. ISBN (Student edition) ISBN (Teacher edition). Mathematics - Problems, exercises, etc.. Mathematics. I. Title. Reproduction and communication for educational purposes The Australian Copyright Act 968 allows a maximum of one chapter or 0% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. Cover photograph and internal design images: Digital Vision Cartography by MAPgraphics Pty Ltd, Brisbane Illustrated by the Wiley Art Studio Printed in China by Printplus Limited

3 Contents Introduction vi About ebookplus viii Acknowledgements ix CHAPTER Credit and borrowing Are you ready? Flat rate interest Exercise A 6 Home loans 9 Exercise B 0 Quick Questions 6 The cost of a loan 6 Exercise C 9 Investigation Researching home loans Credit cards Exercise D 6 Investigation Researching credit cards 8 0 Quick Questions 8 Loan repayments 9 Exercise E Summary Chapter review 6 Practice examination questions 9 CHAPTER Further applications of area and volume Are you ready? Area of parts of the circle Exercise A 5 Area of composite shapes 8 Exercise B 50 0 Quick Questions 5 Simpson s rule 5 Exercise C 55 Surface area of cylinders and spheres 57 Exercise D 59 Investigation Packaging 6 Volume of composite solids 6 Exercise E 65 Investigation Maximising volume 67 0 Quick Questions 68 Error in measurement 69 Exercise F 7 Summary 7 Chapter review 75 Practice examination questions 78 CHAPTER Applications of trigonometry 79 Are you ready? 80 Review of right-angled triangles 8 Exercise A 85 Bearings 86 Exercise B 89 Investigation Trigonometric ratios for obtuse angles 9 The sine rule 9 Investigation Derivation of the sine rule 9 Exercise C 95 Exercise D 99 0 Quick Questions 0 Area of a triangle 0 Exercise E 0 The cosine rule 06 Investigation Derivation of the cosine rule 06 Exercise F 09 Exercise G 5 0 Quick Questions 7 Radial surveys 8 Exercise H Investigation Conducting a radial survey Summary Chapter review 5 Practice examination questions 7 CHAPTER Interpreting sets of data 9 Are you ready? 0 Measures of location and spread Exercise A 6 Skewness 0 Exercise B 0 Quick Questions 5 Displaying multiple data sets 5 Investigation Examining exam results 6 Exercise C 50

4 iv Comparison of data sets 5 Exercise D 55 Investigation Developing a two-way table 59 Summary 60 Chapter review 6 Practice examination questions 65 CHAPTER 5 Algebraic skills and techniques 67 Are you ready? 68 Substitution 69 Exercise 5A 7 Algebraic manipulation 7 Exercise 5B 7 0 Quick Questions 75 Equations and formulas 75 Exercise 5C 78 Solution by substitution 80 Exercise 5D 8 Investigation Repeated enlargements 8 0 Quick Questions 8 Scientific notation 8 Exercise 5E 86 Summary 87 Chapter review 88 Practice examination questions 89 CHAPTER 6 Multi-stage events 9 Are you ready? 9 Tree diagrams 9 Exercise 6A 95 Counting techniques 96 Investigation Ordered arrangements 96 Investigation Tree diagrams and ordered arrangements 98 Investigation Committee selections 99 Investigation Unordered selection 00 Exercise 6B 00 Probability and counting techniques 0 Investigation Popular gaming 0 Exercise 6C 0 0 Quick Questions 0 Probability trees 05 Exercise 6D 09 Summary Chapter review Practice examination questions 5 CHAPTER 7 Applications of probability 7 Are you ready? 8 Expected outcomes 9 Investigation Rolling a die 9 Exercise 7A Financial expectation Exercise 7B 5 0 Quick Questions 6 Two-way tables 7 Exercise 7C 9 Summary Chapter review Practice examination questions 5 CHAPTER 8 Annuities and loan repayments 7 Are you ready? 8 Future value of an annuity 9 Exercise 8A 0 Quick Questions 5 Present value of an annuity 6 Exercise 8B 8 Future and present value tables 50 Exercise 8C 5 0 Quick Questions 5 Loan repayments 55 Exercise 8D 57 Investigation Types of loan arrangements 59 Summary 60 Chapter review 6 Practice examination questions 6 CHAPTER 9 Modelling linear and non-linear relationships 65 Are you ready? 66 Linear functions 67 Exercise 9A 7 Investigation Conversion of temperature 7 Quadratic functions 7 Exercise 9B 78 Investigation Maximising areas 80 0 Quick Questions 80

5 v Other functions 8 Exercise 9C 8 Investigation Compound interest 85 Variations 85 Exercise 9D 88 Graphing physical phenonema 89 Exercise 9E 9 Investigation Force of gravity 9 Summary 9 Chapter review 95 Practice examination questions 97 CHAPTER 0 Depreciation 99 Are you ready? 00 Modelling depreciation 0 Investigation Depreciation of motor vehicles 0 Exercise 0A 0 Straight line depreciation 07 Exercise 0B 09 Declining balance method of depreciation 0 Exercise 0C Investigation Rates of depreciation 0 Quick Questions Depreciation tables Exercise 0D 9 Summary Chapter review Practice examination questions 5 CHAPTER The normal distribution 7 Are you ready? 8 z-scores 9 Exercise A Comparison of scores Exercise B 6 0 Quick Questions 8 Investigation Comparison of subjects 8 Distribution of scores 9 Exercise C Investigation Examining a normal distribution Summary Chapter review 5 Practice examination questions 6 CHAPTER Correlation 9 Are you ready? 50 Scatterplots 5 Exercise A 55 Investigation Collecting bivariate data 57 Regression lines 57 Exercise B 59 Exercise C 65 Investigation Relationship between variables 69 0 Quick Questions 69 Correlation 70 Investigation Causality 7 Exercise D 7 Summary 78 Chapter review 79 Practice examination questions 8 CHAPTER Spherical geometry 8 Are you ready? 8 Arc lengths 85 Exercise A 86 Great circles and small circles 89 Exercise B 90 0 Quick Questions 9 Latitude and longitude 9 Exercise C 96 Investigation Important parallels of latitude 97 Distances on the Earth s surface 97 Exercise D 99 0 Quick Questions 0 Time zones 0 Investigation Australian time zones 0 Exercise E 0 Investigation The keepers of time 05 Summary 06 Chapter review 07 Practice examination questions 09 Glossary Formula sheet Answers 7 Index 9

