mathematics How would you define Spending wisely? How can you recognise a person who spends wisely?

Transcription

1 11 11A Spending wisely 11B Discount, profit and loss 11C Simple interest What do you know? Financial mathematics 1 List what you know about money? Create a concept map to show your list. 2 Share what you know with a partner and then with a small group. 3 As a class, create a large concept map that shows your class s knowledge of money. Digital doc Hungry brain activity Chapter 11 doc-6236 opening question How would you define Spending wisely? How can you recognise a person who spends wisely?

2 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 located on your ebookplus. Digital doc SkillSHEET 11.2 doc-6237 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) Digital doc SkillSHEET 11.3 doc-6238 Multiplying and dividing a quantity (money) by a whole number 2 Calculate each of the following. a $23.50 ì 26 b $ ó 12 c $ ì 52 d $ ó 52 Digital doc SkillSHEET 11.4 doc-6239 Multiplying and dividing a quantity (money) by a fraction 3 Calculate each of the following. a $76.42 ó b $ ó Digital doc SkillSHEET 11.8 doc-6240 Converting a percentage to a decimal 4 Convert each of the following percentages to a decimal. a 34% b 79% c 4% d 67.2% e 8.25% f 17.5% Digital doc SkillSHEET 11.9 doc-6241 Finding a percentage of a quantity (money) 5 Find each of the following. a 10% of $350 b 25% of $1424 c 18% of $9000 d 12.5% of $4570 Digital doc SkillSHEET doc-6242 Expressing one quantity as a percentage of another 6 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, $ Maths Quest 9 for the Australian Curriculum

3 11a Spending wisely The basic principal of spending money wisely is to never spend more than you earn. Earning money with a part-time job Many Year 9 students have part-time jobs, either after school or on weekends or both. Worked Example 1 Both Adele and Beth have part-time jobs. Adele works at a fast-food restaurant for 13 hours per week earning $12.90 per hour, while Beth works at her father s electronics store for 11 hours per week earning $14 per hour. a How much will each girl have earned after 6 months (26 weeks)? b How long will it take each girl to earn $2000 for a summer holiday overseas? THINK Write a 1 Calculate the amount of money Adele earns in a week. Calculate the amount of money Adele earns in 26 weeks. 2 Calculate the amount of money Beth earns in a week. Calculate the amount of money Beth earns in 26 weeks. a Hourly wage = $12.90 Hours worked = 13 Adele s wages per week = ì 13 = $ Adele s wages for 6 months = 26 ì = $ Hourly wage = $14.00 Hours worked = 11 Beth s wages per week = ì 11 = $154 Beth s wages for 6 months = 26 ì 154 = $ Answer the question. In 6 months, Adele earns $ and Beth earns $ b 1 Calculate the number of weeks it would take Adele to earn $2000 based on a weekly wage of $ (step 1 above). Divide 2000 by Calculate the number of weeks it would take Beth to earn $2000 based on a weekly wage of $154 (step 2 above). Divide 2000 by 154. b Time for Adele to earn $2000 = 2000 ó = 11.9 weeks Time for Beth to earn $2000 = 2000 ó 154 = 13.0 weeks 3 Answer the question. It would take Adele 12 weeks and Beth 13 weeks to each earn $2000. Spending with a budget When a young person begins to earn money via a part-time job it is important to develop a budget, so that: money spent is less than that earned any expensive items can be saved for over a number of weeks expenses can be accounted for. Chapter 11 Financial mathematics 349

