Preliminary General Maths Text Book

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MATHS

_Quest

General Mathematics

PRELIMINARY COURSE

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Typeset in 10.5/12.5 pt Times

Cataloguing-in-Publication data Rowland, Robert, 1963–.

Maths quest general mathematics: preliminary course. 2nd ed.

For secondary school students.

ISBN 978 0 7314 0570 1 (student edition) ISBN 978 0 7314 0571 8 (teacher edition) 1. Mathematics — Textbooks. I. Title. 510

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Introduction viii

About eBookPLUS x

Acknowledgements xi

CHAPTER 1 Earning money 1

Are you ready? 2

Calculating salary payments 3

Exercise 1A 4

Calculating wages 6

Exercise 1B 8

10 Quick Questions 1 11

Commission and royalties 11

Exercise 1C 14 Payment by piece 16 Exercise 1D 17 10 Quick Questions 2 18 Working overtime 19 Exercise 1E 21

Investigation — Investigating government payments 24

Additions to and deductions from gross

pay 25

Exercise 1F 27

Investigation — Examining bank fees and taxes 30 10 Quick Questions 3 31 Budgeting 31 Exercise 1G 35 Summary 40 Chapter review 41

Practice examination questions 43

CHAPTER 2 Units of measurement 45

Units of measurement 47

Exercise 2A 50

Relative error 52

Exercise 2B 54

Investigation — Measuring heights 56 10 Quick Questions 1 56 Significant figures 57 Exercise 2C 60 Rates 61 Exercise 2D 65 Percentage change 67 Exercise 2E 68 10 Quick Questions 2 69 Using ratios 69 Exercise 2F 72 Summary 74 Chapter review 75

Practice examination questions 76

CHAPTER 3 Applications of area and

volume 77

Review of area 79

Exercise 3A 81

Investigation — Maximising an area of land 84

Calculating irregular areas from a field

diagram 85

Investigation — Land survey 86 Exercise 3B 87 10 Quick Questions 1 88 Solid shapes 89 Exercise 3C 91 Surface area 92 Exercise 3D 94 10 Quick Questions 2 96 Volume of a prism 97

Investigation — Exploring the volume of a prism 97

Exercise 3E 99

Volume of other solids 103

Exercise 3F 105 Summary 108 Chapter review 109

Practice examination questions 112

CHAPTER 4 Basic algebraic skills 113

Are you ready? 114

General number patterns 115

Exercise 4A 117

Number pattern notation 119

Exercise 4B 122 10 Quick Questions 1 124

Adding and subtracting like terms 125

Exercise 4C 126

Substitution 127

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10 Quick Questions 2 130

Multiplication and division of algebraic expressions 131

Exercise 4E 133

Solving linear equations 134

Exercise 4F 137

Equations arising from substitution 139

Exercise 4G 141 Summary 143 Chapter review 144

CHAPTER 5 Statistics and society 147

Analysing data 149

Investigation — Why statistical investigation? 149

Investigation — A statistical investigation – 1 149

Statistical processes 150

Investigation — Posing questions 150 Investigation — A statistical investigation – 2 150

Exercise 5A 152

Exercise 5B 155

Exercise 5C 159

Quality control 160

Exercise 5D 162

Privacy and ethical issues 163

Investigation — Privacy issues 163 Investigation — Organisations that use statistics 164

Summary 165 Chapter review 166

CHAPTER 6 Data collection and

sampling 167

Target populations and sampling 169

Investigation — Gallup poll 169 Investigation — Identifying the target population 169

Exercise 6A 172

Investigation — Census or sample 174

Population characteristics 174

Investigation — Population characteristics 175

Exercise 6B 177

Investigation — Choosing a sample 179 10 Quick Questions 1 179

Bias 180

Investigation — Bias in statistics 181 Investigation — Biased sampling 182 Investigation — Spreadsheets creating misleading graphs 182 Exercise 6C 184 Investigation — Bias 185 Types of data 186 Exercise 6D 188 10 Quick Questions 2 191 Estimating populations 191

Investigation — Estimating a population 192 Exercise 6E 193

CHAPTER 7 Modelling linear

relationships 199

Graphing linear functions 201

Exercise 7A 204

Investigation — Graph of height versus age 205

Gradient and intercept 205

Exercise 7B 209

Drawing graphs using gradient and intercept 211

Exercise 7C 214 10 Quick Questions 1 215

Graphing variations 216

Exercise 7D 217

Investigation — Currency conversions 218

Step and piecewise functions 218

Exercise 7E 220

Simultaneous equations 221

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CHAPTER 8 Investing money 229

Calculation of simple interest 231

Exercise 8A 234 10 Quick Questions 1 236

Graphing simple interest functions 236

Exercise 8B 239

Calculation of compound interest 241

Calculating compound interest from a table of

compounded values 248

Exercise 8D 251

Graphing compound interest functions 253

Exercise 8E 255

Share dividends 257

Exercise 8F 258

Graphing share performance 260

Exercise 8G 262

Investigation — Researching share prices 263

Inflation and appreciation 264

Exercise 8H 265 Summary 267 Chapter review 268

CHAPTER 9 Displaying single data

sets 271

Frequency tables 273

Exercise 9A 276

Types of graphs 277

Exercise 9B 280

Investigation — Choice of graph 283 Investigation — Producing graphs using technology 283

Statistical graphs 283

Range and interquartile range 292

Exercise 9D 297 Stem-and-leaf plots 302 Exercise 9E 306 Five-number summaries 308 Exercise 9F 312 Summary 315 Chapter review 316