6 Introduction Maths Quest General Mathematics HSC course is the second book in a series specifically designed for the General Mathematics Stage 6 Syllabus starting in 000. This course replaces the current syllabuses for Mathematics in Society (98) and Mathematics in Practice (989). There are five new areas of study: Financial mathematics Data analysis Measurement Probability Algebraic modelling. This resource contains: a student textbook with accompanying ebookplus and a teacher edition with accompanying eguideplus. Student textbook Full colour is used throughout to produce clearer graphs and diagrams, to provide bright, stimulating photos and to make navigation through the text easier. Clear, concise theory sections contain worked examples, highlighted important text and remember boxes. Worked examples in a Think/Write format provide a clear explanation of key steps and suggest a presentation for solutions. Exercises contain many carefully graded skills and application problems, including multiple-choice questions. Cross-references to relevant worked examples appear beside the first matching question throughout the exercises. Investigations, including spreadsheet investigations, provide further learning opportunities through discovery. Sets of 0 Quick Questions allow students to quickly review the concepts just learnt before proceeding further in the chapter. A glossary of mathematical terms is provided to assist students understanding of the terminology introduced in each unit of the course. Words in bold type in the theory sections of each chapter are defined in the glossary at the back of the book. Each chapter concludes with a summary and chapter review exercise, containing questions in a variety of forms (multiple-choice, short-answer and analysis) that help consolidate students learning of new concepts. Practice examination questions provide a ready source of problems for students to use to gain further confidence in each topic.

7 vii Technology is fully integrated, in line with Board of Studies recommendations. As well as graphics calculators, Maths Quest features spreadsheets, dynamic geometry software and several graphing packages. Not only does the text promote these technologies as learning tools, but demonstration versions of the programs (with the exception of Microsoft Excel) are also included, as well as hundreds of supporting files available online. Graphics calculator tips are incorporated throughout the text. All formulas, which are given on the examination formula sheet, are marked with the symbol. Programs included Graphmatica: an excellent graphing utility Equation grapher and regression analyser: like a graphics calculator for the PC GrafEq: graphs any relation, including complicated inequalities Poly: for visualising D polyhedra and their nets Tess: for producing tessellations and other symmetric planar illustrations TI Connect: calculator screen capture and program transfer CASIO Software FA-: calculator screen capture and program transfer Cabri Geometry II: dynamic geometry program Adobe Acrobat Reader.0 Teacher edition with accompanying eguideplus The teacher edition textbook contains everything in the student textbook and more. To support teachers assisting students in class, answers appear in red next to most questions in the exercises. Each exercise is annotated with relevant study design dot points. A readily accessible Work program lists all available resources and provides curriculum coverage information. The accompanying online resources contain everything in the student ebookplus and more. Four tests per chapter, fully worked solutions to WorkSHEETs, the work program and other curriculum advice in editable Word 000 format are provided. Maths Quest is a rich collection of teaching and learning resources within one package. Maths Quest General Mathematics HSC course provides ample material, such as exercises, analysis questions, investigations, worksheets and technology files, from which teachers may set assessment tasks.

8 Next generation teaching and learning About ebookplus This book features ebookplus: an electronic version of the entire textbook and supporting multimedia resources. It is available for you online at the JacarandaPLUS website ( These additional resources include: Word documents designed for easy customisation and editing interactive activities and a wealth of ICT resources weblinks to other useful resources and information on the internet. Using the JacarandaPLUS website To access your ebookplus resources, simply log on to There are three easy steps for using the JacarandaPLUS system. Step. Create a user account The first time you use the JacarandaPLUS system, you will need to create a user account. Go to the JacarandaPLUS home page ( and follow the instructions on screen. Step. Enter your registration code Once you have created a new account and logged in, you will be prompted to enter your unique registration code for this book, which is printed on the inside front cover of your textbook. LOGIN Once you have created your account, you can use the same address and password in the future to register any JacarandaPLUS books. Step. View or download ebookplus resources Your ebook and supporting resources are provided in a chapter-by-chapter format. Simply select the desired chapter from the drop-down list and navigate through the tabs to locate the appropriate resource. Troubleshooting Go to the JacarandaPLUS help page at Contact John Wiley & Sons Australia, Ltd. support@jacplus.com.au Phone: 800 JAC PLUS ( ) Key to the icons in the book In this student text, the resource icons, such as the one shown, indicate that digital support materials for this section of the text are available online at JacarandaPLUS. WorkSHEET Minimum requirements Internet Explorer 7, Mozilla Firefox.5 or Safari. Adobe Flash Player 9 Javascript must be enabled (most browsers are enabled by default).

