Mathscape 9 Extention

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(2) First published 2004 by MACMILLAN EDUCATION AUSTRALIA PTY LTD. 627 Chapel Street, South Yarra 3141 Visit our website at www.macmillan.com.au Associated companies and representatives throughout the world. Copyright © Clive Meyers, Graham Barnsley, Lloyd Dawe, Lindsay Grimison 2004 All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia (the Act) and subsequent amendments, 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 the prior written permission of the copyright owner. Educational institutions copying any part of this book for educational purposes under the Act must be covered by a Copyright Agency Limited (CAL) licence for educational institutions and must have given a remuneration notice to CAL. Licence restrictions must be adhered to. For details of the CAL licence contact: Copyright Agency Limited, Level 19, 157 Liverpool Street, Sydney, NSW 2000. Telephone: (02) 9394 7600. Facsimile: (02) 9394 7601. Email: [email protected] National Library of Australia cataloguing in publication data Meyers, Clive. Mathscape 9 extension : working mathematically. For secondary school students. ISBN 0 7329 8085 2. 1. Mathematics – Textbooks. I. Grimison, Lindsay. II. Barnsley, Graham J. III. Dawe, Lloyd. IV. Title. 510 Publisher: Ben Dawe Project editor: Jasmin Chua Editors: Marta Veroni and Lisa Schmidt Illustrator: Stephen Francis Cover and text designer: Dimitrios Frangoulis Typeset in 11/13 pt Times by Palmer Higgs Cover image: Photolibrary.com Printed in Australia Internet addresses At the time of printing, the Internet addresses appearing in this book were correct. Owing to the dynamic nature of the Internet, however, we cannot guarantee that all these addresses will remain correct. Publisher’s acknowledgments The authors and publisher would like to gratefully credit or acknowledge the following for permission to reproduce copyright material: AAP Image for photo, p. 248; Coo-ee Picture Library for photo, p. 506; Corbis for photos, pp. 20, 148; Corbis Digital Stock for photos, pp. 254, 282; Digital Vision for photos, pp. 1, 25, 78, 203, 311, 488; Fairfax Photos/AFR for photo, p. 433; Getty Images for photo, p. 479; Image 100 for photo, p. 511; istockphoto.com for photos, pp. 305, /Jeannette Meier Kamer 408; Mary Evans Picture Library for photo, p. 71; National Library of Australia for photos, by permission, pp. 111, 397; Photodisc for photos, pp. 117, 155, 211, 338, 343, 440, 554; World Bank for table, 2001 World Development indicators <http://devdata.worldbank.org/ hnpstats/DCselection.asp>, p. 249. While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful owner in each case..

(3) iii. Contents Green indicates material is exclusively Stage 4. All other material is Stage 5.1/5.2/5.3.. Preface. vi. How to use this book. vii. Chapter 1 Rational numbers. 1. 1.1 1.2 1.3. 2 4 8 10 10 13 15 15 19. Significant figures The calculator Estimation Try this: Fermi problem 1.4 Recurring decimals 1.5 Rates Try this: Desert walk 1.6 Solving problems with rates Try this: Passing trains Focus on working mathematically: A number pattern from Galileo 1615 Language link with Macquarie Chapter review. Chapter 2 Algebra 2.1 2.2 2.3 2.4. 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16. Describing simple patterns Try this: Flags Substitution Adding and subtracting algebraic expressions Multiplying and dividing algebraic expressions Try this: Overhanging the overhang The order of operations The distributive law The highest common factor Adding and subtracting algebraic fractions Multiplying and dividing algebraic fractions Generalised arithmetic Try this: Railway tickets Properties of numbers Generalising solutions to problems using patterns Binomial products Perfect squares Try this: Proof Difference of two squares Miscellaneous expansions. 20 22 23. 25 26 31 32 33 36 38 38 40 42 44 47 49 53 54 56 60 63 66 67 69. Focus of working mathematically: A number pattern from Blaise Pascal 1654 Language link with Macquarie Chapter review. Chapter 3 Consumer arithmetic 3.1 3.2 3.3 3.4 3.5 3.6. Salaries and wages Other methods of payment Overtime and other payments Wage deductions Taxation Budgeting Try this: Telephone charges 3.7 Best buys 3.8 Discounts Try this: Progressive discounting 3.9 Profit and loss Focus on working mathematically: Sydney market prices in 1831 Language link with Macquarie Chapter review. 71 74 74. 78 79 83 87 90 93 98 101 102 104 107 108 111 113 114. Chapter 4 Equations, inequations and formulae 117 4.1 4.2. One- and two-step equations 118 Equations with pronumerals on both sides 121 4.3 Equations with grouping symbols 123 4.4 Equations with one fraction 124 4.5 Equations with more than one fraction 126 4.6 Inequations 129 4.7 Solving worded problems 134 Try this: A prince and a king 137 4.8 Evaluating the subject of a formula 138 4.9 Equations arising from substitution 141 Try this: Floodlighting by formula 143 4.10 Changing the subject of a formula 144 Focus on working mathematically: Splitting the atom 149 Language link with Macquarie 151 Chapter review 152.

(4) iv. Mathscape. 9 Extension. Chapter 5 Measurement 5.1 5.2 5.3. Length, mass, capacity and time Accuracy and precision Pythagoras’ theorem Try this: Pythagorean proof by Perigal 5.4 Perimeter 5.5 Circumference Try this: Command module 5.6 Converting units of area 5.7 Calculating area Try this: The area of a circle 5.8 Area of a circle 5.9 Composite areas Try this: Area 5.10 Problems involving area Focus on working mathematically: The solar system Language link with Macquarie Chapter review. 155 156 162 165 170 170 175 180 181 183 191 192 195 200 200 203 206 206. Chapter 6 Data representation and analysis 211 6.1 6.2 6.3 6.4. Graphs Organising data Analysing data Problems involving the mean Try this: The English language 6.5 Cumulative frequency 6.6 Grouped data Focus on working mathematically: World health Language link with Macquarie Chapter review. Chapter 7 Probability 7.1 7.2. Probability and its language Experimental probability Try this: Two-up 7.3 Computer simulations Try this: The game of craps 7.4 Theoretical probability Try this: Winning chances Focus on working mathematically: A party game Language link with Macquarie Chapter review. 212 219 225 233 236 236 242 248 251 252. 256 257 260 266 266 271 272 275 276 278 279. Chapter 8 Surds. 282. 8.1 8.2. Rational and irrational numbers Simplifying surds Try this: Greater number 8.3 Addition and subtraction of surds 8.4 Multiplication and division of surds Try this: Imaginary numbers 8.5 Binomial products with surds 8.6 Rationalising the denominator Try this: Exact values Focus on working mathematically: Fibonacci numbers and the golden mean Language link with Macquarie Chapter review. Chapter 9 Indices 9.1 9.2. 283 288 291 291 294 297 298 301 304. 305 308 309. 311. Index notation Simplifying numerical expressions using the index laws 9.3 The index laws 9.4 Miscellaneous questions on the index laws 9.5 The zero index Try this: Smallest to largest 9.6 The negative index 9.7 Products and quotients with negative indices Try this: Digit patterns 9.8 The fraction index 9.9 Scientific notation 9.10 Scientific notation on the calculator Focus on working mathematically: Mathematics is at the heart of science Language link with Macquarie Chapter review. Chapter 10 Geometry 10.1 10.2 10.3. 10.4 10.5 10.6. Angles Parallel lines Triangles Try this: The badge of the Pythagoreans Angle sum of a quadrilateral Special quadrilaterals Try this: Five shapes Polygons. 312 313 315 320 322 323 323 326 328 329 333 335 338 340 340. 343 344 350 356 363 363 367 374 374.

(5) Contents. Try this: How many diagonals in a polygon? Try this: An investigation of triangles 10.7 Tests for congruent triangles 10.8 Congruent proofs Try this: Triangle angles 10.9 Deductive reasoning and congruent triangles Focus on working mathematically: Does a triangle have a centre? Language link with Macquarie Chapter review. Chapter 11 The linear function 11.1 11.2. The number plane Graphing straight lines (1) Try this: Size 8 11.3 Graphing straight lines (2) 11.4 Gradient of a line Try this: Hanging around 11.5 The linear equation y = mx + b Try this: Latitude and temperature Focus on working mathematically: Television advertising Language link with Macquarie Chapter review. Chapter 12 Trigonometry 12.1 12.2. Side ratios in right-angled triangles The trigonometric ratios Try this: Height to base ratio 12.3 Trigonometric ratios using a calculator 12.4 Finding the length of a side 12.5 Problems involving finding sides Try this: Make a hypsometer 12.6 Finding the size of an angle 12.7 Problems involving finding angles 12.8 Angles of elevation and depression Try this: Pilot instructions 12.9 Bearings Try this: The sine rule Focus on working mathematically: Finding your latitude from the Sun Language link with Macquarie Chapter review. 379 380 381 387 392 392 397 401 402. 408 409 412 417 417 422 427 427 433 433 436 436. 440 441 444 448 448 451 456 460 461 464 467 470 471 478 479 483 484. Chapter 13 Simultaneous equations. 488. 13.1 13.2 13.3. Equations with two unknowns The graphical method The substitution method Try this: Find the values 13.4 The elimination method Try this: A Pythagorean problem 13.5 Solving problems using simultaneous equations Focus on working mathematically: Exploring for water, oil and gas— the density of air-filled porous rock Language link with Macquarie Chapter review. Chapter 14 Co-ordinate geometry 14.1 14.2 14.3. 502. 506 508 509. 511. The distance between two points The midpoint of an interval The gradient formula Try this: A line with no integer co-ordinates 14.4 General form of the equation of a line 14.5 The equation of a line given the gradient and a point 14.6 The equation of a line given two points Try this: Car hire 14.7 Parallel lines Try this: Temperature rising 14.8 Perpendicular lines 14.9 Regions in the number plane 14.10 Co-ordinate geometry problems Focus on working mathematically: Finding the gradient of a ski run Language link with Macquarie Chapter review. Answers. 489 492 496 498 499 502. 512 516 520 525 525 530 533 536 536 540 540 544 549 554 558 559. 563. v.

