Chapter 1 - Financial Mathematics PDF

Chapter Financial 1 mathematics What you will learn Strand: Number and Algebra 1A Review of percentages...

Chapter Financial 1 mathematics What you will learn Strand: Number and Algebra 1A Review of percentages Substrand: FINANCIAL MATHEMATICS 1B Applying percentages 1C Income Maths@work: So many ways to earn a living! In this chapter you will learn to: 1D The PAYG income tax system Keeping in touch with numeracy solve financial problems to do with purchases 1E Simple interest solve financial problems to do with earning, 1F Compound interest spending and investing money. 1G Investments and loans This chapter is mapped in detail to the NSW Syllabus 1H Using spreadsheets for investments, for the Australian Curriculum in the teacher resources loans and depreciation at: www.cambridge.edu.au/goldnsw10 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Additional resources Additional resources for this chapter can be downloaded from Cambridge GO: Saving and www.cambridge.edu.au/goldnsw10 borrowing money for drilling for Gold: Building knowledge and skills expensive items Skillsheets: Extra practise of important skills Spreadsheets: For many people, a car is the first expensive item Models for activities using spreadsheets that involves long-term saving and borrowing. There Worksheets: is also a need to budget for the ongoing costs of Consolidation of the topic petrol, insurance and repairs, as well as the hidden chapter Test: costs, such as depreciation and interest on the loan. Preparation for an examination 3 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  4 Chapter 1 Financial mathematics 1 Find the following totals. Pre-test a $15.92 + $27.50 + $56.20 b $134 + $457 + $1021 c $457 × 6 1 d $56.34 × 1 e $87 560 ÷ 52 (to the nearest cent) 2 2 Express the following fractions as percentages. 1 3 1 17 9 a b c d e 2 4 5 25 20 3 Round the following decimals to 2 decimal places. a 16.7893 b 7.347 c 45.3444 d 6.8389 e 102.8999 4 Copy and complete the following table. Gross income ($) Deductions ($) Net income ($) Net income = gross 4976 456.72 a income − deductions 92 411 b 62 839 c 18 472.10 79 431.36 5 Calculate the following annual incomes for each of these people. a Tom: $1256 per week b Viviana: $15 600 per month c Anthony: $1911 per fortnight d Crystal: $17.90 per hour, for 40 hours per week, for 50 weeks per year 6 Without a calculator, find: a 10% of $400 b 5% of $5000 c 2% of $100 d 25% of $844 e 20% of $12.80 f 75% of $1000 7 Find the simple interest earned on the following amounts. a $400 at 5% p.a. for 1 year b $5000 at 6% p.a. for 1 year Simple interest I = PRN c $800 at 4% p.a. for 2 years 8 Complete the following table. Cost price Deduction Sale price $34 $16 a $460 $137 b $500 c $236 d $45 $67 e $12.65 $45.27 9 The following amounts include the 10% GST. By dividing each one by 1.1, find the original costs before the GST was added. a $55 b $61.60 c $605 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 5 1A Review of percentages Stage 5.2 5.2◊ It is important that we are able to work 5.1 with percentages in our everyday lives. 4 Banks, retailers and governments use percentages every day to work out fees, prices and interest. Let’s start: Which option should Jamie choose? Jamie currently earns $38 460 p.a. (i.e. per year) and is given a choice of two different pay rises. Which should she choose and why? Choice A Choice B Increase of $20 a week Increase of 2% on p.a. salary Key ideas A percentage means ‘out of 100’. It can be written using the symbol % or as a fraction or a decimal. Percentage A convenient way of Drilling for Gold 75 3 For example: 75 per cent = 75% = or = 0.75 writing fractions with 1A1 1A2 100 4 denominators of 100 at the end of this Denominator The part section To convert a fraction or a decimal to a percentage, multiply by of a fraction that sits 100% below the dividing line 100% or. 1 To convert a percentage to a fraction, write it with × 100% a denominator of 100 and simplify. 15 3 Fraction % For example: 15% = = 100 20 ÷ 100% To convert a percentage to a decimal, divide by 100%. × 100% For example: 15% = 15 ÷ 100 = 0.15 Decimal % To find a percentage of a quantity, write the percentage as a fraction or a decimal, then multiply by the quantity. ÷ 100% 35 For example: 35% of $600 = × $600 100 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  6 Chapter 1 Financial mathematics Exercise 1A Understanding 1 Write the following with denominators of 100. 2 17 49 Make sure you a b c 5 20 50 have an equivalent 2 7 9 fraction: =. d e 5 100 25 10 2 Complete the following. 7 3 a 7% = b 0.9 = % c = % 5 3 Use mental strategies to find: a 10% of $7.50 b 20% of $400 c 50% of $98 Drilling 1 for Gold d 75% of $668 e 25% of $412 f 2% of $60 10% = 1A3 10 at the end 1 ‘Of’ means times. of this section g 5% of $750 h 33 % of $1200 i 30% of $15 3 Fluency Example 1 Converting to a percentage without a calculator Write each of the following as a percentage. 19 3 a b c 0.07 20 8 Solution Explanation ×5 Write using a denominator of 100. Alternatively, multiply the fraction by 100%. 19 95 a = 19 100 5 20 100 1 × % = 19 × 5% = 95% 20 1 ×5 19 Note: With a calculator, enter × 100. 20 25 3 3 100 Multiply the fraction by 100%. b = × % 8 82 1 Cancel common factors, then simplify. 75 = % 2 = 37.5% c 0.07 = 0.07 × 100% Multiply the decimal by 100%. = 7% 7 Alternatively, 0.07 = = 7%. 100 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 7 4 Convert each fraction to a percentage. Check your answers with a calculator. First, write using 1 1 1 a denominator of a b c 100 or, alternatively, 2 5 4 multiply by 100%. 1 1 7 d e f 10 100 25 15 3 5 g h i 50 4 8 19 99 47 j k l 25 100 50 5 Write these decimals as percentages. a 0.17 b 0.73 c 0.48 d 0.09 To multiply by e 0.06 f 0.13 g 1.13 h 1.01 100%, move the i 0.8 j 0.9 k 0.99 l 0.175 decimal point two places to the right. Example 2 Writing percentages as simple fractions without a calculator Write each of the following percentages as a simple fraction. 