9 Acknowledgements The Maths Quest project began in 997, and the first edition of this book was printed in 00. In that time we believe that Maths Quest has become the best resourced mathematical database in Australian education. I would like to thank all of those people who have supported us with our first edition. I hope that we have been able to help you in achieving your goals and have also played a part in your successes. Technology has evolved greatly since our first edition was published. The second edition has evolved from being a textbook in its first edition, into an interactive resource for both students and teachers. I would like to thank everyone at John Wiley & Sons Australia, Ltd for giving me the opportunity to do this. There are three people in particular that I would like to single out for special mention: Jennifer Nolan, whose support for the Maths Quest project and for me personally has made everything possible; Ingrid Kemp, the newest addition to our team. Ingrid has brought a new set of eyes to our project and kept the ball rolling thanks, Ingrid; and finally, Keith Hartmann, who has tirelessly reviewed all of the new material and has completed all of the answer checking. Thanks, Keith hope you re enjoying retirement! Finally and most importantly to my family thank you. Without your support this book and online resources would never have been completed. The author and publisher would like to thank the following copyright holders, organisations and individuals for their assistance and for permission to reproduce copyright material in this book. Illustrative material Banana Stock: p. 0 Brand X Pictures: p. Comstock: p. Corbis Corporation: p. 67, p. 7, p. 77, p. 00, p. 6, p. 77 Digital Stock: p. 0/Corbis Corporation, p. 7/Corbis Corporation, p. / Corbis Corporation, p. 58/Corbis Corporation, p. 8/Corbis Corporation, p. 67/Corbis Corporation, p. 79/Corbis Corporation, p. 9, Corbis Corporation, p. 0/Corbis Corporation, p. 7/Corbis Corporation, p. /Corbis Corporation, p. 97/Corbis Corporation, p. 08/Corbis Corporation Digital Vision: p. 9, p. (lower), p. 6, p., p. 9, p. 8, p. 88, p. 9 Image Addict: p. IT StockFree: p. 66, p. 98 John Wiley & Sons Australia: p. xi/photo by Jo Patterson, p. 9/Photo by Werner Langer, p. 99/Taken by Kari-Ann Tapp MAPgraphics: p. 9/Map by MAPgraphics Pty Ltd, Brisbane. Photodisc: p., p. 7, p. 8, p. 7, p., p. 7, p. 5 (all), p. 55, p. 69, p. 79, p. 05, p. 6, p. 8, p. 5, p. 9, p. 9, p. (upper), p. 58, p. 59, p. 7, p. 9, p. 95, p. 96, p. 05, p. 08, p. 0, p., p., p. 5, p. 7, p. 9, p., p. 5, p., p. 7, p. 9, p. 6, p. 65, p. 79, p. 89, p. 07, p., p., p. 7, p., p. 6, p. 7, p. 9, p. 5, p. 56, p. 57 (both), p. 68, p. 70, p. 7, p. 80, p. 96 Purestock: p. 05 Stockbyte: p. 90

10 x Software The authors and publisher would like to thank the following software providers for their assistance and for permission to use their materials. However, the use of such material does not imply that the providers endorse this product in any way. Third party software registered full version ordering information Full versions of third party software may be obtained by contacting the companies listed below. Texas Instruments TI Connect and TI-GRAPHLINK software TI Connect and TI-GRAPHLINK software reproduced with permission of the publisher Texas Instruments Incorporated. TI Connect software available from Texas Instruments Web: Note: The TI Connectivity cable can be purchased from educational booksellers or calculator suppliers. Casio FA- Software used with permission of Casio Computer Co. Ltd. 00 All rights reserved. Distributed by Shriro Australia Pty Ltd 7 Chaplin Drive Lane Cove NSW 066 Web: and find the calculator product range If you are interested in this product after expiry, please contact Shriro Australia Pty Ltd. Graphmatica Reproduced with permission of ksoft, Inc. 5 Montecillo Dr., Walnut Creek, CA ksoft@graphmatica.com Web: Software included is for evaluation purposes only. The user is expected to register share-ware if use exceeds 0 days. Order forms are available at txt Cabri Geometry II PLUS Reproduced with permission of Cabrilog. Cabrilog 6, Robert Schuman Place 8000 Grenoble FRANCE Web: Due to copyright restrictions, the demo version of Cabri Geometry II Plus must not be used in the classroom for presentation on a regular basis.. For site licences contact Cabrilog Grenoble-France at «sales@cabri.com» or

11 xi GrafEq and Poly Evaluation copies of GrafEq and Poly have been included with permission from Pedagoguery Software, Inc. Web: Microsoft Excel, Microsoft Word and Microsoft PowerPoint Microsoft Excel, Microsoft Word and Microsoft PowerPoint are registered trademarks of the Microsoft Corporation in the United States and/or other countries. Screenshots reproduced throughout with permission from Microsoft. Every effort has been made to trace the ownership of copyright material. Information that will enable the publisher to trace the copyright holders or to rectify any error or omission in subsequent reprints will be welcome. In such cases, please contact the Permission Section of John Wiley & Sons Australia, who will arrange for the payment of the usual fee. About the author Robert Rowland has been teaching Mathematics for over 0 years and currently holds the position of Head teacher, Teaching and learning at Ulladulla High School. He taught at Cabramatta High School from 985 to 988 before taking up his appointment at Ulladulla High School in 989. Robert has successfully taught all levels of Mathematics to Year as well as Computing Studies 7 and Information Processes and Technology. Robert is the coauthor of New South Wales Maths Year 9 Standard and New South Wales Maths Year 0 Standard as well as being the author of Maths Quest General Mathematics Preliminary Course and Maths Quest General Mathematics HSC Course.