(6) vi. Mathscape. 9 Extension. Preface Mathscape 9 Extension is a comprehensive teaching and learning resource that has been written to address the new Stage 5.1/5.2/5.3 Mathematics syllabus in NSW. Our aim was to write a book that would allow more able students to grow in confidence, to improve their understanding of Mathematics and to develop a genuine appreciation of its inherent beauty. Teachers who wish to inspire their students will find this an exciting, yet very practical resource. The text encourages a deeper exploration of mathematical ideas through substantial, well-graded exercises that consolidate students’ knowledge, understanding and skills. It also provides opportunities for students to explore the history of Mathematics and to address many practical applications in contexts that are both familiar and relevant. From a teaching perspective, we sought to produce a book that would adhere as strictly as possible to both the content and spirit of the new syllabus. Together with Mathscape 10 Extension, this book allows teachers to confidently teach the Stage 5.1/5.2/5.3 courses knowing that they are covering all of the mandatory outcomes. Content from Stage 4 has been included in each chapter, where appropriate. This will allow teachers to diagnose significant misconceptions and identify any content gaps. For those students who have achieved the relevant Stage 4 outcomes, this material could be used as a review to introduce the Stage 5.1/5.2/5.3 topics, or to revise important concepts when they occur. However, for those students who have not achieved these outcomes by the start of Year 9, this material will be new work. All content is clearly listed as either Stage 4 or Stage 5.1/5.2/5.3 in the contents section at the front of the book. A detailed syllabus correlation grid has been provided for teachers on the Mathscape 9/9 Extension School CD-ROM. Mathscape 9 Extension has embedded cross-curriculum content, which will support students in achieving the broad learning outcomes defined by the Board of Studies. The content also addresses the important key competencies of the Curriculum Framework, which requires students to collect, analyse and organise information; to communicate mathematical ideas; to plan and organise activities; to work with others in groups; to use mathematical ideas and techniques; to solve problems; and to use technology. A feature of each chapter which teachers will find both challenging and interesting for their students is the ‘Focus on working mathematically’ section. Although the processes of working mathematically are embedded throughout the book, these activities are specifically designed to provoke curiosity and deepen mathematical insight. Most begin with a motivating real-life context, such as television advertising, or the gradient of a ski run, but on occasion they begin with a purely mathematical question. (These activities can also be used for assessment purposes.) In our view, there are many legitimate, time-proven ways to teach Mathematics successfully. However, if students are to develop a deep appreciation of the subject, they will need more than traditional methods. We believe that all students should be given the opportunity to appreciate Mathematics as an essential and relevant part of life. They need to be given the opportunity to begin a Mathematical exploration from a real-life context that is meaningful to them. To show interest and enjoyment in enquiry and the pursuit of mathematical knowledge, students need activities where they can work with others and listen to their arguments, as well as work individually. To demonstrate confidence in applying their mathematical knowledge and skills to the solution of everyday problems, they will need experience of this in the classroom. If they are to learn to persevere with difficult and challenging problems, they will need to experience these sorts of problems as well. Finally, to recognise that mathematics has been developed in many cultures in response to human needs, students will need experiences of what other cultures have achieved mathematically. We have tried to address these values and attitudes in this series of books. Our best wishes to all teachers and students who are part of this great endeavour. Clive Meyers Lloyd Dawe Graham Barnsley Lindsay Grimison.

(7) vii. How to use this book Mathscape 9 Extension is a practical resource that can be used by teachers to supplement their teaching program. The exercises in this book and the companion text (Mathscape 10 Extension) provide a complete and thorough coverage of all content and skills in the Stage 5.1/5.2/5.3 course. The great number and variety of questions allow for the effective teaching of more able students. Each chapter contains: • a set of chapter outcomes directed to the student • all relevant theory and explanations, with important definitions and formulae boxed and coloured • step-by-step instructions for standard questions • a large number of fully worked examples preceding each exercise • extensive, thorough and well-graded exercises that cover each concept in detail • chapter-related, problem-solving activities called ‘Try this’ integrated throughout • a language skills section linked to the Macquarie Learners Dictionary • novel learning activities focusing on the process of working mathematically • a thorough chapter review.. Explanations and examples The content and skills required to complete each exercise have been introduced in a manner and at a level that is appropriate to the students in this course. Important definitions and formulae have been boxed and coloured for easy reference. For those techniques that require a number of steps, the steps have been listed in point form, boxed and coloured. Each exercise is preceded by several fully worked examples. This should enable the average student to independently complete the majority of relevant exercises if necessary.. The exercises The exercises have been carefully graded into three distinct sections: • Introduction. The questions in this section are designed to introduce students to the most basic concepts and skills associated with the outcome(s) being covered in the exercise. Students need to have mastered these ideas before attempting the questions in the next section. • Consolidation. This is a major part of the exercise. It allows students to consolidate their understanding of the basic ideas and apply them in a variety of situations. Students may need to use content learned or skills acquired in previous exercises or topics to answer some of these questions. The average student should be able to complete most of the questions in this section, although the last few questions may be a little more difficult. • Further applications. Some questions presented in this section will be accessible to the average student; however, the majority of questions are difficult. They might require a reverse procedure, the use of algebra, more sophisticated techniques, a proof, or simply time-consuming research. The questions can be open-ended, requiring an answer with a justification. They may also involve extension or off-syllabus material. In some questions, alternative techniques and methods of solution other than the standard method(s) may be introduced, which may confuse some students. Teachers need to be selective in the questions they choose for their students. Some students may not need to complete all of the questions in the Introduction or Consolidations sections of each exercise, while only the most able students should usually be expected to attempt the questions in the Further applications section. Those questions not completed in class might be set as homework at the teacher’s discretion. It is not intended that any student would attempt to answer every possible question in each exercise.. Focus on working mathematically The Working Mathematically strand of the syllabus requires a deeper understanding of Mathematics than do the other strands. As such, it will be the most challenging strand for students to engage with and for teachers to assess. The Working Mathematically outcomes listed in the syllabus have been carefully integrated into each chapter of the book; however, we also decided to include learning activities in each chapter that will.

(8) viii. Mathscape. 9 Extension. enable teachers to focus sharply on the processes of working mathematically. Each activity begins with a reallife context and the Mathematics emerges naturally. Teachers are advised to work through them before using them in class. Answers have not been provided, but notes for teachers have been included on the Mathscape 9/9 Extension School CD-ROM, with suggested weblinks. Teachers may wish to select and use the Learning activities in ‘Focus on working mathematically’ for purposes of assessment. This too is encouraged. The Extension activities will test the brightest students. Suggestions are also provided to assess the outcomes regarding Communication and Reflection.. Problem solving Each chapter contains a number of small, chapter-related, problem-solving activities called ‘Try this’. They may be of some historical significance, or require an area outside the classroom, or require students to conduct research, or involve the use of algebra, while others relate the chapter content to real-life context. Teachers are advised to work through these exercises before using them in class.. Technology The use of technology is a clear emphasis in the new syllabus. Innovative technology for supporting the growth of understanding of mathematical ideas is provided on the Mathscape 9/9 Extension School CD-ROM, which is fully networkable and comes free-of-charge to schools adopting Mathscape 9 Extension for student use. Key features of the CD-ROM include: • spreadsheet activities • dynamic geometry • animations • executables • student worksheets • weblinks for ‘Focus on working mathematically’.. Language The consistent use of correct mathematical terms, symbols and conventions is emphasised strongly in this book, while being mindful of the students’ average reading age. Students will only learn to use and spell correct mathematical terms if they are required to use them frequently in appropriate contexts. A language section has also been included at the end of each chapter titled ‘Language link with Macquarie’, where students can demonstrate their understanding of important mathematical terms. This might, for example, include explaining the difference between the mathematical meaning and the everyday meaning of a word. Most chapters include a large number of worded problems. Students are challenged to read and interpret the problem, translate it into mathematical language and symbols, solve the problem, then give the answer in an appropriate context. Clive Meyers Lloyd Dawe Graham Barnsley Lindsay Grimison.

(9) Rational numbers. 1. This chapter at a glance . evaluate numerical expressions using a calculator estimate the result of a calculation state the number of significant figures in a number round off a number correct to a given number of significant figures determine the effect of rounding during calculations on the accuracy of the results convert fractions to recurring decimals convert recurring decimals to fractions express a rate in its simplest form convert rates from one set of units to another solve problems involving rates.. Rational numbers. Stage 5.1/5.2/5.3 After completing this chapter, you should be able to:. 1.

(10) 2. Mathscape. 1.1. 9 Extension. Significant figures. No quantity, such as length, mass or time, can be measured exactly. For a measurement to be of use, we need to know how accurate it is. That is, we must be confident that each digit in the measurement is significant. A significant figure is a number that is correct within some stated degree of accuracy. The rules for significant figures are: All non-zero digits are significant. Zeros between non-zero digits are significant. Zeros at the end of a decimal are significant. Zeros before the first non-zero digit in a decimal are not significant. Zeros after the last non-zero digit in a whole number may or may not be significant. When rounding off correct to a specified number of significant figures, choose the number that is closest in value to the given number and which also contains the required number of significant figures.. EG +S. Example 1 State the number of significant figures in each number. a. 4.009. b 137.20. c. 0.001 64. d 5000. Solutions a In 4.009, the two non-zero digits (i.e. 4 and 9) are significant and the two zeros between these digits are significant. ∴ The number has 4 significant figures. b In 137.20, the four non-zero digits (i.e. 1, 3, 7 and 2) are significant and the zero at the end of the decimal is significant. ∴ The number has 5 significant figures. c In 0.001 64, the three non-zero digits (i.e. 1, 6 and 4) are significant; however, the zeros at the beginning of the decimal are not significant. ∴ The number has 3 significant figures. d In 5000, the non-zero digit (i.e. 5) is significant. Either some, all or none of the final zeros could possibly be significant. This would need to be determined from the context in which the number occurs. If we knew that the number had been rounded off correct to: i 1 significant figure, then only the 5 would be significant ii 2 significant figures, then only the 5 and the first zero would be significant iii 3 significant figures, then only the 5 and the first two zeros would be significant iv 4 significant figures, then all of the digits would be significant..