CHAPTER 10 Summary statistics 321

Calculating the mean 323

Investigation — Average — what does it mean? 323

Exercise 10A 328

Standard deviation 333

Exercise 10B 337

Median and mode 341

Best summary statistics 350

Exercise 10D 351

Investigation — Wage rise 354

Investigation — Best summary statistics and comparison of samples 354

CHAPTER 11 Similarity of two-dimensional

figures 363

Are you ready? 364

Similar figures and scale factors 365

Exercise 11A 367

Investigation — Enlarging a figure 369 Investigation — Investigating scale factors 369

Investigation — Similar triangles 370

Solving problems using similar figures 371

Exercise 11B 372

Investigation — Scale drawing of the classroom 373

House plans 374

Exercise 11C 376

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CHAPTER 12 Taxation 383

Calculating allowable deductions 385

Exercise 12A 388 Taxable income 390 Exercise 12B 392 10 Quick Questions 1 395 Medicare levy 395 Exercise 12C 397

Investigation — Medicare levy 397

Calculating tax 398

Exercise 12D 402 10 Quick Questions 2 404

Calculating GST and VAT 405

Exercise 12E 407

Graphing tax functions 409

Exercise 12F 409 Summary 411 Chapter review 412

CHAPTER 13 Right-angled triangles 415

History of mathematics — Pythagoras of Samos (circa 580 BC–500 BC) 417

Pythagoras’ theorem 418

Exercise 13A 421

Calculating trigonometric ratios 423

Investigation — Looking at the tangent ratio 423

Investigation — Looking at the sine ratio 425 Investigation — Looking at the cosine

ratio 426

Exercise 13B 429 10 Quick Questions 1 430

Finding an unknown side 431

Finding angles 438

Exercise 13D 442

Angles of elevation and depression 445

Exercise 13E 448

Investigation — Calculation of heights 449

Proportional diagrams 450

Investigation — Checking with a proportional diagram 450

Investigation — Using proportional diagrams 450

CHAPTER 14 The language of chance 455

Informal description of chance 457

Exercise 14A 460

Investigation — Common descriptions of chance 462

Sample space 462

Exercise 14B 464

Investigation — Matching actual and expected results 465

10 Quick Questions 1 466

Tree diagrams 467

Exercise 14C 470

Investigation — Two-stage experiments 471

Equally likely outcomes 472

Exercise 14D 474 10 Quick Questions 2 475

Using the fundamental counting principle 476

Exercise 14E 479 Summary 481 Chapter review 482

CHAPTER 15 Relative frequency and

probability 485

Relative frequency 487

Exercise 15A 489

Investigation — Researching relative frequencies 491

Single event probability 492

Exercise 15B 494

Investigation — Comparing probabilities with actual results 497

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Writing probabilities as decimals and percentages 499 Exercise 15C 500 Range of probabilities 502 Exercise 15D 504 10 Quick Questions 2 506

Investigation — Graphing results 506

Complementary events 507

Exercise 15E 509 10 Quick Questions 3 511

Glossary 515

Answers 521

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Maths Quest General Mathematics — Preliminary course is the first book in a series specifically designed for the General Mathematics Stage 6 Syllabus starting in 2000. This course replaces the current syllabuses for

Mathematics in Society (1981) and Mathematics in Practice (1989).

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

pro-vide bright, stimulating photos and to make navigation through the text easier. Clear, concise theory sections contain worked examples, highlighted

impor-tant 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 10 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’ under-standing 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, con-taining 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

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Technology is fully integrated, in line with Board of Studies

recommen-dations. 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 on the bonus accompanying online resources.

Graphics calculator tips are incorporated throughout the text.

All formulae, which are given on the HSC 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 3D polyhedra and their nets

Tess: for producing tessellations and other symmetric planar illustrations TI Connect: calculator screen capture and program transfer

CASIO Software FA-123: calculator screen capture and program transfer Cabri Geometry II: dynamic geometry program

Adobe® Acrobat® Reader 4.0

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 rel-evant study design dot points. A readily accessible Work program lists all available resources and provides curriculum coverage information.

The accompanying teacher eGuidePLUS contains everything in the student eBookPLUS and more. Two tests per chapter, fully worked solutions to

WorkSHEETs, the work program and other curriculum advice in editable Word 2000 format are provided.

Maths Quest is a rich collection of teaching and learning resources within

one package.

Maths Quest General Mathematics Preliminary course, Second edition, provides ample material, such as exercises, analysis questions, investi-gations, worksheets and technology ﬁles, from which teachers may set assessment tasks.

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The Maths Quest project began in 1997, and the ﬁrst edition of this book was printed in 2000. 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 the ﬁrst textbook 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 whom 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, who 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 — I 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.

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Cabri Geometry™ II PLUS

Reproduced with permission of Cabrilog. Cabrilog 6, Robert Schuman Place 38000 Grenoble FRANCE Web: http://www.cabri.com

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About the author

Robert Rowland has been teaching Mathematics for over 20 years and cur-rently holds the position of Head teacher, Teaching and learning at Ulladulla High School. He taught at Cabramatta High School from 1985 to 1988 before taking up his appointment at Ulladulla High School in 1989. Robert has suc-cessfully taught all levels of Mathematics to Year 12 as well as Computing Studies 7–12 and Information Processes and Technology. Robert is the co-author of New South Wales Maths Year 9 Standard and New South Wales

Maths Year 10 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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In this

_chapter

1A Calculating salary

payments

1B Calculating wages

1C Commission and royalties

1D Payment by piece

1E Working overtime

1F

Additions to and

deductions from gross pay

1G Budgeting

syllabus

_reference

Financial Mathematics 1

• Earning money

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READY?

areyou

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 Preliminary Course CD-ROM or ask your teacher for a copy.