1 a 37% b 58% c 6 % 2 Solution Explanation 37 a 37% = Write the percentage with a denominator of 100. 100 58 b 58% = Write the percentage with a denominator of 100. 100 58 29 Simplify by cancelling, using the HCF of 58 and 100, = 100 50 which is 2. 29 58 29 = 100 50 50 1 1 6 c 6 %= 2 Write the percentage with a denominator of 100. 2 100  1 13 Double the numerator 6  and the denominator (100) so =  2 200 that the numerator is a whole number. 6 Write each percentage as a simple fraction. Use a calculator to check your answers. a 71% b 80% c 25% d 55% 1 e 40% f 88% g 15% h 16 % Write with a 2 denominator of 100, then simplify 1 1 1 1 if possible. i 17 % j 2 % k 5 % l 52 % 2 2 2 2 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  8 Chapter 1 Financial mathematics 1A Example 3 Writing a percentage as a decimal Convert these percentages to decimals. a 93% b 7% c 30% Solution Explanation a 93% = 0.93 Divide the percentage by 100. This is done by moving the decimal point two places to the left. 93 ÷ 100 = 0.93 b 7% = 0.07 Divide the percentage by 100. 7 ÷ 100 = 0.07 c 30% = 0.3 Divide the percentage by 100. 30 ÷ 100 = 0.30 Write 0.30 as 0.3. 7 Convert to decimals. a 61% b 83% c 75% d 45% e 9% f 90% g 50% h 16.5% i 7.3% j 200% k 430% l 0.5% Example 4 Finding a percentage of a quantity, with a calculator Find 42% of $1800. Solution Explanation 42% of $1800 Remember that ‘of ’ means multiply. = 0.42 × 1800 42 Write 42% as a decimal or a fraction: 42% = = 0.42 = $756 100 Then multiply by the amount. If using a calculator, enter 0.42 × 1800. 42 Without a calculator: × 1800 = 42 × 18 100 8 Use a calculator to find: a 10% of $250 b 50% of $300 c 75% of $80 d 12% of $750 e 9% of $240 f 43% of 800 grams 1 g 90% of $56 h 110% of $98 i 17 % of 2000 m 2 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 9 Problem-solving and Reasoning 9 A 300 g jar of spread contains 15 g of saturated fat. a What fraction of the spread is saturated fat? b What percentage of the spread is saturated fat? 15 g out of 300 g. 10 About 80% of the mass of a human body is water. If Hugo is 85 kg, how many kilograms of water are in his body? 11 Rema spends 12% of the 6.6 hour school day in Maths. How many minutes are spent in the Maths classroom? 12 In a cricket match, Brett spent 35 minutes bowling. 1 His team’s total fielding time was 3 hours. 2 What percentage of the fielding time, correct to 2 decimal places, did Brett spend bowling? First, convert hours to minutes, then write a fraction comparing times. 13 Malcolm lost 8 kg and now weighs 64 kg. What percentage of his original weight did he lose? 14 47.9% of a local council’s budget is spent on garbage collection. If a rate payer pays $107.50 per quarter in total rate charges, how much do they contribute in a year to garbage collection? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  10 Chapter 1 Financial mathematics 1A Enrichment: Australia’s statistics 15 Below is data on Australia’s population growth, as gathered by the Australian Bureau of Statistics for September 2012. Use this data, or download more recent data from the ABS website. Population at end Change over of September Change over previous year quarter 2012 previous year (%, 1 decimal (’000) (’000) place) New South Wales 7314.1 86.0 Victoria 5649.1 94.8 Queensland 4584.6 91.4 South Australia 1658.1 16.4 Western Australia 2451.4 81.7 Tasmania 512.2 0.5 Northern Territory 236.3 4.2 Australian Capital 376.5 7.4 Territory Australia 22 782.3 382.4 a Calculate the percentage change for each State and Territory shown using the previous year’s population, You will need to calculate the previous year’s population; and complete the table. e.g. for NSW, 7314.1 − 86.0. b What percentage of Australia’s overall population, correct to 1 decimal place, is living in: i NSW? ii Victoria? iii WA? c Use a spreadsheet to draw a sector graph (i.e. a pie chart) showing the populations of the eight States and Territories given in the table. What percentage of the total is represented by each State/Territory? d In part c, what is the angle size of the sector representing Victoria? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 11 1A1: Missing information Find the missing piece of information in each of the following. Drilling for Gold exercise Complete the worksheet or write the answers in your exercise book. Guess the missing information in the Use your calculator to find the missing following equations. Check your answers information in the following equations. with a calculator. Give answers correct to 1 decimal place where necessary. 1 10% of ______ = 12 13 17.5% of $250 = ________ 2 10% of ________= $15 14 20% of _______ = 29.4 cm 3 _______ of $70 = $7 15 ______ of $31 = $5.27 4 _______ of $35 = $7 16 17% of 440 = ______ 5 15% of 20 = ________ 17 22% of ______ = $136.40 6 _______ of 100 = 35 18 _______ of 180 mL = 99 mL 7 20% of _______ = 10 mL 19 12.5% of $180 = _______ 8 20% of _______ = 20 20 14% of _______ = 56 9 _______ of $120 = $30 21 6.5% of 2100 m = ______ 10 25% of 180 g = ________ 22 _______ of $600 = $48 11 15% of _______ = 9 cm 23 20.25% of _______ = 162 m 12 _______ of 80 m = 60 m 24 7.5% of _______ = $37.50 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  12 Chapter 1 Financial mathematics 1A2: Nine ways with percentages Use the worksheet or write your working and answers in your exercise book. Drilling for Gold exercise Question Non-calculator method Calculator method 1 Calculate 25% of 40. 2 Express 25 as a percentage of 40. 3 25% of a number is 40. What is the number? 4 Increase 40 by 25%. 5 Decrease 40 by 25%. 6 A number is increased by 25% to give 40. Not applicable. What is the number? 7 A number is decreased by 25% to give 40. Not applicable. What is the number? 8 A number is increased from 25 to 40. What is the percentage increase? 