(11) Chapter. EG +S. Rat io n al n u mbers. Example 2 Round off 47.503 correct to: a c. 4 significant figures 2 significant figures. b 3 significant figures d 1 significant figure. Solutions a 47.503 = 47.50 (4 significant figures) c 47.503 = 48 (2 significant figures). EG +S. 1:. b 47.503 = 47.5 (3 significant figures) d 47.503 = 50 (1 significant figure). Example 3 Round off 39.99 correct to: a. 3 significant figures. b 2 significant figures. c. 1 significant figure. Solutions a 39.99 = 40.0 (3 significant figures) b 39.99 = 40 (2 significant figures; both the 4 and the 0 are significant figures) c 39.99 = 40 (1 significant figure; only the 4 is significant) Exercise. 1.1. 1 State the number of significant figures in each of the following. a 45 b 7281 c 859 e 607 f 3012 g 4001. d 132 494 h 20 809. 2 State the number of significant figures in each decimal. a 5.28 b 7.152 c 38.5 e 0.4 f 0.005 g 0.0371 i 5.062 j 13.007 k 58.0208 m 9.30 n 0.10 o 1.4700 q 3.030 r 32.0040 s 409.010 00. d h l p t. 254.883 0.003 469 0.001 09 0.004 080 0.010 203 00. ■ Consolidation. 3 Round off each of the following correct to 1 significant figure. a 83 b 27 c 65 e 136 f 415 g 250 i 9450 j 26 449 k 539 499. d 94 h 3810 l 850 000. 4 Round off each of these numbers correct to 2 significant figures. a 128 b 171 c 234 e 1459 f 4026 g 8350 i 45 718 j 76 285 k 285 195. d 675 h 12 042 l 644 003. 3.

(12) 4. 9 Extension. Mathscape. 5 Round off each of the following decimals correct to the number of significant figures indicated in the brackets. a 3.67 [1] b 0.484 [1] c 0.0731 [2] d 6.2085 [4] e 11.784 [2] f 0.3 [2] g 25.156 [3] h 49.066 28 [5] i 91.045 [3] j 144.387 [2] k 7.3855 [4] l 10.9367 [2] m 2018.68 [3] n 3693.21 [2] o 4002.142 [5] p 9187.549 [6] 6 Round off the following correct to: i 1 significant figure ii 2 significant figures iii 3 significant figures a 99.35 b 194.97 c 998.763 d 499.861 ■ Further applications. 7 Write down a possible number that is approximately equal to: a 130, correct to 2 significant figures b 2.47, correct to 3 significant figures. 1.2. The calculator. As a wide variety of calculators is available, there are differences in the way they operate. The examples here have been worked with the use of a direct logic calculator. That is, the calculations are performed in the logical order in which they appear. For example, to evaluate followed by the 9, then 9 on a direct logic calculator, we press the square root key press = . For models that do not use direct logic, we enter the 9, then press the square root key. You will need to familiarise yourself with how your calculator works.. EG +S. Example 1 Evaluate each of the following. a e. 2 6 --- + 5 --3 7 3 41. b −78 − 96. c. 15.982. d. 69.4. 3.524. g. 5. h. 1 ------------0.274. f. Solutions Calculator steps: a b c. 6 a b--c- 7 + 5 a b--c- 2 a b--c- 3 = + − 78 − 96 = 15.98 x2 =. /. d e. Answer: -----6 11 21. –174 255.3604. 69.4 =. 8.330 666 24. 41 =. 3.448 217 24. 3. f. 3.52. g. 5. x. 117.3. 4 =. 153.522 012 2. 117.3 =. 2.593 340 858. xy. h 0.274 x−1 =. 3.649 635 036.

(13) Chapter. EG +S. Rat io n al n u mbers. Example 2 Evaluate each of these, correct to 2 decimal places, using the grouping symbols keys ( and ) . 86.9 + 213.7 -----------------------------5.6 × 8.3. a. b. 342.5 – 114.8. Solutions Calculator steps: 86.9 + 213.7 = ÷. a. Answer ( 5.6 × 8.3 ). ( 342.5 − 114.8 ). b. EG +S. 1:. =. =. 6.47 15.09. Example 3 72.6 + 153.9 - , correct to 3 decimal Use the memory function on the calculator to evaluate ----------------------------2 12.5 × 0.98 places. Solution i Evaluate the denominator first and store the answer in the memory. 12.5 x2 × 0.98 = Min ii Evaluate the numerator, then divide the answer by the number stored in the memory. 72.6 + 153.9 = ÷ MR = Answer: 1.479 (3 decimal places) Exercise. 1.2. Evaluate 72 × 12.43 correct to 3 decimal places, without rounding off during the calculation. b Round off 72 to the nearest integer, then multiply by 12.43. c Round off 72 to 1, 2 and 3 decimal places then multiply by 12.43. What do you notice? d What effect does rounding off too early have on the accuracy of an answer?. 1 a. 2 Evaluate each of these using the fraction key a b--c- , then give the answers as decimals, correct to 2 decimal places. 3 1 a --- + -----b 8 7--9- – 3 4--5c 2 5--6- × 4 2--78 11 3 Evaluate each of the following using the sign change key + − . a −98 − 156 b −49 + 32 − 77 c. /. −156 ÷ −12. 4 Evaluate each of these correct to 4 significant figures using the square key x2 . a. 7.82. b (–12.7)2. c. ( 4 7--8- ). 2. 5.

(14) 6. Mathscape. 9 Extension. 5 Evaluate each of these correct to the nearest tenth using the square root key root key 3 . a d. 23 3. 70. b. 85 + 72.6. e. 3. 110.4 ÷ 2.96. 90 × 16.45. c f. and cube. 3. 36.7 + 152.6. 6 Evaluate each of the following correct to 1 decimal place using the power key x y . a 6.53 b 3.724 c 4.085 d. 3 -) ( 2 ----11. 6. e. 1.857 × 4.3. f. 8.94 − 3.15. 7 Evaluate each of these correct to the nearest hundredth using the root key a. 4. 11. b. 5. 68.2. d. 5. 96 × 12.5. e. 3 – 6 2.4. c. 7. 212.9. f. 4. 7 1--5-. x. .. 8 Evaluate each of the following correct to 3 significant figures using the reciprocal key x−1 or. 1 --x. a. 1 --7. b. 1 ------------0.245. c. d. 1 ------3. e. 1 --------------3 51.4. f. . 1 ---------------2 0.065 1 ------------4 1.98. 9 Evaluate each of these correct to 2 decimal places using the pi key π . 9π a π + 16.82 b 7π c -----2 1 e --f 5π d π2 π 10 Evaluate each of the following correct to the nearest tenth using the grouping symbols keys ( and ) where necessary. 73 + 115 172 19.3 × 54.7 a --------------------b --------------------c --------------------------14 8.5 × 3.1 6.4 + 9.8 d. 12 × 11 × 10 -----------------------------7×8×9. 3. e. 9.4 -----------------------5.1 × 7.25. f. 135 + 18.7 -----------------------------11 – π. ■ Consolidation. 11 Find the value of each expression, correct to 2 decimal places. a. 10.652 × 8.3. b. 83 ---------4 2.6. c. 101 --------7.

(15) Chapter. d g. 3. 42 × 7.5. e. 28 ------5. 74.9 + 87.2. h. 5. 7.9. k. 3. 25 + 50.3 – 19.6. n. 10 + 20 -------------------------15 30 m -------------------2+ 3 j. 2. 2. 1:. Rat io n al n u mbers. f. 34 − 4.13. i. 16.8 ------------4 13.9. l. 1 -----------------------0.06 × 7. – 250 --------------2 2 5 –8. o. 4. 1 -----------------5 5 2 – 2. r. 116.7 + 99.8 ---------------------------------2 2.1. u. 5 + 8 1--8-. x. ( – 8.4 ) ---------------------------– 6.3 + 11.4. c. ( 13.6 ). f. – 40.6 + 15.35 ---------------------------------6.2 × 7.7 1 --------------------------------2 0.86 – 0.29. 6. 3. 3. p. 24 + 23 ---------------------2 2 16 – 15. q. s. (1.7 +1.16)6. t. v. 10 + 3 ------------------10 – 3. 1 w ---------------------------23 0.1 + 0.2. 2. 18 + 7. 82.6 × 16.1. 3. 4. 3. 3. 12 Evaluate, correct to the nearest tenth. 3. 1 ---------------3– 5 1 1 1 ------- + ------- + ------5 3 2. a. π × 14. b. d. 0.92 18.9 ---------- + ---------2.3 5.14. e. g. 17 + 18 --------------------------3 17 + 3 18. h 4.6(19.83 − 7.12)3. i. j. 3 7 ⎛ 9 2---⎞ ÷ ⎛ 1 1---⎞ ⎝ 3⎠ ⎝ 2⎠. k. 100 ------------------------------------------10 + 3 10 + 4 10. l. 7. 124.37 – 19.66 -----------------------------------9.7 + 11.75. ■ Further applications. 13 Use the memory function on the calculator to evaluate each of these, correct to 1 decimal place. 3. a d. 2. 7.6 + 39 ---------------------------3 1.4 × 0.995. 21.4 3.9 × 15.6 -------------------------------3- + ---------6.09 10.58 – 1.33. 5. 3. b. 9.47 ⎛ 11.6 ----------⎞ ÷ ⎛ ----------⎞ ⎝ 2.3 ⎠ ⎝ 1.02⎠. e. 1 57.5 – 13.6 ------------------------------- × ------------2 15 × 98.2 12.4. 3. c. 4. f. 8.1 13.4 ------------------------ + ---------------------21.9 + 2.64 7 × 0.16 4. 17.5 × 5.3 3 1 ------------------------ – ---------------2 6.7 0.075. 7.