Converting units of time

1 Convert each of the following to the units shown in brackets.

a 2 years (months) b 3 years (weeks)

c 42 weeks (fortnights) d 60 months (years) Multiplying and dividing a quantity (money) by a whole number

2 Calculate each of the following.

a $23.50 × 26 b $31 432.70 ÷ 12

c $528.72 × 52 d $45 600 ÷ 52

Converting a percentage into a decimal

3 Convert each of the following percentages to a decimal.

a 34% b 79% c 4%

d 67.2% e 8.25% f 17.5%

Finding a percentage of a quantity (money)

4 Find each of the following.

a 10% of $350 b 25% of $1424

c 18% of $9000 d 12.5% of $4570 Multiplying a quantity (money) by a decimal

5 Calculate each of the following.

a $8.56 × 1.5 b $12.90 × 2.5

Adding periods of time

6 Jessica has worked the following hours in one week. Thursday 6.30 pm to 9.00 pm

Friday 5.45 pm to 9.00 pm Saturday 8.00 am to 2.30 pm How many hours has she worked?

Expressing one quantity as a percentage of another

7 For each of the following pairs, express the first quantity as a percentage of the second quantity.

a $56, $400 b $13, $20 c $125, $625

Increasing a quantity by a percentage

8 Increase each of the following by the percentage indicated.

a $560 by 10% b $1120 by 5% c $2560 by 15%

1.1

1.2

1.3

1.4

1.5

1.6

1.8

1.9

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Calculating salary payments

Methods of payment

A payment received by an employee for doing a job is called

income. There are many different ways people are paid for performing a job. In this section we are going to look at some of these methods of payment: salaries, wages,

commission, royalties, piecework and overtime.

Salaries

Many people employed in professional occupations are paid a salary. Such employees include teachers,

lawyers, accountants and some doctors.

A salary is a fixed amount of money that is paid to employees to do their jobs. The amount paid does not change, regardless of the number of hours worked.

Salaries are usually calculated on an annual basis. A salary is therefore usually stated as an amount per

annum, which means per year. Salaries are paid in weekly, fortnightly or monthly amounts. To make calcu-lations about salaries, you will need to remember the following information.

1 year = 52 weeks

= 26 fortnights = 12 months

We reverse this calculation when we are given the weekly, fortnightly or monthly pay of a person and are then asked to calculate the annual salary.

Dimitri works as an accountant and receives an annual salary of $46 800. Calculate the amount that Dimitri is paid each fortnight.

THINK WRITE

There are 26 fortnights in a year, so we divide $46 800 by 26.

Fortnightly pay = $46 800 ÷ 26

Evaluate. Fortnightly Pay= $1800

1 2

1 WORKED

E

xample

Grace is a solicitor who is paid $3500 per month. Calculate Grace’s annual salary.

THINK WRITE

There are 12 months in a year, so multiply $3500 (monthly pay) by 12.

Annual salary = $3500 × 12

Evaluate. Annual salary= $42 000

1 2

2 WORKED

E

xample

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To compare a salary payment with other forms of income it may be necessary to calcu-late the equivalent daily or hourly payment. To do this, we need to know the number of days or hours worked per week.

Calculating salary payments

1 Toni is paid a salary of $44 200 per annum. Calculate Toni’s fortnightly pay.

2 Roger is paid a salary of $49 920 per annum. Calculate Roger’s weekly pay.

3 Frieda is paid a salary of $54 000 per annum. Calculate Frieda’s monthly pay.

4 Wendy works as an office secretary and is paid a salary of $38 740 per annum. Calculate Wendy’s pay if she is paid:

a weekly b fortnightly c monthly.

5 Darren earns a salary of $43 000 per annum. Calculate Darren’s fortnightly pay, correct to the nearest cent.

Charlotte works as a laboratory technician and is paid an annual salary of $41 560. If Charlotte works an average of 42 hours per week, calculate her equivalent hourly rate of pay.

THINK WRITE

Calculate the weekly pay by dividing the salary by 52.

Weekly pay = $41 560 ÷ 52 = $799.23 Calculate the hourly

rate by dividing the weekly pay by 42. Hourly rate = $799.23 ÷ 42 = $19.03 1 2

3 WORKED

E

xample

1. A salary is a fixed payment made for doing a job.

2. A salary is usually calculated on an annual basis and can be paid in weekly, fortnightly or monthly instalments.

3. To calculate information about equivalent daily or hourly rates of pay, we need information about the number of days and hours worked by the employee.

remember

1A

Skill SHEET

1.1

Converting units of time Skill SHEET

1.2

Multiplying and dividing a quantity (money) by a whole number WORKED Example 1 EXCEL Spreads_heet Payroll calculations

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6 Copy and complete the table below for food production employees.

7 Maxine is paid a salary. She receives $460 per week. Calculate Maxine’s annual salary.

8 Thao receives $1250 per fortnight. Calculate Thao’s annual salary.

9 Deidre is paid monthly and receives $5800. Calculate Deidre’s annual salary.

10

Which of the following people receives the greatest salary?

A Goran, who receives $530 per week.

B Bryan, who receives $1075 per fortnight.

C Wayne, who receives $2330 per month.

DRon, who receives $27 900 per annum.

11 Fiona receives a salary of $29 700 per annum. If Fiona works an average of 40 hours per week, calculate the equivalent hourly rate of pay.

12 Jade receives a salary of $33 000 per annum.

a Calculate Jade’s weekly pay, correct to the nearest cent.

b Jade works an average of 36 hours each week. Calculate the hourly rate to which Jade’s salary is equivalent. Give your answer correct to the nearest cent.