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13 Credit and borrowing syllabusreference Financial mathematics Credit and borrowing In this chapter A B C D E Flat rate interest Home loans The cost of a loan Credit cards Loan repayments

14 reyou READY? Are you ready? Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy Converting a percentage to a decimal Convert each of the following percentages to decimals. a 0% b % c 8% d.% e 0.% f 7 -- % g -- % h 0.0% Finding a percentage of a quantity (money) Find: a 0% of $5000 b 5% of $790 c 7.% of $5 000 d 0.5% of $600 e -- % of $8 000 f 0.06% of $ Calculating simple interest Calculate the simple interest earned on an investment of: a $7000 at 9% p.a. for years b $ at 6.5% p.a. for years c $ at 7 -- % for -- years d $ at 5.% p.a. for 9 months Finding values of n and r in financial formulas Find the value of n and r for each of the following investments a Interest of 6% p.a. for 5 years with interest calculated annually b Interest of 9% p.a. for years with interest calculated six-monthly c Interest of 8.8% p.a. for years with interest calculated quarterly d Interest of 7.% p.a. for 0 years with interest calculated monthly e Interest of % p.a. for June with interest calculated daily Calculating compound interest 5 Use the formula A = P( + r) n to calculate the amount to which each of the following investments will grow. a $7000 at 9% p.a. for years with interest compounded annually b $ at 6.% p.a. for 6 years with interest compounded six-monthly Calculate the amount of compound interest earned on an investment of: c $8 000 at 9.% p.a. for years with interest compounded annually d $ at 8.% p.a. for 0 years with interest compounded quarterly Substitution into a formula 6 Evaluate each of the following by substituting into the given formula. m a If d = ---, find d when m = 0 and v =. v b If A = -- (x + y)h, find A when h = 0, x = 7 and y =. c If s = ut + -- at, find s when u = 0.8, t = 5 and a =..

15 Chapter Credit and borrowing Flat rate interest During the preliminary course we calculated the simple interest earned on investments. Flat rate interest is the borrowing equivalent of simple interest. Flat rate interest applies to many small loans and hire purchase agreements. When money is borrowed from a lending institution such as a bank at a flat rate of interest, the total amount of interest is calculated as a percentage of the initial amount borrowed and then this is multiplied by the term of the loan. The term of the loan is the length of time which the loan is agreed to be repaid over. The formula for calculating the amount of flat interest to be paid on a loan is the same formula as for simple interest (I): I = Prn where P = initial quantity r = percentage interest rate per period expressed as a decimal n = number of periods As you work through the financial mathematics strand there are several formulas that use the same pronumerals. While the initial quantity (P) will be the principal in an investing scenario, it will represent the amount borrowed in a loans situation. All of these formulas use the same pronumerals and all of them require r to be expressed as a decimal. It should be part of your normal practice when doing such questions to convert the interest rate, expressed as a percentage, to a decimal. In simple or flat rate interest, r will always be a rate per annum or per year and there will be no variation on this regardless of how often interest is paid. Similarly, n will always be the number of years of the investment or loan. WORKED Example Calculate the flat interest to be paid on a loan of $0 000 at 7.5% p.a. flat interest if the loan is to be repaid over 5 years. THINK WRITE Convert the interest rate to a decimal. r = = Write the formula. Substitute the values of P, r (as a decimal) and n. I = Prn Calculate. = $7500 = $ Once the interest has been calculated, we can calculate the total amount that must be repaid in a loan. This is calculated by adding the principal and the interest.

16 Maths Quest General Mathematics HSC Course Alvin borrows $8000 to buy a car at a flat rate of 9% p.a. interest. Alvin is to repay the loan, plus interest, over years. Calculate the total amount that Alvin is to repay on this loan. THINK 5 WORKED Example WRITE Convert the interest rate to a decimal. r = 9 00 = 0.09 Write the interest formula. I = Prn Substitute the values of P, r and n. = $ Calculate the interest. = $880 Calculate the total repayments by adding the interest and principal. Total repayments = $ $880 Total repayments = $0 880 Graphics Calculator tip! Calculate simple interest Your Casio graphics calculator can perform a number of financial functions by using the TVM mode. One of the options in this mode is to calculate simple interest. Examples such as worked example above are more simply done using the arithmetic method as shown above. However, for some of the more complex questions later in this chapter it is worth familiarising yourself with this method.. From the MENU select TVM.. Press F to select Simple Interest.. The calculator has two modes of calculating interest: 60 day mode or 65 day mode. You need to make sure that it is on 65 day mode. If not, press SHIFT [SET UP], select Date Mode and press F for 65.. Press EXE to return to the previous screen and enter the data for worked example. n = 65 (as n is in days) I% = 9 PV = 8000 (principal or present value is entered as a negative)

17 Chapter Credit and borrowing 5 5. The calculator gives you two options. Press F (SI) for simple interest Press F (SFV) for future value (in other words, the principal plus interest). In this example we want the total repayments, so we press F (SPV). Most loans are repaid on a monthly basis. Once the total amount to be repaid has been calculated, this can be divided into equal monthly, fortnightly or weekly instalments. WORKED Example Narelle buys a computer on hire purchase. The cash price of the computer is $000, but Narelle must pay a 0% deposit with the balance paid at 8% p.a. flat rate interest in equal monthly instalments over years. a Calculate the deposit. b Calculate the balance owing. c Calculate the interest on the loan. d Calculate the total amount to be repaid. e Calculate the amount of each monthly instalment. THINK WRITE a Find 0% of $000. a Deposit = 0% of $000 Deposit = $00 b Subtract the deposit from the cash price to find the amount borrowed. b Balance = $000 $00 Balance = $700 c Write the interest formula. c I = Prn Substitute for P, r and n. I = $ Calculate the interest. I = $68 d Add the interest to the amount borrowed. d Total repayments = $700 + $68 Total repayments = $8 e Divide the total repayments by 6 (the number of monthly instalments in years). e Monthly repayments = $8 6 Monthly repayments = $9.00 If given the amount to be repaid each month, we can calculate the interest rate. The interest on the loan is the difference between the total repaid and the amount borrowed. This is then calculated as a yearly amount and written as a percentage of the amount borrowed.