(16) 8. Mathscape. 1.3. 9 Extension. Estimation. Calculators do not make arithmetic errors. But sometimes we get incorrect answers when we use a calculator. This is because we may have: • • • • • •. left out a decimal point pressed the wrong key by mistake not pressed the equals key at the right time not understood the question set the calculator in the wrong mode not pressed the second function key.. By estimating the answer before using a calculator, we can work out whether the calculator answer is reasonable. An estimate is more than a guess. It is an approximate answer that is worked out logically. It does not have to be very close to the correct answer but it should be of the same order of magnitude. That is, if the estimate is in the tens, the correct answer should not be in the hundreds or the thousands. For example, before evaluating 19.855 × 4.84 with a calculator, we could estimate that the answer would be close to 20 × 5, that is, 100. If the calculator gives the answer as 9609.82, we might have made an error when entering the data. In fact, a decimal point was omitted, since the correct answer is 96.0982. It is also possible, of course, that our estimate is incorrect. NOTE: Many different estimates can be given to calculations depending on the way that each individual number is rounded off.. EG +S. Example Estimate the answer to each of these calculations. a. 386 × 19. b 154.5 ÷ 11.2. c. 17.74 × 0.493. d. 41.68 × 21.19 --------------------------------6.904. Solutions a. 386 × 19 ⯐ 400 × 20 = 8000. Exercise. b. 154.5 ÷ 11.2 ⯐ 150 ÷ 10 = 15. c. 17.74 × 0.493 ⯐ 18 × =9. 1 --2. d. 41.68 × 21.19 --------------------------------6.904 42 × 20 ⯐ -----------------7 = 6 × 20 = 120. 1.3. 1 Round off each number correct to 1 significant figure and hence estimate the value of: a 48 × 33 b 385 × 11 c 69 × 114 d 19 952 × 9 e 223 ÷ 52 f 642 ÷ 22 g 38 840 ÷ 375 h 8445 ÷ 23 i 54 × 186 j 2751 ÷ 63 k 297 × 42 l 96 959 ÷ 4367.

(17) Chapter. 1:. Rat io n al n u mbers. 2 Estimate the answer, as an integer, to each of the following calculations. a 8.7 + 19.4 + 12.1 b 96.5 − 27.3 + 15.046 c 24.2 × 3.75 × 5.3 d 24.8 × 3.88 e 32.42 ÷ 7.93 f 126.7 ÷ 9.82 g 5.34 × 11.92 × 8.15 h 53.5 ÷ 6.12 × 8.046 i 189.4 − 47.75 − 283.19 ■ Consolidation. 3 Estimate the answer to each of these. a (14.797 + 32.88) ÷ 8.1 b (348.5 − 102.7) × 4.193. c. 495.13 ÷ (9.96 × 10.02). 4 Find the approximate value of: a 18.8 + 6.84 × 3.125 c 20.4 ÷ 3.95 + 19.87 × 5.02. b 183.4 − 31.2 ÷ 5.17 d 2117 − 12.13 × 8.4 × 4.96. 5 Estimate: a 16.45 × 0.482. c. b 43.65 × 0.252. 13.82 × 1.55. d 8.094 × 1.26. 6 Estimate the answer for each of these, giving the answer as an integer. a b c 3 29.1 d 23.67 84.77. 3. 119.8. 7 Estimate the value of each calculation. a. 4.76 × 9.27 --------------------------2.89. b. 73.4 × 15.2 --------------------------4.57. c. 50.6 + 73.1 -----------------------------15.8 – 4.593. d. 106.2 ----------------27.046. 8 The crowds at each day of a test cricket match played at the SCG between Australia and England were as follows: • Day 1—34 356 • Day 2—29 875 • Day 3—26 234 • Day 4—18 558 • Day 5—9063 Round off each day’s crowd to the nearest 5000 spectators and hence estimate the total match attendance. 9 A group of 4 people having dinner in a restaurant ordered the following meals from the menu: • Tamara: spaghetti bolognaise $18.75 • Luke: steak Diane $21.75 • Amanda: fettuccine boscaiola $19.20 • Barry: veal parmigiana $20.60 They also ordered 2 bottles of wine at $11.45 each and 4 coffees at $3.25 each. a Estimate the total cost of the meal, allowing for a small tip. b Approximately how much would each person expect to pay if they shared the bill equally? 10 Therese decided to re-carpet her lounge room using carpet squares of side length 50 cm. The lounge room is rectangular in shape and measures 5.2 m by 6.8 m. a Estimate the area of the room in square metres. b How many carpet squares are needed to cover an area of 1 m2?. 9.

(18) 10. Mathscape. 9 Extension. c. Estimate the number of carpet squares that are needed to cover the entire lounge room floor. d If the carpet squares are sold in packs of 40 at $385 per pack, estimate the total cost of the re-carpeting. e Should re-carpeting decisions be based on estimates or accurate measurements? Explain. ■ Further applications. 11 a Evaluate 4 and 9 . Hence, find estimates for 5 and 7 , correct to 1 decimal place. b Evaluate 100 and 121 . Hence, find estimates for 110 , 105 and 115 , correct to 1 decimal place. 12 Consider the statement 2n = 12. a Show by substitution that: i 3⬍n⬍4 ii 3.5 ⬍ n ⬍ 3.6 iii 3.58 ⬍ n ⬍ 3.59 iv 3.584 ⬍ n ⬍ 3.585 b Hence, estimate the value of n, correct to 3 decimal places. 13 By substituting and then refining estimates, find the approximate value of n in each of the following, correct to 3 decimal places. a 2n = 20 b 3n = 36 c 5n = 100 TRY THIS. Fermi problem. A Fermi problem is a problem solved by making a good estimation. Try these problems: 1. How many telephone calls are made in one day in Australia?. 2. What would be the total value of all the books in every library in Australia?. 1.4. Recurring decimals. A recurring decimal has an infinite number of decimal places, with one or more of the digits repeating themselves indefinitely. Recurring decimals are written with a dot above the first and last digits in the repeating sequence. 0.616 161 … = 0.6˙ 1˙ 1.288 888 … = 1.28˙ a A rational number is a number that can be written in the form --- , where a and b are integers b (whole numbers) and b ≠ 0. Every recurring decimal can be expressed as a fraction, so recurring decimals are rational numbers. For example:. 0.444 444 … = 0.4˙ 0.329 329 … = 0.3˙ 29˙.

(19) Chapter. 1:. Rat io n al n u mbers. To convert a fraction to a recurring decimal divide the numerator by the denominator. To convert a recurring decimal to a fraction: let the decimal be x multiply both sides by the smallest power of 10 so that the recurring part of the decimal becomes a whole number subtract the first equation from the second solve the resulting equation.. EG +S. EG +S. Example 1 Convert each of these fractions to a recurring decimal. 5 7 a --b -----9 11 Solutions a 0.5 5 5… b 0.6 3 6 3… 11 7.04070407 9 5.050505 5 7 ∴ --- = 0. 5˙ ∴ ------ = 0. 6˙ 3˙ 9 11. c. c. Example 2 Convert each recurring decimal to a fraction in simplest form. a 0. 8˙ b 0. 1˙ 5˙ c Solutions a Let x = 0. 8˙ … ∴ 10x = 8. 8˙ … Subtract from ∴ 9x = 8 8 ∴ x = --9. Exercise. b Let x = 0. 1˙ 5˙ … ∴ 100x = 15. 1˙ 5˙ … Subtract from ∴ 99x = 15 15 ∴ x = -----99 5 = -----33. c. 1 -----12 0.08 3 3… 12 1.0040404 1 ∴ ------ = 0.08 3˙ 12. 0.2 4˙ Let x = 0.2 4˙ … ∴ 10x = 2. 4˙ … ∴ 100x = 24. 4˙ … Subtract from ∴ 90x = 22 22 ∴ x = -----90 11 = -----45. 1.4. This exercise should be completed without the use of a calculator, unless otherwise indicated. 1 Write each of these as a recurring decimal. a 0.222 … b 0.777 … c. 0.6444 …. d 0.3555 …. 11.

(20) 12. Mathscape. 9 Extension. e 0.272 727 … i 0.146 146 … m 1.666 …. f 0.919 191 … j 0.029 029 … n 3.818 181 …. g 0.484 848 … k 0.152 152 … o 8.274 274 …. h 0.030 303 … l 0.698 698 … p 13.955 555 …. ■ Consolidation. 2 Use short division to convert each of these fractions to a recurring decimal. 1 1 2 4 a --b --c --d --3 9 3 9 1 2 3 1 e -----f -----g --h -----15 6 11 11 5 7 5 11 i -----j -----k --l -----12 22 6 12 3 a. Convert 1 2--3- to a decimal using a calculator.. b Does the calculator round off the answer at the last digit? 4 Express each of the following as a recurring decimal. 1 5 1 a --b --c -----7 7 13 5 a. d. 4 -----13. Write down the recurring decimal for 1--9- .. b Hence, write down recurring decimals for 2--9- , 5--9- , c What meaning should be given to 0.9˙? Why?. 7 --9. and 8--9- .. 6 Convert each of these recurring decimals to a fraction or mixed numeral, in simplest form. a 0. 2˙ b 0. 7˙ c 0. 3˙ d 0. 6˙ ˙ ˙ ˙ ˙ ˙ ˙ e 0. 1 9 f 0. 3 5 g 0. 2 7 h 0. 7˙ 5˙ i 0.1 5˙ j 0.4 8˙ k 0.7 3˙ l 0.9 4˙ ˙ ˙ ˙ ˙ m 2. 1 n 1. 6 0 o 7.8 3 p 3.41 6˙ ■ Further applications. 7 a. Write down the recurring decimal for 1--3- .. b Hence, express 8 a. Express. 1 -----30. 1 11 ------ and --6 30. and. 1 --------- as 300. recurring decimals.. as recurring decimals.. 1 1 11 b Show that --- + --- = ------ by adding fractions. 6 5 30 1 1 11 c Show that --- + --- = ------ by adding decimals. 6 5 30.