13 Karina is on an annual salary of $35 776. Letitia is on a wage and is paid $16.00 per hour.

a Calculate Karina’s weekly pay.

b If Karina works an average of 42 hours per week, calculate whether Karina or Letitia receive the better rate of pay.

14 Garry earns $42 500 per year while his friend Henry earns $18.50 per hour. Calculate the number of hours that Henry will need to work each week to earn more money than Garry does.

Annual salary Weekly pay Fortnightly pay Monthly pay

$30 000 $39 500 $42 250 $54 350 $86 475 WORKED Example 2 multiple choice WORKED Example 3

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Calculating wages

Most people in the workforce earn a wage. A wage is paid at an hourly rate.

The hourly rate at which a person is usually paid is called an ordinary rate. The wage for each week is calculated by multi-plying the ordinary rate by the number of hours worked during that week.

To compare two people’s wages, we can’t just look at the amount of money each receives in a pay packet. We must also consider the number of hours each has worked. Wages are compared by looking at the hourly rate. To calculate the hourly rate of an employee we need to divide the wage by the number of hours worked.

Using a similar method we are able to calculate the number of hours worked by an employee, given their wage and hourly rate of pay. The number of hours worked is found by dividing the wage by the hourly rate.

In some cases, wages are increased because an allowance is paid for working in unfavourable conditions. An allowance is an additional payment made when the working conditions are difficult or unpleasant.

Sadiq works as a mechanic and is paid $13.65 per hour. Calculate Sadiq’s wage in a week where he works 38 hours.

THINK WRITE

Multiply $13.65 (the hourly rate) by 38 (the number of hours worked).

Wage = $13.65 × 38

Wage= $518.70

4 WORKED

E

xample

Georgina works 42 hours as a data entry operator for a computer company. Her wage for the week totalled $483.84. Calculate Georgina’s hourly rate of pay.

THINK WRITE

Divide $483.84 (the wage) by 42 (number of hours worked).

Hourly rate = $483.84 ÷ 42

Hourly rate= $11.52

5

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For example, a road worker may be paid an allowance for working in the rain. In these cases, the allowance must be multiplied by the number of hours worked in the unfavourable conditions and this amount added to the normal pay.

This type of allowance is also paid to casual workers. When you are employed on a casual basis you do not receive any holiday pay and you do not get paid for days you have off because you are sick. The casual rate is a higher rate of pay to compensate for this.

Ryan is a road worker and is paid $9.45 per hour for a 35-hour week. For working on wet days he is paid a wet weather allowance of 86c per hour. Calculate Ryan’s pay if for 12 hours of the week he works in the rain.

THINK WRITE

Calculate Ryan’s normal pay by multiplying $9.45 (hourly rate) by 35 (number of hours worked).

Normal pay = $9.45 × 35 = $330.75 Calculate the wet weather allowance by

multiplying 0.86 (the wet weather allowance) by 12 (number of hours worked in the wet).

Allowance = $0.86 × 12 = $10.32 Add the normal pay to the wet weather

allowance to calculate the total pay.

Total pay = $330.75 + $10.32 = $341.07 1 2 3

6 WORKED

E

xample

1. A wage is money earned at an hourly rate.

2. To calculate a wage we multiply the hourly rate by the number of hours worked during the week.

3. To calculate an hourly rate we divide the wage by the number of hours worked. 4. To calculate the number of hours worked we divide the wage by the hourly

rate.

5. Allowances are paid for working under unfavourable conditions. The total allowance should be calculated and then added to the normal pay.

6. A casual rate is a higher rate of pay for casual workers to compensate them for having no holidays and receiving no sick leave.

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Calculating wages

1 Allan works in a newspaper printing mill and is paid $12.95 per hour. Calculate Allan’s wage in a week where he works 40 hours.

2 Copy and complete the table below by calculating the wage of each of the workers.

3 Alicia is an apprentice chef. In the first year of her apprenticeship she earns $11.80 per hour. Calculate Alicia’s wage in a week where she works:

a 36 hours

b 48 hours

c 42.5 hours.

4 Domonic is a fully qualified chef. He earns $13.50 per hour. Calculate Domonic’s wage in a week where he works:

a 32 hours

b 37 hours

c 44.5 hours.

5 Katherine works as a casual waitress. Casual workers earn 20% more per hour than full-time workers to compensate for their lack of holidays and sick leave.

a A full-time waitress earns $14.45 per hour. Calculate the casual rate earned by casual waitresses.

b Calculate Katherine’s wage in a week where she works 6 hours on Saturday and 7 hours on Sunday.

6

Which of the following workers earns the highest wage for the week?

A Dylan, who works 35 hours at $13.50 per hour

B Lachlan, who works 37 hours $12.93 per hour

C Connor, who works 38 hours at $12.67 per hour

DCameron, who works 40 hours at $12.19 per hour

Name Hourly rate Hours worked Wage

A. Smith $14.52 40 B. Brown $16.45 38 N. Tran $15.95 37.5 A. Milosevic $20.10 41 L. McTavish $18.04 36

1B

WORKED Example 4 EXCEL Spreads_heet Payroll calculations multiple choice

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7 Calculate the hourly rate of a person who works 40 hours for a wage of $387.20.

8 Julie earns $11.42 per hour. Calculate the number of hours worked by Julie in a week where she is paid $445.38.

9 Copy and complete the table below.

10 Calculate the hourly rate of a casual worker who earns $250.80 for 20 hours work.

11

Which of the following workers is paid at the highest hourly rate?

A Melissa, who works 35 hours for $366.45

B Belinda, who works 36 hours for $376.20

C April, who works 38 hours for $399.76

DNicole, who works 40 hours for $419.60

12

Which of the following people worked the greatest number of hours?