4 Worked Example 2 Carlotta earns $13.40 per hour for a 9.5-hour part-time job each week. Her expenses are as follows: casual expenses: $28 per week school supplies: $12 per week cosmetics: $11 per week. Carlotta wishes to save up for a fancy leather jacket costing $200 and a book about Australian Aborigines costing $37. How many weeks will it take her? Think write 1 Calculate Carlotta s weekly wages. Hourly wage = $13.40 Hours worked = 9.5 Weekly wages = ì 9.5 = $ Calculate Carlotta s weekly expenses. Add up the cost of casual, school supplies and cosmetics. 3 Calculate how much Carlotta saves each week after her regular expenses. Subtract the expenses from the wages. 4 Calculate the value of the items she is saving for. Add up the cost of the jacket and the book. 5 Calculate the length of time it takes to save up $237 based on the result from step 3. Divide 237 by Weekly expenses = = $51 Weekly savings = = $76.30 Cost of items = = $237 Saving time required = 237 ó = 3.11 weeks 6 Answer the question. It will take Carlotta 4 weeks to save for the jacket and the book (rounded up to the nearest whole week). Saving and borrowing Sometimes it may be necessary to borrow money in order to afford to purchase an expensive item. Normally banks will not lend money to teenagers, so one has to go to parents, relatives or (wealthy) friends. Worked Example 3 It is 1 August, just 19 weeks before Christmas. Two Year 9 boys, Dean and Don both work as maths tutors for 6 hours a week earning $20 per hour. They each wish to buy themselves a fancy mountain bicycle for Christmas, costing $780 each. Dean has a budget of $40 per week for general expenses and can save the rest, while Don spends $90 each week. a How much money will each boy be able to save in 19 weeks? b How much will either boy have to borrow from their respective parents, in order to purchase the bike before Christmas? c If either boy needs to borrow money, how many weeks, at their current rate of savings, will it take to pay back the loan? THINK Write a 1 Calculate Dean s and Don s weekly wages. a Hourly wage = $20 Hours worked = 6 Weekly wages = 20 ì 6 = $ Maths Quest 9 for the Australian Curriculum

5 2 Calculate how much each boy can save after subtracting their weekly expenses. 3 Calculate how much each boy can save after 19 weeks at the current rate of savings. Dean saves: = $80 per week Don saves: = $30 per week After 19 weeks, Dean saves 19 ì 80 = $1520 Don saves 19 ì 30 = $570 4 Answer the question. Dean saves $1520, while Don saves $570 in 19 weeks. Don does not have enough money to buy the bicycle. b 1 Calculate how much money either needs to borrow based on the result of part a above. Subtract their savings from the cost of the bicycle. b Since Dean saves $1520 he does not need to borrow any money. Don needs to borrow = $ Answer the question. Don needs to borrow $210 from his parents. c 1 Calculate the length of time it takes to repay the loan based on the fact that Don can save $30 per week (part a, step 2). c Time taken to repay loan = 210 ó 30 = 7 weeks 2 Answer the question. It will take Don 7 weeks (after he buys the bike in December) to repay his parents. Savings goals One of the main reasons for saving money is to purchase items in the future. These are called savings goals. There are three main types of savings goals: Short-term: To purchase an item within a few months Medium-term: To purchase an item within about 6 months to 2 years. Long-term: To purchase an item 2 or more years in the future Worked Example 4 Savings goals: Kim, a Year 9 girl has three savings goals: university tuition fees, a new jumper costing $75 and a new laptop computer costing $675 that she will need for Year 10. a Identify the goals as short-, medium- or long-term b Kim has a part-time job working 10 hours per week at $15 per hour. She wishes to put 10% of her earnings towards the jumper, $40 per week towards the computer and the rest towards her university tuition. Calculate how long it will take her to save for the jumper. c After the jumper is paid for, she wishes to put $25 per week towards the computer and the rest towards university. Calculate how long it will take her to save for the computer. d After the computer is paid for, all the money goes towards university. How much will she have saved for university during the year (52 weeks)? a THINK Identify the type of goals based upon the length of time required to meet them. a Write The shortest goal is the jumper. The longest goal is university tuition. The medium-term goal is the computer. b 1 Calculate Kim s weekly wages. b Hourly wage = $15 Hours worked = 10 Weekly wages = 15 ì 10 = $150 Chapter 11 Financial mathematics 351