(21) Chapter. 9 a. Express. 2 --3. Rat io n al n u mbers. as a recurring decimal.. b Use the fact that. 1.5. 1:. 1 -----15. =. 2 -----30. =. 2 --3. ×. 1 -----10. to express. 1 -----15. as a recurring decimal.. Rates. A rate is a comparison of two unlike quantities. This is different from a ratio, in that a ratio is a comparison of two or more like quantities. In particular, a rate is a measure of how one quantity is changing with respect to another. In a ratio, units are not written, whereas in a rate, the units must be written if the rate is to have any meaning. A rate is a comparison of two unlike quantities. Equivalent rates can be formed by changing the units in either or both quantities. For example, a rate of 5 cm/s is equivalent to 50 mm/s since, in both cases, the object moves the same distance (5 cm or 50 mm) in equal amounts of time (1 s). To be in simplest form, a rate must be expressed as a quantity per 1 unit of another quantity. For example, a rate of 60 km/h is in simplest form because it represents a change in distance of 60 km for every 1 hour of time.. EG +S. Example 1 Express each of the following statements as a rate in simplest form. a. $150 in 3 hours. b 48 L in 12 min. Solutions a $150 in 3 hours ÷3 ÷3 = $50 in 1 hour = $50/h. EG +S. b. 48 L in 12 min ÷ 12 ÷ 12 = 4 in 1 min = 4 L/min. Example 2 Convert: a. 2.4 kg/day to g/day. Solutions a 2.4 kg in 1 day = 2400 g in 1 day = 2400 g/day. b 3.5 cm3/s to cm3/min b. 3.5 cm3 in 1 s × 60 × 60 = 210 cm3 in 1 min = 210 cm3/min. c. 18 m/s to km/h. c. 18 m in 1 s × 60 × 60 = 1080 m in 1 min × 60 × 60 = 64 800 m in 1 h = 64.8 km/h. 13.

(22) 14. Mathscape. Exercise. 9 Extension. 1.5. 1 Express each statement as a rate in simplest form. a 30 m in 3 s b 80 km in 2 h d 42 kg over 7 m2 e 32 g in 4 s g 108 km on 9 L h $180 in 4 h j $12 for 8 kg k 119 runs in 34 overs 1 m 240 beats in 2 --2- min n 72 kL in 1.5 h. c f i l o. 2 Complete these equivalent rates. a 3 cm/s = _____ cm/min b 5 g/min = _____ g/h c d 7.5 L/h = _____ L/day e 0.9 km/min = _____ km/h f. 45 L in 5 min 200 trees in 8 h 90c for 5 min 150 crates in 4 days 13 km on 1.25 L $2.30/kg = $_____ /t 0.4 kg/m2 = _____ kg/ha. 3 Complete these equivalent rates. a 2 L/min = _____ mL/min c 3.8 cm/s = _____ mm/s e 14.6 t/day = _____ kg/day. b 9 m/s = _____ cm/s d $1.15/g = _____ c/g f 2.35 ha/week = _____ m2/week. 4 Complete these equivalent rates. a 70 mm/s = _____ cm/s c 4900 mL/day = _____ L/day e 25 g/m3 = _____ kg/m3. b 850 cm/min = _____ m/min d 24c/min = $ _____ /min f 59 600 L/year = _____ kL/year. ■ Consolidation. 5 Complete the following equivalent rates. a 75 cm/s = _____ m/min c 9 m/mL = _____ km/L e 81.25 mL/h = _____ L/day. b 8c/g = $ _____ /kg d 150 kg/h = _____ t/day f 142 m/min = _____ km/h. 6 Complete the following equivalent rates. a 25 m/s = _____ km/h c 27.5 g/s = _____ kg/h e 0.8 m/min = _____ km/day g 72 km/h = _____ m/s. b d f h. 40 mL/s = _____ L/h 5 mm/min = _____ m/day 2.4c/mm = $ _____/m 12.24 t/day = _____ kg/min. 7 Convert these annual interest rates to monthly rates. a 12% p.a. b 6% p.a. c 18% p.a. 8 Convert these monthly interest rates to annual rates. a 0.75% per month b 0.9% per month. d 4.2% p.a. c. 1.25% per month. 9 Calculate the daily interest rate on a credit card if the annual rate is 15.33% p.a. 10 Convert: a $240/week to an equivalent monthly rate b $1352/month to an equivalent fortnightly rate.

(23) Chapter. 1:. Rat io n al n u mbers. c $2.80/week to an equivalent quarterly rate d $44.20/quarter to an equivalent fortnightly rate. ■ Further applications. 11 Complete these equivalent rates. a 5c/cm2 = $_____/m2 b 60 mL/m2 = _____ L/km2 c. 1.2 g/cm3 = _____ t/m3. 12 Complete this equivalent rate: $25/L = _____ c/cm3. TRY THIS. Desert walk. James is a cross-country walker. He comes to a 60 km stretch of desert where there is neither water nor food. He can walk 20 km per day and he can carry enough food and water for 2 days. How many days will it take him to cross the desert, and how many kilometres will he travel if he has to build up depots of food and water? Difficult part If he was considering a 100 km trip across the desert, how many days’ supply of food would be necessary?. 1.6. Solving problems with rates. We use many different types of rates every day, often without realising it. For example: • driving speed • petrol consumption rates • electricity rates. • bank interest rates • sporting strike rates • pollution rates. • currency exchange rates • rates of pay • medical recovery rates. As most adults drive a car, the concept of speed plays a very important role in our daily lives. We need to know how fast to drive in order to reach a particular destination on time. It is also important to know at what speed we can safely drive under various conditions, such as on narrow roads, in wet weather, near pedestrian crossings and so on. Informally, we think of speed as a measure of how fast an object is travelling. Formally, however, speed is defined as the rate of change of distance with respect to time. If we know the distance that an object has travelled from one point to another and the amount of time that it took to get there, then we can calculate how fast it was travelling. You should already be familiar with the following formulae relating speed, distance and time.. 15.

(24) 16. Mathscape. 9 Extension. Distance Speed = --------------------Time D S = ---T. Distance = Speed × Time D=S×T. Distance Time = --------------------Speed D T = ---S. There is an important distinction that needs to be made between average speed and instantaneous speed. The formulae above are usually associated with average speed, since the speed of the object may vary at different times throughout its journey. It may start moving slowly, speed up at times and slow down or even stop at other times. If, however, a speed camera had been used to measure the speed of the object at a single moment in time, then it would have measured the instantaneous speed of the object. The instantaneous speed at a split second may therefore differ from the average speed over the entire journey. The degrees and minutes key on the calculator can be used to simplify the working in some questions, particularly when the time is given in hours and minutes or minutes and seconds.. EG +S. Example 1 a The entry price to an amusement park is $7.50 per child. Find the total entry cost for a group of 90 children. b A farmer used 145 kg of super phosphate to cover an area of 5 ha. How many kilograms were used per hectare? Solutions a The entry cost for 1 child = $7.50 ∴ cost for 90 children = 90 × $7.50 = $675 b 145 kg covers an area of 5 ha ÷5 ÷5 ∴ 29 kg covers an area of 1 ha. EG +S. Example 2 A car can travel 138 km on 15 L of petrol. How far can it travel on a full tank of 35 L?. b 145 kg covers an area of 5 ha ÷5 ÷5 ∴ 29 kg covers an area of 1 ha. Solution Using the unitary method, 138 km on 15 L ÷ 15 ÷ 15 = 9.2 km on 1 L × 35 × 35 = 322 km on 35 L ∴ The car can travel 322 km on a full tank of 35 L of petrol..

(25) Chapter. EG +S. 1:. Rat io n al n u mbers. Example 3 a Jenny ran 600 metres in 80 seconds. What was her running speed? b A man drove at an average speed of 60 km/h for 7 hours. How far did he drive? c Shona’s average walking speed is 5 km/h. How long would it take her to walk 20 km? Solutions D a S = ---T 600 = --------80 = 7.5 m/s Exercise. b D=S×T = 60 × 7 = 420 km. D c T = ---S 20 = -----5 =4h. 1.6. An author writes at a rate of 3 pages per hour. How many pages would she write in 6 hours? b A shearer was able to shear 18 sheep per hour. How many sheep could he shear in 2 1--2- hours? c If petrol costs 97.4 cents/L, find how much it would cost to fill the tank in a car if the tank holds 42 L. d A tap is dripping at the rate of 3 mL per minute. How many litres of water will be lost in 2 days? e The crew on a fishing boat put out the nets every 2 hours and catch an average of 240 kg of fish. How many tonnes would the crew expect to catch if they fish for 10 hours?. 1 a. 2 a. Trevor earns $15.20 per hour as a sales assistant. How many hours would he need to work in order to earn $562.40? b Janine has a typing speed of 54 words per minute. How long would it take her to type a 1350 word article? c A cricket side scored 243 runs in 50 overs during a limited overs cricket match. Calculate the average scoring rate in runs per over. d A plumber charged $200 for 2 1--2- hours labour to repair a broken pipe. Find the plumber’s hourly rate. e A machine prints 150 newspapers per minute. How long would it take to print 18 000 newspapers?. ■ Consolidation. 3 a. Georgina drove 12 km in 10 minutes. At the same speed, how far would she drive in 30 minutes? b Gino’s pulse rate was 100 beats per minute. How many times would his heart beat in 15 seconds?. 17.