18 6 Maths Quest General Mathematics HSC Course WORKED Example Theresa borrows $ 000 to buy a car. This is to be repaid over 5 years at $0 per month. Calculate the flat rate of interest that Theresa has been charged. THINK WRITE Calculate the total amount that is repaid. Total repayments = $0 60 Total repayments = $9 00 Subtract the principal from the total repayments to find the interest. Interest = $9 00 $ 000 Interest = $700 Calculate the interest paid each year. Interest per year = $700 5 Interest per year = $0 Write the annual interest as a $0 Interest rate = % percentage of the amount borrowed. $ 000 Interest rate = % remember. Flat rate interest is the borrowing equivalent of simple interest. It is calculated based on the initial amount borrowed.. The simple interest formula is used to calculate the amount of flat rate interest to be paid on a loan. The simple interest formula is I = Prn.. The total amount to be repaid on a loan is the principal plus interest. To calculate the amount of each instalment, we divide the total amount by the number of repayments.. When given the amount of each instalment, we can calculate the flat rate of interest. SkillSHEET SkillSHEET SkillSHEET. WORKED Converting Example a percentage to a decimal. Finding a percentage of a quantity. Calculating simple interest WORKED Example A Flat rate interest Calculate the amount of flat rate interest paid on each of the following loans. a $5000 at 7% p.a. for years b $8000 at 5% p.a. for years c $5 000 at 0% p.a. for 5 years d $9500 at 7.5% p.a. for years e $500 at 0.% p.a. for 8 months Roula buys a used car that has a cash price of $7500. She has saved a deposit of $000 and borrows the balance at 9.6% p.a. flat rate to be repaid over years. Calculate the amount of interest that Roula must pay. Ben borrows $000 for a holiday. The loan is to be repaid over years at.5% p.a. flat interest. Calculate the total repayments that Ben must make. Calculate the total amount to be paid on each of the following flat rate interest loans. a $500 at 8% p.a. over years b $ 500 at.6% p.a. over 5 years c $500 at.5% p.a. over 8 months d $00 at % p.a. over month e $ at 7% p.a. over 5 years

19 Chapter Credit and borrowing 7 WORKED Example 5 Mr and Mrs French purchase a new lounge suite, which has a cash price of $5500. They purchase the lounge on the following terms: 0% deposit with the balance to be repaid at 9% p.a. flat interest over years. Calculate: a the deposit b the balance owing c the interest to be paid d the total amount that they pay for the lounge. 6 Yasmin borrows $5000 from a credit union at a flat interest rate of 8% p.a. to be repaid over years in equal monthly instalments. Calculate: a the interest that Yasmin must pay on the loan b the total amount that Yasmin must repay c the amount of each monthly repayment. 7 Ian borrows $000 from a pawnbroker at 0% p.a. interest. The loan is to be paid over year in equal weekly payments. a Calculate the interest on the loan. b Calculate the total that Ian must repay. c Calculate Ian s weekly payment. 8 The Richards family purchase an entertainment system for their home. The total cost of the system is $8000. They buy the system on the following terms: 5% deposit with the balance repaid over years at % p.a. flat interest in equal monthly instalments. Calculate: a the deposit b the balance owing c the interest on the loan d the total repayments e the amount of each monthly repayment. 9 Sam buys an electric guitar with a cash price of $00. He buys the guitar on the following terms: one-third deposit, with the balance at 5% p.a. flat interest over years in equal monthly instalments. Calculate the amount of each monthly repayment. 0 multiple choice The amount of flat rate interest on a loan of $0 000 at 0% p.a. for years is: A $000 B $000 C $ 000 D $ 000 multiple choice A refrigerator with a cash price of $800 is bought on the following terms: 0% deposit with the balance paid in equal monthly instalments at % p.a. flat interest. The total cost of the refrigerator when purchased on terms is: A $7.80 B $6.00 C $97.80 D $06.00 Simple interest EXCEL Spreadsheet GC program Interest Interest Casio GC program TI

20 8 Maths Quest General Mathematics HSC Course WORKED Example Andy borrows $000, which is to be repaid over years at $0 per month. Calculate the flat rate of interest that Andy has been charged. Sandra buys a used car with a cash price of $ 000 on the following terms: 0% deposit with the balance paid at $89. per week for years. Calculate: a the deposit b the balance owing c the total cost of the car d the flat rate of interest charged. Calculate the flat rate of interest charged on a lounge suite with a cash price of $5000 if it is purchased on the following terms: 5% deposit with the balance paid at $0. per month for years. Computer Application Flat rate interest loan calculator EXCEL Spreadsheet Flat interest Access the spreadsheet Flat Interest from the Maths Quest General Mathematics HSC Course CD-ROM. This spreadsheet will demonstrate how to calculate a deposit, the total repayments on a loan and the size of each repayment. Monthly payment calculator Consider a $5000 loan to be repaid at 9% p.a. flat rate interest over years.. On the sheet titled Monthly Payments, in cell B5 enter the amount which has been borrowed ($5000), or the balance owing on a purchase after the deposit has been paid.. In cell B7 enter the interest rate as a percentage (9%).. In cell B9 enter the number of years over which the loan is to be repaid ().