9 A number is decreased from 40 to 25. What is the percentage decrease? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 13 1A3: Percentage of a quantity (mentally) Drilling for Gold exercise Use the worksheet, or write the answers in your exercise book. 400 800 600 240 120 20 90 50 440 5000 Choose a number from the above table Imagine the number is here. This is 100%. 1 What is 50% of the number? 6 What is 5% of the number? 2 What is 75% of the number? 7 What is 2.5% of the number? 3 What is 25% of the number? 8 What is 15% of the number? 4 What is 12.5% of the number? 9 What is 70% of the number? 5 What is 10% of the number? 10 What is 90% of the number? After you have finished with each number, check your answers using a calculator. 11 What is 1% of the number? 12 What is 0.5% of the number? 13 What is 3% of the number? 14 What is 3.5% of the number? 15 What is 2% of the number? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. 14 Chapter 1 Financial mathematics 1B Applying percentages Stage 5.2 5.2◊ There are many applications of 5.1 percentages. Prices are often 4 increased by a percentage to create a profit or decreased by a percentage when on sale. Let’s start: Discounts Discuss as a class: Which is better: 20% off or a $20 discount? If a discount of $20 or 20% off resulted in the same price, what would be the original price? Why are percentages used to show discounts rather than a fixed amount? Key ideas To increase an amount by a given percentage: – Add the percentage increase to 100%. – Multiply the amount by this new percentage. For example: to increase by 25%, multiply by 100% + 25% = 125% = 1.25. × 1.25 ease by 25% to incr Original price Increased price (100%) (125%) When the Goods and Services Tax (GST) was introduced, prices were increased by 10%. × 1.1 Goods and Services Tax (GST) In Australia, Price Price a 10% government (excluding GST) (including GST) levy included in the 100% 110% purchase price of merchandise. ÷ 1.1 For example: × 1.1 pre-GST post-GST $120 $132 ÷ 1.1 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 15 To decrease an amount by a given percentage: – Subtract the percentage from 100%. – Multiply the amount by this new percentage. For example: to decrease by 25%, multiply by 100% − 25% = 75% = 0.75. × 0.75 crease by 25% to de Original price Decreased price (100%) (75%) To find a percentage change, use: change in price Percentage change = × 100% original price Profit The amount of money made by selling When goods are purchased by a store, the cost to the owner is something for more than its cost called the cost price. Loss The amount of The price on display in the shop is the selling price. money lost by selling something for less than Profit is the amount of money made on a sale. its cost Discount An amount Profit = selling price - cost price subtracted from a price A loss is made when the selling price is less than the cost price. Loss = cost price - selling price Mark-up is the amount added to the cost price to produce the selling price. Selling price = cost price + mark-up The percentage profit or percentage loss can be found by dividing the profit or loss by the cost price and multiplying by 100%: profit / loss %Profit / Loss = × 100% cost price Discount is the amount by which an item is marked down. Discount = % discount × original price New price = original price - discount ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  16 Chapter 1 Financial mathematics Exercise 1B Understanding 1 By what percentage do you multiply to increase an amount by: a 10%? b 20%? c 50%? d 2%? e 18%? Increase 100% + percentage 2 By what percentage do you multiply to decrease an amount by: Decrease a 5%? b 30%? c 15%? d 50%? e 17%? 100% - percentage 3 Decide how much profit or loss is made in each of the following situations. a cost price = $15 selling price = $20 b cost price = $17.50 selling price = $20 c cost price = $250 selling price = $234 d cost price = $147 selling price = $158 e cost price = $3.40 selling price = $1.20 4 Copy and complete the following, assuming the GST is 10%. pre-GST × post-GST a $150 $ pre-GST post-GST b $ $275 Fluency Example 5 Increasing by a given percentage, with a calculator Drilling for Gold 1B1 Increase $370 by 8%. 1B2 at the end of this Solution Explanation section $370 × 1.08 = $399.60 100% + 8% = 108% Write 108% as a decimal (or fraction) and multiply by the amount. Remember that money has 2 decimal places. 5 a Increase $90 by 5%. b Increase $400 by 10%. c Increase $55 by 20%. d Increase $490 by 8%. To increase by e Increase $50 by 12%. f Increase $7000 by 3%. 5%, multiply by g Increase $49.50 by 14%. h Increase $1.50 by 140%. 100% + 5% = 1.05. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 17 Example 6 Decreasing by a given percentage, with a calculator Decrease $8900 by 7%. Solution Explanation $8900 × 0.93 = $8277.00 100% - 7% = 93% Write 93% as a decimal (or fraction) and multiply by the amount. Remember to put the units in your answer. 6 a Decrease $1500 by 5%. b Decrease $400 by 10%. c Decrease $470 by 20%. d Decrease $80 by 15%. To decrease by e Decrease $550 by 25%. f Decrease $49.50 by 5%. 5%, multiply by g Decrease $119.50 by 15%. h Decrease $47.10 by 24%. 100% - 5% = 0.95. Example 7 Calculating profits and percentage profit The cost price for a new car is $24 780 and it is sold for $27 600. a Calculate the profit. b Calculate the percentage profit, to 2 decimal places. Solution Explanation a Profit = selling price – cost price Write the rule. = $27 600 - $24 780 Substitute the values and evaluate. = $2820 profit Write the rule. b Percentage profit = × 100% cost price 2820 = × 100% Substitute the values and calculate. 244 780 = 11.38% Round your answer as instructed. 