(26) 18. Mathscape. 9 Extension. c. A fruit picker claimed that he could pick 1200 apples per hour. How many apples could he pick in 20 minutes? d A bank teller can serve 20 customers per hour. How many customers can she serve in 45 minutes? e A tap drips 12 times in 20 seconds. How many times would it drip in 30 seconds? 4 Use the unitary method to answer the following questions. a Dianne paid $3.75 for 3 kg of oranges. How much would she have paid for 7 kg? b In a walking race, Paul took 40 minutes to walk 8 km. How long would it take him to walk 13 km? c Susan’s car uses petrol at the rate of 10.6 L/100 km. How much petrol would she use on a journey of 250 km? d If it takes 1 1--2- hours to remove 36 t of sugar from a silo, how long it would take to remove 30 t? e George delivered 400 pamphlets in 50 minutes. How many pamphlets would he deliver in 2 1--2- hours? f If sausages are being sold for $2.80 per kilogram, find the cost of purchasing 350 grams of sausages. 5 The following currency conversions show the value of 1 Australian dollar (A$1) in US$, euro and NZ$. A$1 = US$0.6075 A$1 = 0.5636 euro A$1 = NZ$1.0887 Use these currency conversions to convert: a A$20 into US$ b A$50 into euro c A$175 into NZ$ d A$250 into euro e A$600 into NZ$ f A$4500 into US$ 6 Use the currency conversions in Q5 to convert the following amounts into Australian dollars. Give your answers correct to the nearest cent. a NZ$30 b US$95 c 110 euro d NZ$200 e US$565 f 782 euro g NZ$1400 h US$2378 7 a b c d e f. Dave drove 350 km in 5 hours. What was his average speed? A plane travelled 1960 km in 7 hours. What was the speed of the plane? Jennifer ran at a speed of 8 km/h for 1 1--2- hours. How far did she run? A ship sailed at 42 km/h for 25 hours. What distance did it sail? Morgan rode his motor bike a distance of 340 km at a speed of 85 km/h. How long was the trip? A satellite orbits the Earth at a speed of 22 500 km/h. How long will it take for the satellite to travel a distance of 78 750 km?.

(27) Chapter. 1:. Rat io n al n u mbers. 8 Use the degrees and minutes key on your calculator to answer the following questions. a How far will a bus travel in 4 h 25 min at an average speed of 90 km/h? b Calculate the average speed of a battleship which sails 600 km in 11 h 45 minutes. Answer correct to the nearest km/h. c How long will it take for a plane to fly 615 km at a speed of 180 km/h? Answer correct to the nearest minute. ■ Further applications. 9 The speed of ships and sometimes of aircraft is usually measured in knots. A knot is a speed of 1 nautical mile per hour, where 1 nautical mile is equivalent to 1852 metres. a Express 1 knot in km/h. b If an aircraft is travelling at 120 knots, how long would it take to travel 5000 km? c If another aircraft is travelling at 760 knots, how many kilometres will it travel in 6 hours? 10 The petrol consumption (C) of a car is measured in litres of petrol (L) used per 100 km (K) travelled. a Write down a formula connecting C, L and K. b Calculate the petrol consumption of a car that travels 1038 km in a month and uses 95 L of petrol. c Meera is planning a tour of the Australian outback and expects to travel 10 000 km. Her vehicle’s petrol consumption is expected to average 12 L/100 km. If the average price of petrol in the outback is $1.12 per litre, calculate the expected cost of petrol for this trip. TRY THIS. Passing trains. A slow train leaves Canberra at 9:17 am and arrives at Goulburn at 12:02 pm. On the same day, the express leaves Canberra at 9:56 am and arrives in Goulburn at 11:36 am. At what times does the express pass the slow train if each is travelling at a constant speed? HINT: A travel graph would give an approximate time.. 19.

(28) 20. Mathscape. 9 Extension. 0FF. M. FOCUS. ON. WORKING. MAT. F O C U S O N W O R K I HNE G MATHEMATICALLY MATICALLY. I CAAL LL LYY O C U S O N W 0 R K I N G M AATTHHEEMMAATTI C G N I K R O OCUS ON. A. W. NUMBER PATTERN FROM. GALILEO 1615. Galileo looking through a telescope in his observatory. Introduction Galileo Galilei (1564–1642), the famous Italian mathematician, is better known for his scientific achievements than his mathematical ones. For example, in 1610 he made a series of telescopes that enabled him to discover four of the moons of Jupiter, to see mountains on the Moon, and to prove that the Milky Way was made up of stars. The four moons of Jupiter he discovered centuries ago are today called the Galilean satellites in his honour. Their names are Io, Europa, Ganymede and Callisto. We now know, thanks to space probes, that Jupiter has, in fact, 16 moons, 13 of which have been discovered from Earth. 1 In this activity, however, you will investigate a number pattern for the fraction --- . In 1615, 3 Galileo wrote one of the earliest manuscripts describing this pattern, so we can see how interested he was in pure mathematics. First, we search for a pattern among specific cases using inductive reasoning, and then we use algebra to generalise the pattern using deductive reasoning..

(29) Chapter. 5. 8C. HALLENGE. This is suggested as a group activity for extension stage 5 classes as an exercise in collaborative learning. 1 Investigate the pattern of odd numbers 1 + 3 + 5 + 7 + 9 + 11 + … 2 Notice that the partial sums 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, … are perfect squares. 3 See if you can find the pattern for the sum of 2 terms, 3 terms, 4 terms, … 4 Make a hypothesis about the sum of n terms. 5 Make a hypothesis about the sum of 2n terms. 6 If there were n terms in the numerator, how many would there be in the denominator? How many altogether? 7 Look carefully at the following patterns: 1 + 3 + 5 = 32 and 7 + 9 + 11 = (1 + 3 + 5 + 7 + 9 + 11) − (1 + 3 + 5) = 62 − 32 9 1 1+3+5 32 So ------------------------ = ---------------= ------ = --2 2 27 3 7 + 9 + 11 6 – 3 8. 9. 1 See if you can show that the next term is also --- using this same pattern: 3 1+3+5+7 ---------------------------------------- = … 9 + 11 + 13 + 15 From the pattern of your results, see if you can write down an expression for the fraction you would get if there were n terms on top. Ask your teacher for help if you need it, and 1 discuss the possibilities between yourselves. Check that the expression reduces to --- . 3. MATHEMAT IC. 4. 1 1+3 Check that the following statement is true: --- = -----------3 5+7 Notice that the numbers in the numerator and denominator form the pattern of odd numbers 1, 3, 5 and 7. 1+3+5 1 Continue the pattern to obtain ------------------------ . Does it still equal --- ? 7 + 9 + 11 3 Write down the next term of the sequence and continue, checking that in each case the 1 fraction is equivalent to --- . 3 Why is this true? Don’t try a formal proof, but see if you can draw a diagram to show that it must be. Use dots to represent the odd numbers and choose some specific cases. Ask for help as needed.. WORKING. 3. ACTIVITIES. ON. 2. 21. FOCUS. 1. EARNING. Rat io n al n u mbers. ALLY FOCUS ON WORKING MA THEMATICALLY. 2L. 1:.

(30) 22. Mathscape. FOCUS ON WORKING MATHEMATICALLY. E. 9 Extension. L ET’S. COMMUNICATE. Discuss what you have learned from this activity with a classmate or, perhaps, if you have worked in a group for this activity, with the group members. Can you see the value of inductive thinking in mathematics, that is, finding a pattern to suggest a general rule? If you worked in a group, write a short account of whether you enjoyed collaborating with others. Is it a good way to learn?. %R. EFLECTING. Mathematical thinking can be inductive, searching for a pattern to suggest a general rule, or deductive, reasoning in a chain of argument that leads to a mathematical proof. Both are very important in learning mathematics and are often used together. Think over how much of your learning in Year 9 is inductive and how much deductive. Discuss with your teacher as to how the two go together in mathematics lessons.. 1 What is a small word for magnitude? 2 Explain the difference between a guess and an estimate. 3 What is a rational number? 4 When is a digit in a number significant? 5 Read the Macquarie Learners Dictionary entry for rate:. rate noun 1. speed: to work at a steady rate | The car was travelling at a rate of 100 kilometres an hour. 2. a charge or payment: The interest rate on the loan is 10 per cent per year. 3. rates, the tax paid to the local council by people who own land –verb 4. to set a value on, or consider as: The council rated the land at $20 000. | I rate him a very good friend. –phrase 5. at any rate, in any case: We enjoyed ourselves at any rate. 6. at this rate, if things go on like this: At this rate we will soon run out of money.. How is the word ‘rate’ used in this chapter?.