21 Chapter Credit and borrowing 9. The total interest paid on the loan will be displayed in cell B. The formula for this will be displayed in this cell. 5. Cell B shows the total amount to be repaid and cell B5 shows the amount of each repayment. Flat interest rate calculator The worksheet Flat Interest Rate will calculate the flat rate of interest charged given the amount of each repayment. Consider a $5 000 loan that is repaid over 5 years at $50 per month.. In cell B5 enter the amount borrowed ($5 000).. In cell B7 enter the amount of each monthly payment ($50).. In cell B9 enter the total number of monthly payments (60).. Displayed will be the total amount to be repaid (cell B), the total interest paid on the loan (cell B), the amount of interest paid per year (cell B5) and the flat rate of interest (cell B7). Check your answers to the previous exercise using this spreadsheet. Home loans The biggest loan that most people will ever take out will be for a home. These loans are usually for large amounts of money and are taken over long periods of time. Most commonly they are taken over 0, 5, 0 or 5 years but they can be taken over even longer periods of up to 5 years. Home loans are not charged at a flat rate of interest. The interest on these loans is reducible, which means that the interest is calculated on the amount of money owing on the loan at the time rather than on the amount initially borrowed. This is known as a reducing balance loan. The interest on a home loan is usually calculated at the beginning of each month, and payments are calculated on a monthly basis. So each month interest is added to the loan and a payment is subtracted from the balance owing. The balance increases by the amount of interest and then decreases by the amount of each payment.

22 0 Maths Quest General Mathematics HSC Course Consider the case of a person who borrows $ to buy a home at 9% p.a. reducible interest. The monthly repayment on this loan is $500 per month. The interest rate of 9% p.a. converts to 0.75% per month. First month s interest = 0.75% of $ = $875 Balance owing = $ $875 $500 = $9 75 In the second month the interest is calculated on the balance owing at the end of the first month. Second month s interest = 0.75% of $9 75 = $870. Balance owing = $ $870. $500 = $8 75. The progress of this loan can be followed in the following computer application. Computer Application Home loan calculator EXCEL Spreadsheet Home loan Access the spreadsheet Home Loan from the Maths Quest General Mathematics HSC Course CD-ROM. This spreadsheet will allow you to follow the progress of a home loan as it is paid off. Use the Edit and then the Fill and Down functions on columns A, B, C and D. Look down column D to find when the balance owing becomes 0 or when it becomes negative. At this time the loan will have been fully repaid. Examine other loans by changing the data in C, C5 and C6.

23 Chapter Credit and borrowing WORKED Example 5 Mr and Mrs Chan take out a $ home loan at 8% p.a. reducible interest over 5 years. Interest is calculated and added on the first of each month. They make a payment of $775 each month. Calculate: a the interest added after one month b the balance owing after one month. THINK WRITE a Convert 8% p.a. to a monthly rate. a 8% p.a. = -- % per month Calculate -- % of $ to find the Interest = -- % of $ interest for one month. Interest = $ b Add the interest to the principal and subtract the repayment. b Balance owing = $ $ $775 Balance owing = $ Graphics Calculator tip! Calculate the interest in a one-month period We can use the TVM function to calculate the interest for a one-month period but great care needs to be taken. Remembering that the interest is calculated for a number of days, to calculate monthly interest we need to enter n = 65. Consider the method shown below for worked example 5.. From the MENU select TVM.. Press F to select Simple Interest.. n = 65 (as n is in days) I% = 8 PV = Press F (SI) to find the interest for one month. When interest is calculated every year for such a long period of time, as with many home loans, the amount of money required to pay off such a loan can be a great deal more than the initial loan.

24 Maths Quest General Mathematics HSC Course WORKED Example 6 A loan of $0 000 is paid off at 9% p.a. reducible interest over a period of 5 years. The monthly repayment is $ Calculate the total amount made in repayments on this loan. THINK Calculate the number of repayments by multiplying the number of years by. Multiply the monthly repayment by the number of repayments. WRITE No. of repayments = 5 No. of repayments = 00 Total repayments = $ Total repayments = $0.00 remember. The interest on home loans is calculated at a reducible rate. This means that the interest is calculated on the balance owing rather than the initial amount borrowed.. Interest is calculated each month; this is then added to the principal and a payment is made. The interest next month is then calculated on the new amount owing.. To calculate the total amount to be repaid on a home loan, we multiply the monthly payment by the number of repayments made. SkillSHEET SkillSHEET. WORKED Example Finding 5 values of n and r in financial formulas.5 EXCEL Spreadsheet Calculating compound interest Interest B Mr and Mrs Devcich borrow $ to buy a home. The interest rate is % p.a. and their monthly payment is $850 per month. a Calculate the interest for the first month of the loan. b Calculate the balance owing at the end of the first month. The repayment on a loan of $ at 7.5% p.a. over a 5-year term is $668.6 per month. a Calculate the interest for the first month of the loan and the balance owing at the end of the first month. b Calculate the amount by which the balance has reduced in the first month. Home loans

25 Chapter Credit and borrowing c Calculate the interest for the second month of the loan and the balance at the end of the second month. d By how much has the balance of the loan reduced during the second month? The repayment on a loan of $ over a 0-year term at 9.6% p.a. is $08.0 per month. Copy and complete the table below. Month Principal ($) Interest ($) Balance ($) Mr and Mrs Roebuck borrow $ to purchase a home. The interest rate is 9% p.a. and over a 5-year term the monthly repayment is $9.. a Copy and complete the table below. Month Principal ($) Interest ($) Balance ($)