7 Copy and complete the table on profits and percentage profit. Cost Selling Percentage profit Percentage profit = × 100% price price Profit profit cost price a $10 $16 b $240 $300 c $15 $18 d $250 $257.50 e $3100 $5425 f $5.50 $6.49 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  18 Chapter 1 Financial mathematics 1B Example 8 Finding the selling price A retailer buys some material for $43.60 per roll. She wishes to make a 35% profit. a What will be the selling price per roll? b If she sells 13 rolls, what profit will she make? Solution Explanation a Selling price = 135% of $43.60 For a 35% profit the unit price is 135%. = 1.35 × $43.60 Write 135% as a decimal (1.35) and = $58.86 per roll evaluate. b Profit per roll = $58.86 - $43.60 = $15.26 Profit = selling price - cost price Total profit = $15.26 × 13 There are 13 rolls at $15.26 profit = $198.38 per roll. 8 A retailer buys some snow globes for $41.80 each. He wishes to make a 25% profit. a What will be the selling price per snow globe? b If he sells a box of 25 snow globes, what profit will he make? 9 Ski jackets are delivered to a shop in packs of 50 for $3500. If the shop owner wishes to make a 35% profit: a What will be the total profit made on a pack? b What is the profit on each jacket? 10 A second-hand car dealer buys a trade-in car for $1200 and wishes to resell it for a 28% profit. What will be the resale price? Example 9 Finding the discounted price A shirt worth $25 is discounted by 15%. a What is the selling price? b How much is the saving? Solution Explanation a Selling price = 85% of $25 15% discount means there must be 85% = 0.85 × 25 left (100% - 15%). = $21.25 Convert 85% to 0.85 and multiply by the amount. b Saving = 15% of $25 You save 15% of the original price. = 0.15 × 25 Convert 15% to 0.15 and multiply by the original = $3.75 price. or saving = $25 - $21.25 Saving = original price - discounted price = $3.75 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 19 11 Samantha buys a wetsuit from the sports store where she works. Its original price was $79.95. If employees receive a 15% discount: a What is the selling price? b How much will Samantha save? 12 A travel agent offers a 12.5% discount on airfares if you travel during May or June. If the normal return fare to London is $2446: a What is the selling price? b How much is the saving? 13 A store sells second-hand goods at 40% off the recommended retail price. For a lawn mower valued at $369: a What is the selling price? b How much do you save? Problem-solving and Reasoning 14 A pair of sports shoes is discounted by 47%. If the recommended price is $179: a What is the amount of the discount? b What will be the discounted price? 15 Jeans are priced during a sale for $89. If this is a saving of 15% off the selling price, what do the jeans normally 85% is $89 ÷ 85 ÷ 85 sell for? 1% is × 100 × 100 Drilling 100% is. for Gold 16 Discounted tyres are reduced in price by 35%. 1B3 They now sell for $69 each. Determine: a the normal price of one tyre b the saving when you buy one tyre 17 A price tag says ‘$308, inc. GST’. a What is the ‘pre-GST’ price? b How much GST is included in the price? 18 The local shop purchases a carton of containers for $54. Each container is sold for $4. If the carton has 30 containers, determine: a the profit per container b the percentage profit per container, to 2 decimal places c the overall profit per carton d the overall percentage profit, to 2 decimal places ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  20 Chapter 1 Financial mathematics 1B 19 A retailer buys a book for $50 and wants to sell it for a 26% profit. The 10% GST must then be added to the cost of the book. a Calculate the profit on the book. b How much GST is added to the cost of the book? % Increase = increase × 100% c What is the advertised price of the book, including cost price the GST? d Find the overall percentage increase of the final selling price compared to the $50 cost price. Enrichment: Building a gazebo 20 Christopher designs a gazebo for a new house. He buys the timber from a retailer, who sources it at a wholesale price and then adds a mark-up before selling to Christopher at the retail price. The table below shows the wholesale prices, as well as the mark-up for each type of timber. a Determine Christopher’s overall cost for the material, including the mark-up. b Determine the profit the retailer made. c Determine the retailer’s overall percentage profit, to 2 decimal places. d If the retailer pays 27% of their profits in tax, how much tax do they pay on this sale? Quantity Description Cost/unit Mark-up 6 treated pine posts $23 20% 11 300 × 50 oregon beams $75 10% 5 sheet lattice work $86 15% 2 300 × 25 oregon fascias $46 12% 8 laserlite sheets $32 10% ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 21 1B2: Repeated percentage change (with a calculator) Drilling for Gold exercise Calculator short-cut for increasing by a percentage Example: Increase $120 by 5%. The original amount was 100%, so the new amount will be 100% + 5% = 105%. To convert to a decimal, 105 divided by 100 gives 1.05. Calculation: $120 × 1.05 = $126 Calculator short-cut for decreasing by a percentage Example: Decrease $120 by 5%. The original amount was 100%, so the new amount will be 100% − 5% = 95%. To convert to a decimal, 95 divided by 100 gives 0.95. Calculation: $120 × 0.95 = $114 Complete the following questions on the worksheet or in your exercise book. The first one has been done for you as an example. 1 Increase $120 by 5% and then increase the result by 5%. $120 × 1.05 × 1.05 = $132.30 2 Decrease $120 by 5% and then decrease the result by 5%. 3 Increase $120 by 5% and then decrease the result by 5%. 4 Decrease $120 by 5% and then increase the result by 5%. 5 Increase $150 by 8% and then increase the result by 8%. 6 Decrease $150 by 8% and then decrease the result by 8%. 7 Increase $150 by 8% and then decrease the result by 8%. 8 Decrease $150 by 8% and then increase the result by 8%. Calculator short-cut for repeated increase The population of a town is expected to increase by 4% every year. The current population is 50 000. Try this on your calculator: Enter 50 000, then enter =. Now enter × 1.04. Now enter = = = = = repeatedly and watch the population growing. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  22 Chapter 1 Financial mathematics Calculator short-cut for repeated decrease The population of a town is expected to decrease by 4% every year. The current population is 50 000. Drilling for Gold exercise Try this on your calculator: Enter 50 000, then enter =. Now enter × 0.96. Now enter = = = = = repeatedly and watch the population falling. In Questions 9 to 12, start with a population of 50 000. 9 If the population increases by 4% every year, what will it be at the end of the tenth year? 10 If the population decreases by 4% every year, what will it be at the end of the tenth year? 11 If the population increases by 4% every year, how many years will it take for it to double? 12 If the population decreases by 4% every year, how many years will it take for it to be halved? 13 A car is purchased for $40 000 and loses 15% in value every year. How much will it be worth by the end of the fifth year? 14 If I invest $1000 now and it increases by 4% every year, how much will it be worth by the end of the 40th year? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 23 1C Income Stage 5.2 5.2◊ You may have earned money for babysitting 5.1 or delivering newspapers, or have a part-time 4 job. As you move more into the workforce, it is important that you know how to check that you are being paid the correct amount. Let’s start: Casual teaching versus full-time teaching When one of your normal teachers is absent from school, your school might employ a casual teacher to take his or her classes for the day. Teacher A, a casual, was paid $340 for every school day they worked during 2015. Teacher B, a full-time teacher, was paid about $65 000 during 2015. If Teacher A had worked every school day of 2015, how much would they have earned? Is this more or less than Teacher B? If Teacher A had worked only two days every school week of 2015, how much would they have earned? What are some reasons for being a casual teacher compared with a full-time teacher? Research the meaning of the following advantages of being a full-time teacher compared with a casual teacher. Holiday pay Sick pay Salary scale Long service leave Annual leave loading Salary packaging Job security International teacher exchange Key ideas Methods of payment Hourly wages: You are paid a certain amount per hour worked. Wages Earnings paid Commission: You are paid a percentage of the total amount of sales. to an employee based on an hourly rate Some people who work for commission are also paid a set weekly amount, called a retainer. Commission Earnings of a salesperson based Piecework: You are paid according to number of things you make on a percentage of or do. the value of goods or services sold Salary: You are paid a set amount per year. Salary An employee’s fixed agreed yearly Fees: You are paid according to the charges you set; e.g. doctors, income lawyers, contractors. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. 24 Chapter 1 Financial mathematics Terminology Gross income: The total amount of money you earn before taxes and other deductions. Deductions: Money taken from your income before you are paid; e.g. taxation, union fees, superannuation. Net income: The amount of money you actually receive after the deductions are taken from your gross income. Gross income Total income Net income = gross income - deductions before any deductions (e.g. income tax) are made Payments by hourly rate Deductions If you are paid by the hour you will be paid an amount per hour for your Amounts of normal working time. Usually, normal working time is 38 hours per week. money taken If you work overtime the rates may be different. from gross –– Normal: 1.0 × normal rate income –– Time and a half: 1.5 × normal rate Net income –– Double time: 2.0 × normal rate Income remaining after deductions Some people are paid penalty rates; e.g. time and a half for working on have been made from gross Saturdays or double time for working on Sundays. income Penalty rates A higher hourly Leave loading rate for working Some wage and salary earners are paid leave loading. When they are on unsocial hours holidays, they earn their normal pay plus a bonus called leave loading. This is usually 17.5% of their normal pay. 100% + 17.5% = 117.5% = 1.175 × 1.175 A B Normal Holiday pay pay $1500 $1762.50 ÷ 1.175 Leave loading = 17.5% of A = $262.50 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 25 Exercise 1C Understanding 1 If Tao earns $570 for 38 hours’ work, calculate his: a hourly rate of pay ‘Annual’ means b time-and-a-half rate yearly. c double-time rate d annual income, given that he works 52 weeks a year, 38 hours a week 2 Which is better: $5600 a month or $67 000 a year? 1 year = 12 months 3 Callum earns $1090 a week and has annual deductions of $19 838. What is Callum’s net income for the year? Net = total - deductions 1 year = 52 weeks Fluency Example 10 Finding gross and net income (including overtime) Pauline is paid $13.20 per hour at the local stockyard, where she normally works 38 hours per week. She receives time and a half for the next 4 hours worked and double time after that. a What will be Pauline’s gross income if she works 50 hours? b If Pauline pays $220 in tax and $4.75 in union fees, what will be her net income? Solution Explanation a Gross income = 38 × $13.20 First 38 hours is paid at normal rate. + 4 × 1.5 × $13.20 Overtime rate for next 4 hours: time and a half = 1.5 × normal rate + 8 × 2 × $13.20 Overtime rate for next 8 hours: double = $792 time = 2 × normal rate b Net income = $792 - ($220 + $4.75) Net income = gross income - deductions = $567.25 4 Copy and complete this table. Normal Time and Double Hourly hours a half time Gross Net rate worked hours hours income Deductions income a $15 38 0 0 $155 b $24 38 2 0 $220 c $13.15 38 