(31) 1 State the number of significant figures in: a 406 b 7.2009 c 0.0031 d 12.0560 2 Round off each number correct to 1 significant figure. a 76 b 150 c 4278 d 894 000 3 Round off each number correct to 2 significant figures. a 341 b 725 c 15 049 d 369 412 4 Round off each number correct to the number of significant figures shown in the brackets. a 198 [1] b 4316 [1] c 18 209 [1] d 572 [2] e 2154 [2] f 36 587 [2] 5 Round off each decimal correct to the number of significant figures shown in the brackets. a 4.83 [1] b 0.0723 [2] c 3.4661 [3] d 22.018 [3] e 106.84 [2] f 8994.7 [1] 6 Evaluate each of these correct to 2 decimal places, using a calculator. a. 7 5 2--3- − 1 ----10. b −6.3 − 1.29. c e. 5.842. g. 6. d 136.4 f 2.715 1 h ------------0.107. 3. 91. 101.9 8π i -----3 7 Evaluate each of the following, correct to 2 decimal places, using a calculator. 15.7 × 34.15 75.3 × 29.1 a -----------------------------b 12.31 – 5.6 1 c ------------------------------- d 3.45 − (2 3--5- )4 2 3 0.57 + 4.5. CHAPTER REVIEW. e 8. 1: 5 92.8 ---------------4– 2. Rat io n al n u mbers. f. 3 15 + 4 13 --------------------------15 – 13. Estimate the value of each calculation. a 9.84 × 15.2 + 18.77 b 7.97 + 47.3 ÷ 15.49 194.7 × 259.2 c --------------------------------53.6. 9 Write each of these as a recurring decimal. a 0.333 333 … b 0.252 525 … c 0.346 346 … d 5.918 181 … 10 Convert these fractions to recurring decimals. 7 4 7 a --b -----c 1 ----12 9 11 11 Convert these recurring decimals to fractions. a 0.2˙ b 0.7˙ 2˙ c 0.13˙ 1 12 Given that --- = 0.16˙ , express each of the 6 following fractions as a recurring decimal. 1 1 a -----b --------60 600 13 Express each statement as a rate in simplest form. a 80 m in 10 s b $45 for 9 min c 72 L in 3 h d 215 runs for 5 wickets 14 a A car uses 18 L of petrol to travel 150 km. How much petrol would be needed to travel 350 km? b A farmer spreads 25 kg of fertiliser over an area of 4000 m2. How much fertiliser would be needed to cover an area of 1.5 ha?. 23. CHAPTER RE VIEW. Chapter.

(32) 24. Mathscape. 9 Extension. VIEW CHAPTER RE. 15 Convert: a 7 mm/min to mm/h b 75 km/h to km/day c 1.35 L/m2 to mL/m2 d 8.2 m/s to cm/s 16 Convert: a 40 m/min to km/h b 250 mL/h to L/day c 13.5 g/m2 to kg/ha d 5 m/s to km/h. A plane flew 6000 km in 7 1--2- hours. At what speed was the plane travelling? b Karen walked 24 km at 5 km/h, for how long did she walk? c Jude drove at 80 km/h for 4 h 15 min. What distance did he drive? 18 Daryl drove 527 km in 6 h 23 min. Find his speed, correct to 1 decimal place. 17 a. CHAPTER REVIEW.

(33) Algebra. 2. This chapter at a glance . use algebra to find rules for simple number patterns use the method of finite differences to find rules for simple number patterns evaluate algebraic expressions by substituting numbers for pronumerals add and subtract algebraic expressions multiply and divide algebraic expressions simplify algebraic expressions using the order of operations expand algebraic expressions that contain grouping symbols using the distributive law factorise algebraic expressions by removing the highest common factor add and subtract algebraic fractions multiply and divide algebraic fractions link algebra with generalised arithmetic use algebra to prove general properties of numbers use algebra to generalise solutions to problems expand binomial products expand perfect squares using the special identities determine whether a given expression is a perfect square complete a perfect square expand expressions using the difference of two squares identity expand expressions that involve a combination of algebraic techniques.. Algebra. Stage 5.1/5.2/5.3 After completing this chapter, you should be able to:. 25.

(34) 26. Mathscape. 2.1. 9 Extension. Describing simple patterns. Many complex problems can often be solved more easily by using algebra. Algebra lets us replace complex statements with short, simple expressions. Algebra also lets us generalise results that are always true, or are true under certain conditions, so that we do not have to keep solving the same types of problems over and over again.. ■ Finite differences It is not always easy to find the algebraic rule that describes the relationship between variables. The method of finite differences is a simple technique that can be used to help us find this relationship. Finite differences are the differences between the numbers in the bottom row of a table of values. x 1 2 3 4 For example, the numbers in the bottom row of this table are increasing by 3. Therefore, the finite differences in the table are all 3s.. y. 13. 16. 19. 22. +3 +3 +3 To find the rule that links the variables x and y in a linear relationship: write the standard rule in the form y = ∆x + find ∆, the finite differences between the bottom numbers in the table find by substituting into the rule a pair of values from the table. NOTE: This method can only be used for linear relationships when the x-values are consecutive integers (e.g. x = 1, 2, 3…).. EG +S. Example Find the rule that describes the relationship between x and y in this table of values. x. 0. 1. 2. 3. y. 7. 12. 17. 22. Solution Let the rule be in the form y = ∆x + , where ∆ is the difference between each pair of consecutive y-values. Now, the y-values are increasing by 5, ∴ ∆ = 5. If y = 5x + and x = 0 when y = 7, 7 = (5 × 0) + 7=0+ ∴ =7 ∴ The rule is y = 5x + 7.. x. 0. 1. 2. 3. y. 7. 12. 17. 22. +5. +5. +5.

(35) Chapter. Exercise. x. 1. 2. 3. 4. b. y. y = 2x + 5. x. 0. x. 1. 2. 3. 4. 5. y y = 3x − 4. c. Algebra. 2.1. 1 Complete each table of values using the given rules. y=x+3 a. 2:. 5. 6. y = 5x − 7 7. 8. d. y. x. 2. 3. y. 2 For each table of values in Q1, compare the differences between the y-values and the co-efficient of x in the rule. What do you notice? 3 Use the method of finite differences to find a rule for each table of values. a. c. e. g. i. k. x. 1. 2. 3. 4. y. 4. 8. 12. 16. x. 4. 5. 6. 7. y. 11. 13. 15. 17. p. 1. 2. 3. 4. q. 9. 14. 19. 24. a. 4. 5. 6. 7. b. 17. 19. 21. 23. a. 3. 4. 5. 6. b. 18. 24. 30. 36. s. 1. 2. 3. 4. t. 13. 20. 27. 34. b. d. f. h. j. l. x. 0. 1. 2. 3. y. 6. 7. 8. 9. p. 2. 3. 4. 5. q. 5. 8. 11. 14. p. 7. 8. 9. 10. q. 47. 54. 61. 68. a. 0. 1. 2. 3. b. 3. 7. 11. 15. s. 5. 6. 7. 8. t. 17. 22. 27. 32. s. 2. 3. 4. 5. t. 19. 31. 43. 55. 27.

(36) 28. Mathscape. 9 Extension. ■ Consolidation. 4. a. Copy and complete this table of values. Number of pentagons (p). 1. 2. 3. Number of triangles (t) b Write down an algebraic rule that links the number of triangles (t) to the number of pentagons (p). c How many triangles would there be in a figure with 9 pentagons? 5. a. Copy and complete this table of values. Number of squares (s). 1. 2. 3. Number of crosses (c) b Write down an algebraic rule that links the number of crosses (c) to the number of squares (s). c How many crosses would there be in a figure with 20 squares? 6. a. Copy and complete this table of values. Number of circles (c). 1. 2. 3. Number of dots (d) b Write down an algebraic rule that links the number of dots (d) to the number of circles (c). c How many dots would there be in a figure with 15 circles?.

(37) Chapter. 2:. Algebra. 7. a. Copy and complete this table of values. Number of large rhombuses (r). 1. 2. 3. Number of dots (d) b Write down an algebraic rule that links the number of dots (d) to the number of large rhombuses (r). c How many dots would there be in a figure with 40 large rhombuses? 8. a. Copy and complete this table of values. Number of squares (s). 1. 2. 3. Number of dots (d) b Complete this rule that relates the number of dots to the number of squares: d = ∆s + . 9. a. Copy and complete this table of values. Number of rectangles (r). 3. 4. 5. Number of dots (d) b Complete this rule that relates the number of dots to the number of rectangles: d = ∆r + .. 29.

(38) 30. Mathscape. 9 Extension. 10. a. Copy and complete this table of values. Number of circles (c). 3. 4. 5. Number of dots (d) b Complete this rule that relates the number of dots to the number of circles: d = ∆c + . 11. a. Copy and complete this table of values. Number of crosses (c). 2. 3. 4. Number of dots (d) b Complete this rule that relates the number of dots to the number of crosses: d = ∆c + . 12 Use the method of finite differences to find a rule linking the x- and y-values in each table. a. c. e. x. 1. 2. 3. 4. y. −7. −14. −21. −28. x. 1. 2. 3. 4. y. 7. 5. 3. 1. x. −4. −3. −2. −1. y. 10. 9. 8. 7. ■ Further applications. 13. b. d. f. x. 0. 1. 2. 3. y. 5. 4. 3. 2. x. 3. 4. 5. 6. y. 11. 8. 5. 2. x. −2. −1. 0. 1. y. 13. 10. 7. 4.

(39) Chapter. a. 2:. Algebra. Copy and complete this table of values. Number of squares (s). 1. 4. 9. Number of dots (d) b Write down an algebraic rule that links the number of dots (d) to the number of squares (s). c How many dots would there be in a figure with 64 squares? 14. a. Copy and complete this table of values. Number of cans in base (b). 1. 2. 3. Total number of cans (c) b Write down an algebraic rule that links the total number of cans (c) to the number of cans in the base (b). c How many cans would there be in a pile with 10 cans in the base? TRY THIS. Flags. Consider the following diagrams, then complete the table. 1. 2. 3. Pole length. 1. Number of squares. 3. 2. 3. 4. 5. …n. Find a rule relating the number of squares in the flag to the pole length. HINT: The rule is not linear.. 31.