A Su-Li, who earned $439.66 at $11.57 per hour

B Denise, who earned $576.00 at $14.40 per hour

C Vera, who earned $333.20 at $9.52 per hour

DCamille, who earned $707.25 at $17.25 per hour

13 Richard works as an electrical linesman and is paid $10.94 per hour for a 38-hour week. When he has to work at heights he is paid a 46c per hour ‘height allowance’. Calculate Richard’s pay in a week where 15 hours are spent working at heights.

14 Ingrid works as an industrial cleaner and is paid $14.60 per hour for a 35-hour working week. When Ingrid is working with toxic substances she is paid an allowance of $1.08 per hour. Calculate Ingrid’s pay if she works with toxic substances all week.

15 Rema works as a tailor and earns $9.45 per hour.

a Calculate Rema’s wage in a week where she works 37 hours.

b Zhong is Rema’s assistant and earns $8.20 per hour. Find the least time Zhong must work if he is to earn more money than Rema does.

16 Tamarin works 38 hours per week at $12.40 per hour.

a Calculate Tamarin’s weekly wage.

b Zoe earns the same amount each week as Tamarin does, but Zoe works a 40-hour week. Calculate Zoe’s hourly rate of pay.

Name Wage Hours worked Hourly rate

A. White $416.16 36 B. Black $538.80 40 C. Green $369.63 37 D. Brown $813.96 $19.38 E. Scarlet $231.30 $15.42 F. Grey $776.72 $20.44 WORKED Example 5 multiple choice multiple choice WORKED Example 6

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Throughout this chapter we are going to develop a number of spreadsheets that will calculate wages. Work through the following steps.

1. Open a spreadsheet and enter the following information. Alternatively, access the spreadsheet (Wages_1) from the Maths Quest General Mathematics Preliminary

Course CD-ROM.

2. Enter a pay rate of $11.20 per hour for each employee.

3. Enter the hours worked as follows: Frederick Astini, 40; James Carter, 38; Kelly George, 36; Dean Jones, 15; Paul Limbrick, 45.

4. In cell E7 (in the column headed Gross Pay) enter the formula =C7*D7. This will calculate the wage for Frederick Astini (the figure 448 should appear in the cell). 5. Format cell E7 as currency (cell E7 should now show $448.00).

6. Highlight cells E7 to E11 and select the Fill Down option. The wages for each employee should now be calculated and be formatted as currency. (The entries in this column should read $448.00, $425.60, $403.20, $168.00 and $504.00.)

7. If you now change the hours worked by each employee, his or her gross pay should update automatically.

8. Choose the Save As function to save the spreadsheet as Wages_1.

Computer Application

1 Spreadsheets

EXCEL Spreads_heet

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1 Calculate the wage of a person who works 36 hours at a pay rate of $9.56 per hour.

2 Calculate the wage of a person who works 38 hours at $13.65 per hour.

3 Donna works 15 hours on weekends at $14.56 per hour. Calculate Donna’s wage.

4 Calculate what Stephen will earn for working 8 hours at $11.88 per hour.

5 Debbie earns $489.06 for a 38-hour working week. Calculate Debbie’s hourly rate of pay.

6 Damien earns an annual salary of $47 000 and is paid weekly. Calculate Damien’s weekly pay.

7 Simone earns an annual salary of $70 000 and is paid fortnightly. Calculate Simone’s fortnightly pay.

8 Ivan earns an annual salary of $56 480 and is paid monthly. Calculate Ivan’s monthly pay.

9 Penny earns an annual salary of $44 000 and is paid weekly. Calculate Penny’s weekly pay.

10 Penny works an average of 35 hours each week. Calculate the hourly rate to which her salary is equivalent. (Answer to the nearest cent.)

Commission and royalties

Commission is a method of payment used mainly for salespeople. When paid com-mission, a person receives a percentage of the value of goods sold.

A royalty is a payment made to a person who owns a copyright. For example, a musician who writes a piece of music is paid royalties on sales of CDs; an author who writes a book is paid according to the number of books sold. Royalties are calculated in the same way as commission, being paid as a percentage of sales.

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In some cases, commission may operate on a sliding scale. This means that the com-mission rate changes with the value of sales. This type of comcom-mission is commonly used in real estate sales. In these examples, each portion of the commission is calcu-lated separately. The final commission is the sum of each portion.

Jack is a computer salesman who is paid a commission of 12% of all sales. Calculate the commission that Jack earns in a week if he makes sales to the value of $15 000.

THINK WRITE Calculate 12% of $15 000. Commission = 12% of $15 000 Commission = 12 ÷100 × $15 000 Commission= $1800

7 WORKED

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xample

A real estate agent is paid com-mission on his sales at the following rate:

• 5% on the first $75 000

• 2.5% on the balance of the sale

• price.

Calculate the commission earned on the sale of a property for $235 000.

THINK WRITE

Calculate 5% of $75 000. 5% of $75 000 = $3750 Calculate the balance of the sale. Balance = $235 000 − $75 000

Balance= $160 000

Calculate 2.5% of $160 000. 2.5% of $160 000 = $4000 Add up each portion to calculate the

commission. Commission = $3750 + $4000 Commission= $7750 1 2 3 4

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xample

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In some cases, people receive a fixed amount (called a retainer) as well as a com-mission. This is to ensure that the person earns some money even if no sales are made. To calculate this type of pay, you will need to add the retainer to the commission.

In some cases, the commission does not begin to be paid until sales have reached a certain point. Here the commission is calculated only on sales above this fixed amount.

Shelley is a furniture salesperson and is paid $250 per week plus a commission of 2% of all sales. Calculate Shelley’s pay in a week where her sales total $12 250.