26 Maths Quest General Mathematics HSC Course b Mr and Mrs Roebuck decide to increase their monthly payment to $500. Complete the table below. Month Principal ($) Interest ($) Balance ($) WORKED Example 6 c How much less do Mr and Mrs Roebuck owe at the end of one year by increasing their monthly repayment? 5 The repayments on a loan of $ at 8% p.a. reducible interest over 5 years are $80. per month. Calculate the total repayments made over the life of the loan. 6 The Taylors borrow $0 000 over 0 years at 9% p.a. a The monthly repayment on this loan is $59.6. Calculate the total repayments. b The Taylors attempt to pay the loan off quickly by increasing their monthly payment to $500. The loan is then paid off in 6 months. Calculate the total repayments made under this plan. c How much will the Taylors save by increasing each monthly payment? 7 multiple choice The first month s interest on a $ home loan at % p.a. reducible interest is: A $600 B $700 C $ D $ multiple choice A $ loan at 8% p.a. reducible interest over a 5-year term has a monthly payment of $ The total amount of interest paid on this loan is: A $7600 B $ C $ 000 D $ Mr and Mrs Chakraborty need to borrow $ to purchase a home. The interest rate charged by the bank is 7% p.a. Calculate the total interest paid if the loan is taken over each of the following terms: a $ per month over a 5-year term b $775.0 per month over a 0-year term c $898.8 per month over a 5-year term d $6.08 per month over a 0-year term.

27 Chapter Credit and borrowing 5 0 The Smith and Jones families each take out a $ loan at 9.5% p.a. reducible interest. The Smith family repay the loan at $000 per month and the Jones family repay the loan at $000 per month. a How much does each family make in repayments in the first year? b Complete the table below for each family. Smith family Month Principal ($) Interest ($) Balance ($) Jones family Month Principal ($) Interest ($) Balance ($) c After one year how much less does the Jones family owe than the Smith family?

28 6 Maths Quest General Mathematics HSC Course Calculate the amount of flat rate interest payable on a loan of $500 at % p.a. to be repaid over years. Calculate the amount of flat rate interest payable on a loan of $65 at 9.% p.a. to be repaid over -- years. Calculate the total repayments on a loan of $5000 at.5% p.a. to be repaid over years. Susan buys a lounge suite on terms. The cash price of the lounge is $6500 and she pays a 5% deposit. Calculate the amount of the deposit. 5 Calculate the balance that Susan owes on the lounge suite. 6 Calculate the interest that Susan will pay at 7% p.a. flat rate interest for a period of years. 7 Calculate the total amount that Susan will have to repay. 8 Calculate the monthly repayment that Susan will need to make. 9 Harry and Sally borrow $6 000 to purchase a home. The interest rate is % p.a. Calculate the amount of interest payable for the first month. 0 A $ loan that is repaid over 5 years has a monthly repayment of $ Calculate the total amount of interest that is paid on this loan. The cost of a loan Because of the different ways that interest can be calculated, the actual interest rate quoted may not be an accurate guide to the cost of the loan. By using a flat rate of interest, a lender can quote an interest rate less than the equivalent reducible interest rate. To compare flat and reducible rates of interest, we need to calculate the effective rate of interest for a flat rate loan. The effective rate of interest is the equivalent rate of reducible interest for a flat rate loan. The formula for effective rate of interest is: ( + r) E n = n where E = effective rate of interest, expressed as a decimal r = stated rate of flat interest expressed as a decimal n = term of the loan in years Note: This formula for effective rate of interest is not on your formula sheet. This does not mean that you have to memorise it as the formula will be given to you as a part of any question that requires you to use it.

29 Chapter Credit and borrowing 7 WORKED Example 7 Andrea takes out an $8000 loan for a car over 5 years at 6% p.a. flat rate interest. Calculate the effective rate of interest charged on the loan. THINK WRITE Write the formula. ( + r) E = n n Substitute r = 0.06 and n = 5. (.06) E = Calculate. E = Write the interest rate as a percentage. The effective rate of interest is 6.8% p.a. A loan with a reducible rate of interest can be compared to a flat rate of interest if we are able to calculate the total repayments made over the term of the loan. WORKED Example 8 An $ loan at 0% p.a. reducible interest is to be repaid over 5 years at $9. per month. a Calculate the total repayments on the loan. b Calculate the total amount of interest paid. c Calculate the equivalent flat rate of interest on this loan. THINK a Multiply the monthly repayments by the number of months taken to repay the loan. WRITE a Total repayments = $9. 80 Total repayments = $6.80 b Subtract the initial amount borrowed from the total repayments. b Interest = $6.80 $ Interest = $79.80 c Calculate the amount of interest paid per year. c Annual interest = $ Annual interest = $59.5 Write the yearly interest as a percentage of the amount borrowed. $59.5 Flat interest rate = % $ Flat interest rate = 6.% p.a. The most accurate way to compare loans is to calculate the total repayments made in each loan.