(40) 32. Mathscape. 2.2. 9 Extension. Substitution. When we substitute for a pronumeral, we give the pronumeral the value of a number. An algebraic expression can have a number of values, depending on the value(s) that are substituted for each pronumeral.. EG +S. EG +S. Example 1 Evaluate each of the following when x = 3 and y = 7. x+y a 8x − 2y b 2x2 c -----------2 Solutions b 2x2 a 8x − 2y = (8 × 3) − (2 × 7) = 2 × 32 = 24 − 14 = 2×9 = 10 = 18 x+y d 6(x + y) -----------c 2 = 6(3 + 7) 3+7 = 6 × 10 = -----------2 = 60 10 = -----2 =5 Example 2 Evaluate each of these when m = 2 and n = 5. a. m−n+9. Solutions a m−n+9 = 2−5+9 = −3 + 9 =6. EG +S. d 6(x + y). b 3m − 4n b. 3m − 4n = (3 × 2) − (4 × 5) = 6 − 20 = −14. c. mn(m − n). c. mn(m − n) = 2 × 5 × (2 − 5) = 10 × (−3) = −30. Example 3 Evaluate each of the following given that p = 4, q = −3 and r = −6. a. p+q−r. Solutions a p+q−r = 4 + (–3) − (–6) =4−3+6 =1+6 =7. b pqr. c. q( p − r). b. c. q( p − r) = −3(4 − −6) = −3 × 10 = −30. pqr = 4 × (−3) × (−6) = −12 × −6 = 72.

(41) Chapter. Exercise. 2:. Algebra. 2.2. 1 Evaluate each of the following when k = 5. a k+7 b k−2 e 7k + 8 f 12k − 23 i k3 j 3k2 40 k m -----n -----k 15. c 13 − k g 30 − 4k k k2 + 3k k+7 o -----------4. d 3k h k2 l 2k2 − 9k 5k + 11 p -----------------2k – 1. ■ Consolidation. 2 Evaluate each of these when m = 7 and n = 3. a 16 − m + n b mn − 8 c e 13n − 4m f 50 − 2mn g i n2 + 10 j 50 − m2 k 2 3 m 2m + 13 n n − 8m o q 5(m + n) r 12(m − n + 6) s 24 4m + 4n u ------------v -------------------w m–n 5. 6m − n d 2m + 5n 3m + 6n − 11 h 100 − 5m − 3n m2 − n2 l 4n2 2 mn p m2n − mn3 n(8m − 20) t 2n(5m + mn) 3m + 2n 2 ------------------m + 5n x 2 n 3 Find the value of each expression using the substitutions r = 6, s = 2 and t = 11. a s−r b r−t c −s + t d −t − r e r−s−t f s−t+r g −r + s + t h −t + s − r i 3s − t j −5t + 4r k −8r + st l 5s − rt m 100 − rst n rs − st o r2 − 3rt p s 2 − r 2 + t2 2 q t − 5s r r(s − t) s 5(2t − 4r − 9s) t 3s(r2 − t2) ■ Further applications. 4 Evaluate each of the following given that a = −3, b = 8 and c = −6. a a+b b b−c c c+a e a−c+b f c+b+a g b−a−c i 4a − 2b − c j b + 5a + 2c k 3b − 5a + 10c m b(a + c) n c(b − a) o 2a(c + b) q (b − a)(b + c) r a2b s ab − c3 u. 2. b +c. 2.3. 2. v. ab -----c. b – 2c w --------------a–1. d h l p t. a−b −b + c + a −4c + 3b − 7a ac(b − 10) abc. x. 2(a + c ) -----------------------ac. 2. 2. Adding and subtracting algebraic expressions. Algebraic terms with identical pronumerals are called like terms. Only like terms can be added or subtracted.. 33.

(42) 34. Mathscape. 9 Extension. Some examples of: • like terms are 3m and 5m, 7q and −2q, xy and yx, 4t 2 and 9t 2 • unlike terms are 4a and 4b, ef and fg, 6u2 and 11u. To collect the like terms in an algebraic expression: add or subtract the co-efficients keep the same pronumeral(s).. EG +S. EG +S. Example 1 Simplify each of these. a 7s + 3s d 5r 2 + 2r 2. b 12w − 4w e 14gh − 9gh. c f. 6y − y 7pq + 6qp. Solutions a 7s + 3s = 10s d 5r 2 + 2r 2 = 7r 2. b 12w − 4w = 8w e 14gh − 9gh = 5gh. c f. 6y − y = 5y 7pq + 6qp = 13pq. c. 8x + 7y − 5x − 12y. c. 8x + 7y − 5x − 12y = 8x − 5x + 7y − 12y = 3x − 5y. Example 2 Simplify these expressions by collecting the like terms. a. 6e + 13 + 4e + 8. Solutions a 6e + 13 + 4e + 8 = 6e + 4e + 13 + 8 = 10e + 21 Exercise. b 9v 2 + 7v + v 2 − 3v b. 9v 2 + 7v + v 2 − 3v = 9v 2 + v 2 + 7v − 3v = 10v 2 + 4v. 2.3. 1 a Simplify 7x + 3x. b Verify your answer by substituting several values for x. 2 a Simplify 5n + 2n and 2n + 5n. b Does 5n + 2n = 2n + 5n? c Does it matter in which order algebraic expressions are added? 3 a Simplify 5s − 3s and 3s − 5s. b Does 5s − 3s = 3s − 5s? c Does it matter in which order algebraic expressions are subtracted? 4 Simplify each of the following. a 4y + 5y b 12n − 8n e 11z − 11z f 10b − 9b i 6pq + 5pq j 15xy − 8yx. c 2c + c g 3a2 + 4a2 k 2abc + 6abc. d 7k − k h 13g2 − 5g2 l 14m2n + 5m2n.

(43) Chapter. m 3t − 7t q 5pq − 11pq. n −2u + 12u r −10yz + 9zy. o s. −13p + 4p e 2 − 11e 2. 2:. Algebra. p −8j − 7j t −9rs2 + 7rs2. ■ Consolidation. 5 Simplify: a 3a + 4a + 2a b 10b − 3b − b c 9k − 6k + 7k e 3p − 10p + 15p f −6r + 4r + 9r g −x − 7x − 5x i 4e 2 − 7e 2 − 10e 2 j 8a2 − 12a2 + 4a2 k 5ab + ab − 9ab. d 5m − 8m − 4m h −3c + 2c − 11c l −9pq + 6pq + 7pq. 6 Collect the like terms in each expression. a 4q + 3q + 2 b 5g + 8 + 9 e 10c + 8c + d f 9j − 4k + 2j 2 i x + 4x + x j 8m + m2 − 10m. d 13 + 6t − 5t h 12 − 2n − 4n l 4a2b + 6ab2 − 3ab2. c 15u − 7u − 3 g 3a − 5a + 7 k 3w2 + 2w2 + w. 7 Simplify these expressions by collecting the like terms. a k+2+k+3 b 7c + 4 + 5c + 1 d 8m + 5n + m − 4n e 5t + 12 − 2t + 4 g 10g + 4g − 3h + 6h h 11p + 2q − 6q − 4p j 6s + 11 − 6s + 11 k 5y − 9 + 5y + 9 m x + y − 4x − 7y n −6a + 2b + 5a + 10b 2 2 p x + 6x + 2x + 3x q 7a2 + a2 + a − 4a 2 2 s z − 2z + 5z − 3z t d 2 + 7d + 5 − 4d. c f i l o r u. 8p + 3q + p + 7q 8u + 9v − 3u − v 3b − 5c + 2c − 8b 4m − 7n − 10m + 5n −5j − 12k + 15j − 4k 9u − 4u 2 − u 2 + 3u 4mn + 5m − 3mn − 9n. 8 Find, in simplest form, an algebraic expression for the perimeter of each figure. a b c 8n. m. 5k. 6n m+6. d. e. f. 2c − 1. y−5. x−2. c+4. 3c + 11 15 − x. y + 12. ■ Further applications. 9 a b c d e. Subtract 3x 2 − 4x + 10 from 7x 2 + 2x − 4. From 5a2 + 9, take a2 − 2a − 5. Find the difference between 5p + 3 and 2p2 + 6p + 3. By how much does 4k 2 + 7k + 11 exceed k 2 − 2k + 15? Take the sum of t 2 − t + 4 and 2t 2 + 17t + 9 from 4t 2 + 9t + 20.. c−7. 35.

(44) 36. 9 Extension. Mathscape. 2.4. Multiplying and dividing algebraic expressions. Any algebraic terms can be multiplied or divided. They do not have to be like terms. To multiply algebraic terms: multiply the co-efficients multiply the pronumerals. To divide algebraic terms: express the division as a fraction divide the co-efficients divide the pronumerals.. EG +S. EG +S. Example 1 Simplify each of the following: a. b×3. b 4r × 5s. c. 1 --4. × 24w. d 8a × 5a. e. 6xy × 7yz. f. −12u × (−5v). Solutions a b × 3 = 3b. b 4r × 5s = 20rs. c. 1 --4. d 8a × 5a = 40a2. e. f. −12u × (−5v) = 60uv. 6xy × 7yz = 42xy2z. × 24w = 6w. Example 2 Simplify each of the following: a. 15p ÷ 5p. b 21ab ÷ 3a. Solutions a 15p ÷ 5p. b. 15 p = --------5p =3 Exercise. 21ab ÷ 3a 21ab = -----------3a = 7b. c. c. 45t 2 ÷ 9t 45t 2 ÷ 9t 45t 2 = ---------9t = 5t. d 64mn2 ÷ (−8mn) d. 64mn2 ÷ (−8mn) 64mn 2 = ---------------– 8mn = −8n. 2.4. 1 a Simplify 2a × 3b. b Verify your answer by substituting several pairs of values for a and b. 2 a Does 5n × 4n equal 20n or 20n2? b Substitute a value for n to verify your answer..

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