4 1 $300 d $70 40 2 3 $510 e $17.55 35 4 6 $184 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  26 Chapter 1 Financial mathematics 1C Example 11 Calculating shift work Michael is a shift worker and is paid $21.20 per hour for the morning shift, $24.68 per hour for the afternoon shift and $33.56 per hour for the night shift. Each shift is 8 hours. In a given fortnight he works four morning, two afternoon and three night shifts. Calculate his gross income. Solution Explanation Gross income = 4 × $21.20 × 8 4 morning shifts at $21.20 per hour for 8 hours + 2 × $24.68 × 8 2 afternoon shifts at $24.68 per hour + 3 × $33.56 × 8 3 night shifts at $33.56 per hour = $1878.72 Gross income, as tax has not been paid. 5 Greg works shifts at a processing plant. In a given rostered fortnight he works: A fortnight three day shifts ($21.20 per hour) = 2 weeks four afternoon shifts ($24.68 per hour) four night shifts ($33.56 per hour) a If each shift is 8 hours long, determine Greg’s gross income for the fortnight. b If the answer to part a is Greg’s average fortnightly income, what will be his gross income for a year (i.e. 52 weeks)? c If Greg is to be paid monthly, what will be his gross income for a month? Example 12 Calculating income involving commission Erika sells memberships to a gym and receives $225 per week plus 5.5% commission on her sales. Calculate her gross income after a 5-day week. Day 1 2 3 4 5 Sales ($) 680 450 925 1200 1375 Solution Explanation Total sales = $4630 Determine the total sales by adding the daily sales. Commission = 5.5% of $4630 Determine the commission on the total sales at = 0.055 × 4630 5.5% by multiplying 0.055 by the total sales. = $254.65 Gross income = $225 + $254.65 Gross income is $225 plus commission. = $479.65 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 27 6 A real estate agent receives 2.75% commission on the sale of a house valued at $375 000. Find the commission earned. Divide by 100 to convert 2.75% to a decimal. 7 A car salesperson earns $500 a month plus 3.5% commission on all sales. In the month of January their sales total $56 000. Calculate: a their commission for January b their gross income for January 8 Portia earns an annual retainer of $27 000 plus 2% commission on all sales. Find: a her weekly base salary before sales b her commission for a week when her sales totalled $7500 c her gross weekly income for the week mentioned in part b. d her annual gross income if over the year her sales totalled $571 250 Example 13 Calculating holiday pay Rachel is normally paid $1200 per week. When she is on holidays she is paid 17.5% p.a. leave loading. a How much is Rachel’s holiday pay for 1 week? b How much is Rachel’s leave loading for 1 week? Solution Explanation a Holiday pay = $1200 × 1.175 × 1.175 Normal Holiday = $1410 pay pay b Leave loading = $1410 − $1200 Loading is 17.5% of normal pay. = $210 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  28 Chapter 1 Financial mathematics 1C 9 Ashton is normally paid $900 per week. When he is on holidays he is paid leave loading. Leave loading is a Calculate Ashton’s holiday pay for 1 week. an extra 17.5% of b Calculate Ashton’s leave loading for 1 week. normal pay. 10 Mary earns $800 per week. Calculate her holiday pay for 4 weeks, including leave loading. Problem-solving and Reasoning 11 If Simone receives $14 370 on the sale of a property worth $479 000, calculate her rate of commission. What percentage of $479 000 is $14 370? 12 Jonah earns a commission on his sales of fashion items. For goods to the value of $2000 he receives 6% and for sales over $2000 he receives 9% on the amount in excess of $2000. In a given week he sold $4730 worth of goods. Find the commission earned. 13 Mel is taking her holidays. She receives $2937.50. This includes $2937.50 is 117.5% her normal pay and her leave loading. How much is the of Mel’s normal pay. leave loading? 14 Toby earns 1.75% commission on all sales at the electrical goods store where he works. If Toby earns $35 in commission on the 1.75% is $35. Find sale of one television, how much did the TV sell for? 1%, then 100%. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 29 Enrichment: Elmo’s payslip 15 Refer to the payslip below to answer the following questions. During 2015, Elmo received 26 of these payslips. Kuger Incorporated Employee ID: 75403A Page: 1 Name: Elmo Clowner Pay Period: 21/05/2015 Pay method: EFT Tax status: Gen Exempt Bank account name: E. Clowner Bank: Mathsville Credit Union BSB: 102-196   Account No: 00754031 Payment details this pay: Amount Days Payment description Rate/Frequency 2777.16 14.00 Normal time $72 454/annum Before tax deductions: This pay Description 170 Salary sacrifice: car pre-tax deduction Miscellaneous deductions: This pay Description 52.90 Health fund 23.10 Union fees 76.00 Reconciliation details: This pay YTD Description 2607.15 62 571.60 Taxable gross pay 616.00 14 784.00 Less income tax 76.00 1 824.00 Less miscellaneous deductions 1915.15 45 693.60 a For what company does Elmo work? b What is the name of Elmo’s bank and what is his account number? c How much gross pay does Elmo earn in 1 year? d How often does Elmo get paid? e How much, per year, does Elmo salary sacrifice? f How much each week is Elmo’s health fund contributions? g Calculate the union fees for 1 year. h Using the information on this payslip, calculate Elmo’s annual tax and also his annual net income. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  30 Chapter 1 Financial mathematics So many ways to earn a living! There are hundreds of different occupations and careers you could pursue in your working Maths@work life. Some of these options may not exist at the moment but will in the future. When it comes to ‘pay day’, different careers are paid in a variety of ways, such as: Casual pay rates, with the possibility of penalty rates Wages, with the possibility of overtime Salaries Commission, either with or without a retainer Piecework Consignment Royalties Allowances Bonuses Worksheet 1A Here is an example to get you started. For many young people, their first job is working in a fast-food outlet or a shop. There may be some students in your classroom doing this or maybe someone you know. Ask them the following questions, then ask your teacher the same questions. 