THINK WRITE

Calculate the commission of 2% of $12 250.

Commission = 2% of $12 250

Commission= 2 ÷ 100 × 12 250 Commission= $245

Add the $250 to the commission to calculate her pay.

Pay = $250 + $245 Pay= $495 1 2

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xample

Tony is a car salesman. Tony is paid $300 per week and 2% of all sales over $50 000. Calculate Tony’s pay in a week where his sales total $84 000.

THINK WRITE

Calculate the amount on which commission is to be paid.

$84 000 − $50 000 = $34 000 Find 2% of this amount. Commission = 2% of $34 000

Commission = 2 ÷ 100 × $34 000 Commission= $680

Add the $300 to the commission to calculate Tony’s pay.

Pay = $300 + $680 Pay= $980 1 2 3

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xample

1. A commission is earned when a person is paid a percentage of the value of sales made.

2. Some commissions are paid on a sliding scale. In these cases, each portion of the commission is calculated separately and then totalled at the end.

3. Some commissions are paid together with a fixed payment called a retainer. To calculate an employee’s pay, the fixed payment needs to be added to the commission.

4. In some cases where a fixed payment is made, commission may not be paid on all sales, but rather on a section of sales above a certain point.

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Commission and royalties

1 Kylie is an insurance salesperson and she is paid 8% of the value of any insurance that

she sells. Calculate the amount that Kylie is paid for selling insurance to the value of $25 000.

2 Beryl sells exercise equipment and is paid a commission of 10% on all sales. Calculate Beryl’s earnings in a week where her sales total is:

a $2600 b $3270 c $5687.90.

3 Darren’s job is to sell CDs to music stores. If Darren sells CDs to the value of $40 000, calculate his commission if it is paid at a rate of:

a 1% b 3% c 3.4%.

4 Linda is a car salesperson who is paid 1.5% commission. Calculate the amount of money Linda earns in a week where her sales total $95 000.

5 Ken is an author and is paid a royalty on his book sales. The royalty is 12% of the value of all sales of his book. Calculate the value of Ken’s royalty if the value of sales totals $34 500.

6

Ursula is a computer software salesperson. Ursula’s sales total $105 000 and she is paid a commission of 0.8%. How much does Ursula receive in commission?

A $105 B $840 C $8400 D$84 000

7

Asif is a sales representative for a hardware firm. Asif earns $870 commission on sales of $17 400. What rate of commission does Asif receive?

A 0.05% B 0.5%

C 5% D20%

8 A real estate agent charges commission at the following rate: • 5% on the first $75 000

• 2.5% on the balance of the sale price.

Calculate the commission charged on the sale of a property valued at $250 000.

9 Gabrielle is a fashion sales representative. Gabrielle is paid a commission of 5% on the first $3000 of sales each week and 10% commission on the balance. Calculate Gabrielle’s commission in a week where her sales total $9500.

1C

WORKED Example 7 Skill SHEET

1.3

Converting a percentage into a decimal Skill SHEET

1.4

Finding a percentage of a quantity (money) EXCEL Spreads_heet Calculations with percentages multiple choice multiple choice WORKED Example 8

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10 Using the sliding scale for commission shown in question 8, calculate the commission on a property that sells for:

a $90 000 b $140 000 c $600 000.

11 Stanisa is a car salesman who is paid $250 per week plus a commission of 2% of any sales he makes. Calculate Stanisa’s pay in a week where his sales total $35 000.

12 Daniel works as a sales representative for a car accessories firm. Daniel is paid $150 per week plus 4% of any sales. Calculate Daniel’s earnings in a week where his sales total is:

a $6000 b $8500 c $12 475.

13

A group of sales representatives each have $10 000 in sales for a week. Who earns the most money?

A Averil, who is paid a commission of 8%

B Bernard, who is paid $250 plus 6% commission

C Cathy, who is paid $350 plus 4% commission

DDarrell, who is paid $540 plus 2.5% commission

14 Fred and Gina sell life insurance. Fred is paid a commission of 8% and Gina is paid $250 plus 5% commission.

a How much does Fred earn for a week in which his sales are $5000?

b How much does Gina earn for a week in which her sales total $5000?

c In another week Gina earns $650. What is the value of Gina’s sales?

d Fred wishes to earn $650 in a week. How much should his sales be?

15 Mario is a pay television salesman. Mario earns $500 per week plus 5% commission on all sales above $5000. Calculate Mario’s pay in a week where his sales total $7500.

16 Neville is a door-to-door encyclopedia salesman. He is paid $300 per week plus 3% commission on all sales greater than $5000. Calculate Neville’s pay in a week where his sales total is:

a $4000 b $6500 c $8560.

17

A firm employs five sales representatives. Which representative will earn the most in a week where each of their sales totals $12 480?

A Peter, who receives a commission of 4%

B Richard, who receives $100 plus a commission of 3%

C Susan, who is paid $280 plus a commission of 1.8%

DTrevor, who is paid $300 plus a commission of 3.5% on all sales over $6000

18 Andrew and Bonito are sales representatives. Andrew is paid $300 plus a commission of 2.5% on all sales. Bonito is paid $250 plus a 3.5% commission on all sales over $3000.

a Calculate Andrew’s commission in a week where his sales total $6500.

b Calculate Bonito’s commission in a week where his sales total $6500.

c Who will earn the most money in a week where both Andrew and Bonito make $16 000 in sales? WORKED Example 9 multiple choice WORKED Example 10 multiple choice Wor kSHEET

1.1

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Payment by piece

Payment by piece, or piecework refers to payment for the amount of work completed. It is commonly paid for jobs such as car detailing and letterbox delivery.