6 2 Calculate how much she puts towards the jumper at 10% of Calculate how long it will take her to save for the $75 jumper. Amount put aside for jumper = 10% of 150 = $15 Time taken to save for jumper = 75 ó 15 = 5 weeks 4 Answer the question. It will take Kim 5 weeks to save for the jumper. c 1 State the amount of money saved for each goal during the first 5 weeks. 2 Calculate the amount required to purchase the computer. 3 Calculate the time required to purchase the computer at $25 per week. c Over the 5-week period, savings are: $15 per week for the jumper for a total of $75 $40 per week for the computer for a total of $200 $ = $95 per week for university for a total of $475. Kim needs = $475 for the computer. Time taken to save $475 for the computer = 475 ó 25 = 19 weeks 4 Answer the question. Kim will need = 24 weeks to save for the computer. d 1 Calculate the total amount of money earned in a year (52 weeks). d Yearly income = 52 ì 150 = $ Subtract the cost of the computer ($675) and the jumper ($75). Amount towards university tuition = = $ Answer the question. Kim will have saved $7050 towards her university tuition in 52 weeks. Best buys A wise shopper is always on the look-out for the best buy for a particular item. Supermarkets in Australia are now obliged to provide the unit cost on most products; this makes the comparison of prices of similar items simple. Unit price is the price per unit, for example per g, per ml. Worked Example 5 The supermarket displays three brands of frozen mixed vegetables. Brand X: $1.06 for 250 g Brand Y: $1.56 for 350 g Brand Z: $2.11 for 500 g Use the unit price per g to determine which brand is the best buy. Think 1 Determine the unit price (price per g) for Brand X. Don t round off at this stage. write Brand X: No. of lots of g in 250 g = 250 ó = 2.5 Cost per g = $1.06 ó 2.5 = $ Maths Quest 9 for the Australian Curriculum

7 2 Determine the unit price (price per g) for Brand Y. Round to 3 decimal places. 3 Determine the unit price (price per g) for Brand Z. Round to 3 decimal places. 4 Compare the three unit prices and write an answer. Brand Y: No. of lots of g in 350 g = 350 ó = 3.5 Cost per g = $1.56 ó 3.5 = $0.446 Brand Z: No. of lots of g in 500 g = 500 ó = 5 Cost per g = $2.11 ó 5 = $0.422 The unit prices for Brands X, Y and Z are $0.424, $0.446 and $0.422 respectively. This means that Brand Z is the cheapest, followed by Brand X, then Brand Y. remember 1. It is always better to spend less than you earn. 2. Students can save money from part-time jobs in order to buy things they want. 3. Savings goals can be considered as short, medium or long term. 4. Use unit pricing to compare product prices. Exercise 11A Individual pathways Activity 11-A-1 Introduction to unit price doc-4126 Activity 11-A-2 Comparing prices doc-4127 Activity 11-A-3 More complex price comparisons doc-4128 Spending wisely fluency 1 WE 1 Clarissa has a part-time job earning $13.25 per hour for 11 hours per week. a How much does Clarissa earn in 45 weeks? b If she puts all of her earnings towards a home-theatre TV costing $1350, how long will it take her to save for it? 2 Dawn earns $10.95 per hour working in her mother s dress shop. She works from 6 pm to 9 pm on Thursdays and Fridays and 9 am to 1 pm on Saturdays. a How much will Dawn earn in 26 weeks? b Dawn wishes to save for a dress to wear to the semi-formal, costing $260. If she puts 40% of her earnings towards the cost, how many weeks will it take her to save for the dress? 3 Edmond makes $13.40 per hour as a bicycle repairman. He works 8 hours per week. a How much will Edward earn in 2 years (104 weeks)? b Edmond wishes to save up for a new bicycle costing $380. If he puts one-third of his earnings towards this purchase, how long will it take to pay for it? Chapter 11 Financial mathematics 353

8 4 WE 2 Louis, a Year 9 boy earns $12.75 per hour for a 10-hour part-time job each week. His expenses are as follows: casual expenses: $38 per week school supplies: $16 per week car magazines: $8 per week. Louis wishes to save up for a new laser printer ($190) and a book about Australian car manufacturing costing $45. How many weeks will it take him? 5 Greta and Georgette have identical jobs at a video store, earning $15.50 per hour for 9 hours work each week after school. Greta spends all her money, except for $20 that she puts in a savings account at the bank. Georgette spends $50 per week on going out with friends and saves the rest. a How much will each girl have saved after 40 weeks? b The girls wish to go on an overseas holiday costing $2500 each. How long will it take each girl to save this amount? 6 WE3 Horace has a part-time job as a cleaner in an office building after school. He works 7.5 hours per week at a rate of $14 per hour. He wants to buy a bicycle costing $560. He plans on saving his earnings for 3 weeks to put towards the purchase and borrowing the rest from his father. a How much does Horace contribute towards the purchase? b How much does he borrow from his father? 7 WE4 Ingrid, a Year 9 student, wishes to save for a new winter coat ($240) and a car for when she turns 17. She has a part-time job working at Larson E. Daily s used-car lot as a typist. She earns $13.20 per hour for an 8-hour week. She decides to put one-quarter of her earnings towards the coat and the rest towards the car. a Identify Ingrid s savings goals as short, medium or long-term. b How long will it take her to save for the coat? c In 4 months (17 weeks) how much will she have put towards the car? 8 MC Which of these savings goals can be considered long-term? A A new album from itunes B The latest crime thriller novel C Next week s school excursion D A trip to Bali E A donation to the charity Medecins Sans Frontieres 9 MC Jeffrey has a part time job earning $13.60 per hour for a 13 hour week. He has three savings goals and wishes to put money towards them in the ratio of 1 : 2 : 5, with the largest amount going towards a new scooter, the next largest towards a computer and the smallest amount towards new t-shirts. How much does he put towards the computer each week? A $27.20 B $44.20 C $ D $ E None of the above. 354 Maths Quest 9 for the Australian Curriculum