1 Do you get paid if you are too sick to work? 2 Can you take holidays when it suits you? 3 Do you get paid when you are on holidays? 4 Do you always work the same number of hours every week at the same time and get paid the same amount? 5 Is there opportunity for pay rises and promotions within the company? 6 Do you ever receive a bonus on top of your normal pay? 7 Do you get paid more if you sell more? 8 Do you get paid extra for working on weekends and public holidays, such as time and a half or double time? 9 Do you get long-service leave or holiday leave loading? 10 Does your employer pay for your continuing education, training, courses, conferences or pay you while you study? Continue with the activity by downloading the worksheet to help you understand the different ways of being paid, and the advantages and disadvantages of each. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 31 1D The PAYG income tax system Stage 5.2 5.2◊ It has been said that there are only two 5.1 sure things in life: death and taxes! The 4 Australian Taxation Office (ATO) collects taxes on behalf of the government to pay for education, hospitals, roads, railways, airports and services, such as the police force and fire brigades. In Australia, the financial year runs from July 1 to June 30 the following year. People engaged in paid employment are normally paid weekly or fortnightly. Most of them pay some income tax every time they are paid for their work. This is known as the Pay-As-You-Go system (PAYG). At the end of the financial year (June 30), people who earned an income complete an income tax return to determine if they have paid the correct amount of income tax during the year. If they paid too much they will receive a refund. If they did not pay enough, they will be required to pay more. The Australian tax system is very complex and the laws change frequently. This section covers the main aspects only. Let’s start: The ATO website The Australian Taxation Office website has some income tax calculators. Use one to find out how much income tax you would need to pay if your taxable income is: $5200 per annum (i.e. $100 per week) $10 400 per annum (i.e. $200 per week) $15 600 per annum (i.e. $300 per week) $20 800 per annum (i.e. $400 per week) $26 000 per annum (i.e. $500 per week) Does a person earning $1000 per week pay twice as much tax as a person earning $500 per week? Does a person earning $2000 per week pay twice as much tax as a person earning $1000 per week? ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. 32 Chapter 1 Financial mathematics Key ideas U You (the employee and tax payer) B T (The Australian The boss Taxation (your employer) Office or ATO) The PAYG tax system works in the following way. – U works for and gets paid by B every week, fortnight or month. – B calculates the tax that U should pay for the amount earned by U. – B sends that tax to T every time U gets paid. – T passes the income tax to the federal government. – On June 30, B gives U a payment summary to confirm the amount of tax that has been paid to T on behalf of U. – Between July 1 and October 31, U completes a tax return and sends it to T. Some people pay a registered tax agent to do this for them. – On this tax return, U lists the following. All forms of income, including interest from investments. Legitimate deductions shown on receipts and invoices, such as work-related expenses and donations. – Taxable income is calculated using the formula: Taxable income = gross income – deductions – There are tables and calculators on the ATO website, such as the following. Each row in the table is called a tax bracket. Taxable income Tax on this income 0 – $18 200 Nil $18 201 – $37 000 19c for each $1 over $18 200 $37 001 – $80 000 $3572 plus 32.5c for each $1 over $37 000 $80 001 – $180 000 $17 547 plus 37c for each $1 over $80 000 $180 001 and over $54 547 plus 45c for each $1 over $180 000 This table can be used to calculate the amount of tax U should have paid (i.e. the tax payable), as opposed to the tax U did pay during the year (i.e. the tax withheld). – U may also need to pay the Medicare levy. This is a scheme in which all Australian taxpayers share in the cost of running the medical system. For many people this is currently 1.5% of their taxable income. – It is possible that U may have paid too much tax during the year and will receive a tax refund. – It is also possible that U may have paid too little tax and will receive a letter from T asking for the tax liability to be paid. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 33 Exercise 1D Understanding Note: The questions in this exercise relate to the information given in Key ideas, unless stated otherwise. 1 Complete this statement: Taxable income = ________ income minus ___________. 2 Is the following statement true or false? The highest income earners in Australia pay 45 cents tax for every dollar they earn. 3 Joseph paid no income tax. What could his taxable income have been? 4 Ann’s taxable income was $80 000, which puts her at the very top of the middle tax bracket in the tax table. Ben’s taxable income was $80 001, which puts him in a higher tax bracket. Ignoring the Medicare levy, how much extra tax did Ben pay compared to Ann? Fluency 5 Use an online tax calculator on the ATO website to calculate the income tax payable on these taxable incomes. a $30 000 b $60 000 c $150 000 d $200 000 6 Consider the amount of tax payable for these six people. Taxable income $0 $18 200 $37 000 $80 000 $180 000 $200 000 Tax payable $0 $0 $3572 $17 457 $54 547 $63 547 Make a copy of this set of axes, plot the points and then join the dots with straight-line segments. 