The amount earned is calculated by multiplying the rate of payment by the number of pieces of work completed.

In some cases, piecework is paid for multiples, rather than for single units. For example, for letterbox deliveries you may be paid per 1000 deliveries made.

There are also examples where you will be asked to compare payment by piece with other methods of earning income, in particular, wages.

Len has a job washing cars in a car yard. He is paid $2.25 per car washed. Calculate what Len earns in an afternoon where he washes 24 cars.

THINK WRITE

Multiply the pay rate by the number of cars detailed.

Pay = $2.25 × 24

Pay= $54.00

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xample

Holly is delivering brochures to letterboxes in her local area. She is paid $23.00 per thou-sand brochures delivered. Calculate what Holly will earn for a delivery of 3500 brochures.

THINK WRITE

Divide 3500 by 1000 to calculate the number of thousand brochures delivered.

3500 ÷ 1000 = 3.5

Multiply 3.5 by $23.00 to calculate what Holly is paid.

Holly’s pay = 3.5 × $23.00 Holly’s pay= $80.50 1 2

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xample

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Payment by piece

1 Julia works after school at a car yard detailing cars. If Julia is paid $10.85 per car, calculate what she will earn in an afternoon when she details 7 cars.

2 A group of four friends take a job picking fruit over summer. They are paid $4.50 per basket of fruit picked. Calculate the earnings of each person in the group if:

a Ryan picked 23 baskets b Summer picked 21 baskets

c Seth picked 19 baskets d Taylor picked 18 baskets.

3 Natalie advertises that she will do ironing for $12.50 per basket. Calculate Natalie’s earnings for doing 14 baskets of ironing.

4 Matthew charges $15 to mow a lawn. Calculate Matthew’s earnings in a week if he mows 9 lawns.

5 Dean works as a house cleaner. He charges $46.50 to clean a house. If Dean cleans 7 houses, calculate his earnings.

6 Barbara delivers pamphlets to local letterboxes. She is paid $21.80 per thousand pamphlets delivered. Calculate what Barbara will be paid for delivering 15 000 pamphlets.

Tristan has a job picking apples. He is paid $4.40 per basket.

a Calculate Tristan’s pay for picking 21 baskets of apples in one day.

b If it takes Tristan 8 hours to pick these apples, calculate the equivalent hourly rate of pay he has earned.

THINK WRITE

a Multiply 21 (the number of baskets) by $4.40 (the pay per basket).

a Pay = 21 × $4.40

Pay= $92.40

b Divide $92.40 (total pay) by 8 (number of hours worked).

b Hourly rate = $92.40 ÷ 8

Hourly rate= $11.55

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xample

1. Payment by piece is payment to an employee for the amount of work completed.

2. To calculate the amount to be paid, multiply the number of units of work completed by the amount to be paid per unit.

3. Be careful when pay is calculated for completing 100 or 1000 units of work. You will need to first divide by this amount.

4. Remember your work on other methods of payment. You will need it to compare payment by piece with them.

remember

1D

WORKED Example 11 WORKED Example 12

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7 A local business employs four people to deliver advertising to letterboxes. They are paid $18.40 per 1000 deliveries. Calculate the amount each person is paid.

a Jim makes 5000 deliveries. b Georgia makes 7500 deliveries.

c Nicholas makes 4750 deliveries. d Claire makes 6200 deliveries.

8 Raul works in a factory assembling toys. Raul is paid $19.25 per 100 toys assembled. Calculate what Raul is paid in a day where he assembles:

a 300 toys b 650 toys c 540 toys.

9 Carolina works as a bicycle courier. She charges $5.70 per kilometre for her deliveries. Calculate Carolina’s earnings for a 4 km delivery.

10 Keith is a taxi owner/driver. He is paid $3.00 plus $1.60 per kilometre. Calculate the amount Keith will earn for a journey of:

a 5 km b 15.5 km c 10.2 km.

11 Denise works as a fruit picker. She is paid $4.20 for every basket of fruit picked.

a Calculate the amount Denise will earn in a day during which she picks 32 baskets of fruit.

b If it takes Denise 8 hours to pick the fruit, calculate the equivalent hourly rate of pay.

12 Charlie works in a car yard as a detailer. Charlie is paid $11.60 per car.

a What will Charlie earn in an afternoon during which he details 15 cars?

b If it takes Charlie 8 hours to detail the cars, calculate his hourly rate of pay.

c If Charlie could finish in 6 hours, calculate the hourly rate of pay he would earn.

1 Kim works a 37-hour week at a rate of $12.32 per hour. Calculate her weekly wage.

2 Viet works 35 hours a week at an hourly rate of $9.89. Calculate Viet’s weekly wage.

3 Samantha receives an annual salary of $38 500 and is paid weekly. Calculate Samantha’s weekly pay.

4 Tom receives an annual salary of $86 000 and is paid fortnightly. Calculate Tom’s fortnightly pay.

5 Celine is paid $1246.40 per fortnight. Calculate her annual salary.

6 Mick is paid 7% commission on all sales he makes. Calculate his commission for a week in which his sales total $6960.

7 Christine is paid $250 per week plus 2.5% commission on all sales. Calculate Christine’s pay for a week in which her sales total $12 800.

8 Jason has a job picking fruit and is paid $4.85 per basket. Calculate Jason’s pay for a day in which he picks 43 baskets of fruit.

9 Julia has a job delivering pamphlets to letterboxes and is paid $13.40 per 1000 pamphlets delivered. Calculate Julia’s pay for delivering 4500 pamphlets.

10 Cameron is an author who receives a royalty of 8% of the value of sales of his book. Calculate Cameron’s royalty for book sales totalling $23 000.