30 8 Maths Quest General Mathematics HSC Course Allison borrows $6000 and has narrowed her choice of loans down to two options. Loan A: At 8% p.a. flat rate interest over years to be repaid at $65.00 per month. Loan B: At % p.a. reducible interest over years to be paid at $99.9 per month. Which of the two loans would cost Allison less? THINK WORKED Example 9 WRITE Calculate the total repayments on Loan A repayments = $ Loan A. Loan A repayments = $790 Calculate the total repayments on Loan B repayments = $ Loan B. Loan B repayments = $77. Write a conclusion. Loan B would cost $75.56 less than Loan A. In the above example Allison should take Loan B even though it has a much higher advertised interest rate. This of course would depend upon Allison s ability to meet the higher monthly payments. Generally the more quickly that you can pay off a loan the cheaper the loan will be. The savings are particularly evident when examining home loans. Some home loans that offer a lower interest rate allow for you to make only the minimum monthly repayment. This will maximise the amount of interest that the customer will pay. If a person can afford to pay more than the minimum amount, they may be better off over time by paying a slightly higher rate of interest and paying the loan off over a shorter period of time. WORKED Example 0 Mr and Mrs Beasley need to borrow $ and have the choice of two home loans. Loan X: 6% p.a. over 5 years with a fixed monthly repayment of $6.0. No extra repayments are allowed on this loan. Loan Y: 7% p.a. over 5 years with a minimum monthly payment of $ Mr and Mrs Beasley believe they can afford to pay $800 per month on this loan. If they do, the loan will be repaid in 8 years and 9 months. Which loan would you recommend? THINK Calculate the total repayments on Loan X. Calculate the total repayments on Loan Y. Make a recommendation. WRITE Loan X repayments = $ = $9 90 Loan Y repayments = $800 5 = $ Mr and Mrs Beasley should choose Loan Y as they will save $ 90 provided they can continue to pay $800 per month. With loans such as the one in the above example, the savings depend upon the ability to make the extra repayments. If this is doubtful, Loan X would have been the safer option.

31 Chapter Credit and borrowing 9 The other factor to consider when calculating the cost of a loan is fees. Many loans have a monthly management fee attached to them. This will need to be calculated into the total cost and may mean that a loan with a slightly higher interest rate but no fee may be a cheaper option. remember. The actual cost of a loan is calculated by the total cost in repaying the loan. The interest rate is a guide but not the only factor in calculating cost.. A loan that is quoted at a flat rate of interest can be compared to a reducible rate of interest only by calculating the effective rate of interest on the flat rate loan. The effective rate of interest is the equivalent reducible rate of interest and is found using the formula: ( + r) E n = n. By calculating the total repayments on a loan, we can calculate the equivalent flat rate of interest paid on the loan.. A loan that is repaid over a shorter period of time will usually cost less than one where the repayments are made over the full term of the loan. 5. The flexibility of a loan, which includes factors such as whether extra repayments can be made, is important when considering the cost of a loan. 6. When calculating the cost of a loan, any ongoing fees need to be calculated. C The cost of a loan WORKED Example WORKED Example 8 A $5 000 loan is to be repaid at 8% p.a. flat rate interest over a 0-year term. 7 Use the formula E ( + r) n = to calculate the effective rate of interest. n Calculate the effective rate of interest on each of the following flat rate loans. a 0% p.a. over years b 8% p.a. over years c % p.a. over 5 years d 7.5% p.a. over 0 years e 9.6% p.a. over 6 years A bank offers loans at 8% p.a. flat rate of interest. Calculate the effective rate of interest for a loan taken over: a years b years c years d 5 years e 0 years f 0 years. An $ home loan at 9% p.a reducible interest is to be repaid over 5 years at $7. per month. a Calculate the total repayments on the loan. b Calculate the total amount of interest paid. c Calculate the equivalent flat rate of interest on the loan. 5 Calculate the equivalent flat rate of interest paid on a $5 000 loan at % p.a. reducible interest to be repaid over 0 years at $8.90 per month..6 Substitution into a formula Effective rate of interest SkillSHEET EXCEL Spreadsheet

32 0 Maths Quest General Mathematics HSC Course WORKED Example 9 WORKED Example 0 6 Kim borrows $ 000 for a holiday to South-East Asia. She is faced with a choice of two loans. Loan I: At 0% p.a. flat rate of interest over years to be repaid at $600 per month. Loan II: At.5% p.a. reducible interest over years to be repaid at $0. per month. Which loan will cost Kim the least money? 7 Calculate the total cost of repaying a loan of $ at 8% p.a. reducible interest: a over 5 years with a monthly repayment of $77.8 b over 0 years with a monthly repayment of $86. c over 0 years with a monthly repayment of $.8. 8 Masako and Toshika borrow $5 000 for their home. They have the choice of two loans. Loan : A low interest loan at 7% p.a. interest over 5 years with fixed repayments of $8.7 per month. Loan : A loan at 7.5% p.a interest over 5 years with minimum repayments of $9.7 per month. Masako and Toshika believe they can afford to pay $000 per month. If they do, Loan will be repaid in 0 years and months. Which loan should they choose if they could afford to pay the extra each month? 9 multiple choice A loan can be taken out at 8% p.a. flat interest or 9% p.a. reducible interest. Using the ( + r) formula E n = , the number of years of the loan (n) after which the effective n rate of interest on the flat rate loan becomes greater than the reducible rate loan is: A years B years C years D 5 years 0 Glenn and Inge are applying for a $ loan to be repaid over 5 years. a Bank A charges 7.8% p.a. interest, no fees, with the loan to be repaid at $7.9 per month. Calculate the total cost of this loan. b Bank B charges 7.6% p.a. interest, a $600 loan application fee, a $5 per month management fee and repayments of $8.6 per month. Calculate the total cost of this loan. multiple choice A $ loan is to be taken out. Which of the following loans will have the lowest total cost? A 5% p.a. flat rate interest to be repaid over 0 years B 8% p.a. reducible interest to be repaid over 0 years at $606.6 per month C 6% p.a. reducible interest to be repaid over years at $87.9 per month D 6.5% p.a. reducible interest to be repaid over 0 years at $567.7 per month, with a $600 loan application fee and $8 per month account management fee