60 000 50000 40000 Tax payable ($) 30000 20000 10000 0 50000 100000 150000 200000 Taxable income ($) ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  34 Chapter 1 Financial mathematics 1D 7 Jim worked for three different employers. They each paid him $15 000. Based on your graph in the previous question, how much income tax should Jim have paid? Example 14 Calculating income tax payable Richard earned $1050 per week ($54 600 dollars per annum) from his employer and other sources, such as interest on investments. He has receipts for $375 for work- related expenses and donations. a Calculate Richard’s taxable income. b Use this tax table to calculate Richard’s tax payable amount. Taxable income Tax on this income 0 – $18 200 Nil $18 201 – $37 000 19c for each $1 over $18,200 $37 001 – $80 000 $3572 plus 32.5c for each $1 over $37 000 $80 001 – $180 000 $17 547 plus 37c for each $1 over $80 000 $180 001 and over $54 547 plus 45c for each $1 over $180 000 c Richard must also pay the Medicare levy of 1.5% of his taxable income. How much is the Medicare levy? d Add the tax payable and the Medicare levy amounts. e Express the total tax in part d as a percentage of Richard’s taxable income, to 1 decimal place. f During the financial year, Richard’s employer sent a total of $7797 in tax to the ATO. Has Richard paid too much tax or not enough? Calculate his refund or liability. Solution Explanation a Gross income = $54 600 Taxable income = gross income - Deductions = $375 deductions Taxable income = $54 225 b Tax payable: Richard is in the middle tax bracket in the $3572 + 0.325 × ($54 225 - $37 000) table, in which it says: = $9170.13 $3572 plus 32.5c for each $1 over $37 000 Note: 32.5 cents is $0.325. 1.5 c × 54225 = $813.38 Medicare levy is 1.5% of the taxable income. 100 Round your answer to the nearest cent. d $9170.13 + $813.38 = $9983.51 This is the total amount of tax that Richard should have paid. 9983.51 This implies that Richard paid approximately e × 100% = 18.4% 18.4% tax on every dollar. This is sometimes 54 225 read as ‘18.4 cents in the dollar’. (to 1 decimal place) ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 35 f Richard paid $7797 in tax during the This is known as a shortfall or a liability. year. Richard will receive a letter from the ATO He should have paid $9983.51. requesting payment of the difference. Richard has not paid enough tax. $9983.51 - $7797 = $2186.51 He must pay another $2186.51 in tax. 8 Lee has come to the end of her first financial year employed as a website developer. On June 30 she made the following notes about the financial year. Taxable income Gross income from employer $58 725 = all incomes - deductions Gross income from casual job $7500 Interest on investments $75 Donations $250 Work-related expenses $425 Tax paid during the financial year $13 070 a Calculate Lee’s taxable income. b Use the tax table shown in Example 14 to calculate Lee’s tax payable amount. c Lee must also pay the Medicare levy of 1.5% of her taxable income. How much is the Medicare levy? d Add the tax payable and the Medicare levy. e Express the total tax in part d as a percentage of Lee’s taxable income, to 1 decimal place. f Has Lee paid too much tax or not enough? Calculate her refund or liability. Problem-solving and Reasoning 9 Alec’s Medicare levy is $1312.50. This is 1.5% of his taxable income. What is his taxable income? 10 Tara is saving for an overseas trip. Her taxable income is usually about $20 000. She estimates that she will need $5000 for the trip, so she is going to do some extra work to raise the money. How much extra will Tara need to earn in order to save the extra $5000 after tax? 11 When Saled used the tax table to calculate his income tax payable, it turned out to be $23 097. What is his Use the tax table given in Example 14 to determine in taxable income? which tax bracket Saled falls. ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  36 Chapter 1 Financial mathematics 1D 12 Explain the difference between gross income and taxable income. 13 Explain the difference between a tax refund and a tax liability. 14 Gordana looked at the last row of the tax table and said, “It is so unfair that people in that tax bracket must pay 45 cents in every dollar in tax.” Explain why Gordana is incorrect. 15 Consider the tax tables for the two consecutive financial years. Note that the amounts listed first in each table is often called the tax-free threshold (i.e. the amount that a person can earn before they must pay tax). 2011/2012 Taxable income Tax on this income 0 – $6000 Nil $6001 – $37 000 15c for each $1 over $6000 $37 001 – $80 000 $4650 plus 30c for each $1 over $37 000 $80 001 – $180 000 $17 550 plus 37c for each $1 over $80 000 $180 001 and over $54 550 plus 45c for each $1 over $180 000 2012/2013 Taxable income Tax on this income 0 – $18 200 Nil $18 201 – $37 000 19c for each $1 over $18 200 $37 001 – $80 000 $3572 plus 32.5c for each $1 over $37 000 $80 001 – $180 000 $17 547 plus 37c for each $1 over $80 000 $180 001 and over $54 547 plus 45c for each $1 over $180 000 a There are some significant changes between the financial years 2011/2012 and 2012/2013. Describe three of them. b The following people had the same taxable income during both financial years. Find the difference and state whether they were advantaged or disadvantaged by the changes, or not affected at all? i Ali: Taxable income = $5000 ii Xi: Taxable income = $15 000 iii Charlotte: Taxable income = $30 000 iv Diego: Taxable income = $50 000 ISBN 978-1-316-62312-1 © Palmer et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.  Number and Algebra 37 16 Below is the 2012/2103 tax table for people who are not residents of Australia but are working in Australia. Taxable income Tax on this income 0 – $80 000 32.5c for each $1 $80 001 –