WORKED Example 13

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Working overtime

Overtime is paid when a wage earner works more than the regular hours each week. When an employee works overtime a higher rate is paid. This higher rate of pay is called a penalty rate. The rate is normally calculated at either:

time and a half, which means that the person is paid 1 times the usual rate of pay, or

double time, which means that the person is paid twice the normal rate of pay. A person may also be paid these overtime rates for working at unfavourable times, such as at night or during weekends.

To calculate the hourly rate earned when working overtime we multiply the normal hourly rate by the overtime factor, which is 1 for time and a half and 2 for double time.

To calculate the pay for a period of time worked at time and a half or double time, we multiply the normal pay rate by the overtime factor (either 1 or 2) and then by the number of hours worked at that overtime rate.

When we calculate the total pay for a week that involves overtime, we need to calculate the normal pay and then add the amount earned for any overtime.

1 2 ---1 2

---Gustavo is paid $9.78 per hour in his job as a childcare worker. Calculate Gustavo’s hourly rate when he is being paid for overtime at time and a half.

THINK WRITE

Multiply $9.78 (the normal hourly rate) by 1 (the overtime factor for time and a half).

Time and a half rate = $9.78 × 1

Time and a half rate= $14.67

1 2 ---1 2

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xample

1 2

---Adrian works as a shop assistant and his normal rate of pay is $12.84 per hour. Calculate the amount Adrian earns for 6 hours work on Saturday, when he is paid time and a half.

THINK WRITE

Multiply $12.84 (the normal pay rate) by 1 (the overtime factor) and by 6 (hours worked at time and a half).

Pay = $12.84 × 1 × 6 Pay= $115.56 1 2 ---1 2

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Some examples will have more than one overtime rate to consider and some will require you to work out how many hours have been worked at each rate.

Natasha works as a waitress and is paid $11.80 per hour for a 38-hour week. Calculate Natasha’s pay in a week where she works 5 hours at time and a half in addition to her regular hours.

THINK WRITE

Calculate Natasha’s normal pay. Normal pay = $11.80 × 38 = $448.40 Calculate Natasha’s pay for 5 hours at

time and a half.

Time and a half = $11.80 × 1 × 5 = $88.50

Add the normal pay and the time and a half pay together.

Total pay = $448.40 + $88.50 = $536.90 1 2 1₂ ---3

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xample

Graeme is employed as a car assembly worker and is paid $10.40 per hour for a 36-hour week. If Graeme works overtime, the first 6 hours are paid at time and a half and the remainder at double time. Calculate Graeme’s pay in a week where he works 45 hours.

THINK WRITE

Calculate the number of hours overtime Graeme worked.

Overtime = 45 − 36

Overtime= 9 hours

Of these nine hours, calculate how much was at time and a half and how much was at double time.

Time and a half = 6 hours Double time = 3 hours

Calculate Graeme’s normal pay. Normal pay = $10.40 × 36

Normal pay= $374.40

Calculate what Graeme is paid for 6 hours at time and a half.

Time and a half = $10.40 × 1 × 6

Time and a half= $93.60

Calculate what Graeme is paid for 3 hours at double time.

Double time = $10.40 × 2 × 3

Double time= $62.40

Calculate Graeme’s total pay by adding the time and a half and double time payments to his normal pay. Total pay = $374.40 + $93.60 + $62.40 Total pay= $530.40 1 2 3 4 1 2 ---5 6

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Working overtime

1 Reece works in a restaurant and is paid a normal hourly rate of $11.30. Calculate the amount Reece earns each hour when he is being paid time and a half.

2 Carmen works as a waitress and is paid $11.42 per hour. Calculate Carmen’s rate per hour on a Sunday when she is paid double time.

3 Gareth works as a train driver and is normally paid $11.48 per hour. For working on public holidays he is paid double time and a half (overtime factor = 2 ). Calculate Gareth’s hourly rate of pay on a public holiday.

4 Ben works in a hotel and is paid $11.88 per hour. Calculate the total amount Ben will earn for an 8-hour shift on Saturday when he is paid at time and a half.

5 Taylor works as an usher at a concert venue. She is normally paid $13.10 per hour. Calculate Taylor’s pay for 6 hours on Sunday when she is paid double time.

6 Copy and complete the table below.

7

Ernie works as a chef and is paid $9.95 per hour. What will Ernie’s hourly rate be when he is paid time and a half for overtime?

A $11.45 B $14.92 C $14.93 D$19.90 Name Ordinary rate Overtime rate Hours worked Pay

A. Nguyen $8.90 Time and a half 4

M. Donnell $9.35 Double time 6

F. Milosevic $11.56 Time and a half 7 J. Carides $13.86 Time and a half 6.5

Y. Robinson $22.60 Double time 5.5

1. Overtime is paid when you work more than your normal working hours in a week, and you receive a higher rate of pay for the extra hours.

2. Overtime can be paid at:

(a) time and a half — 1 times the normal hourly rate (b) double time — twice the normal hourly rate.

3. To calculate the hourly rate when working overtime, multiply the normal hourly rate by the overtime factor.

4. To calculate the pay that is received for overtime, multiply the normal hourly rate by the overtime factor by the number of hours worked at that overtime rate. 5. To calculate the total pay for a week when overtime has been worked, calculate

the normal pay and the pay for each overtime rate separately, and add them.

1 2

---remember

1E

WORKED Example 14 Ski_llS HE ET

1.5

Multiplying a quantity (money) by a decimal Ski_llS HE_ET

1.6

Adding periods of time Ski_llS HE ET

1.7

Multiplying and dividing a quantity by a fraction 1 2 ---WORKED Example 15 multiple choice