9 10 WE 5 The supermarket displays three different packaged bags of tomatoes. Brand A: $1.95 for 500 g Brand B: $3.15 for 750 g Brand C: $4.62 for 1.2 kg Use the unit price per g to determine which brand is the best buy. Understanding 11 Klaus wishes to purchase a new stereo system which is on sale this week for $1250. He works for a restaurant as a busboy for 16 hours per week earning $11.20 per hour. He decides to put half of last week s earnings towards the stereo and borrow the rest from his mother. a How much does he have to borrow? b He offers to repay his mother 30% of his earnings each week. How long will it take to repay the loan? c During the time he is repaying the loan to his mother, he decides to put half of his remaining weekly earnings towards the purchase of CDs. How much, in total, does he spend on CDs during this period? 12 Francisco and Fiorello both work 12 hours per week, each earning $ per week. a What is their hourly rate of pay? b Both boys wish to save for an expensive surfboard that costs $430. Francisco normally spends $60 per week on entertainment and $15 per week on internet access and the rest he ll put towards his surfboard. Fiorello decides to save 40% of his wages for his surfboard. How long will it take each boy to save for his surfboard? 13 Herman, a Year 9 boy has three savings goals: money towards a large 21st birthday party, a new pair of hi-tech runners ($140) and the latest mobile phone ($320), which won t be released for another 6 months. He has a job working at his uncle s factory earning $14.50 per hour for a 12-hour week. a Identify his savings goals as short, medium or long-term. b Herman decides to put away 20% of his earnings towards the phone, $20 per week towards the runners and the rest for the 21st birthday party. How long will it take him to save for the runners and the phone? c In 40 weeks, how much will he have saved towards his party? 14 Jackson has a part-time job which earns $15 per hour. Over the last 4 weeks he worked 5, 8, 11 and 6 hours per week. a How much did Jackson earn over the last 4 weeks? b His boss offers him a 10% raise. What is hourly rate of pay now? c However, with the new wages, his boss can only offer him a maximum of 6 hours of work per week. Based upon the 4 weeks average number of hours, is he going to make more or less money in the future? 15 There is the misconception that buying in bulk is always cheaper. A certain brand of soap powder costs $4.49 for 750 g. What is the most you could pay for a 2.5-kg packet of another brand of similar soap powder for it to be a better buy? Reasoning 16 Nancy, a Year 9 girl, earns $17.20 per hour for 9 hours as a typist at an accountant s office. Her budget is as follows: 10% for weekly expenses of a casual nature $30 per week for cosmetics Chapter 11 Financial mathematics 355

10 11B 20% per week for new clothes the rest is put in a bank account for university. a How long would it take her to save for a new business suit costing $265 from her clothing budget? b In 52 weeks, how much will she have saved for university? c Which of her goals can be considered long-term? d How much has she spent on casual expenses over a 30-week period? 17 Orrin J. Sampson is a Year 9 boy who wants to save for a 4-wheel drive (4WD) when he is old enough to drive. He works as a football player in a local club earning $120 per match for a 16-match season. During the off-season he works as a fitness trainer for 32 weeks per year at $14.50 per hour for 12 hours a week. a How much does Orrin earn in a year? b If he puts 60% towards the 4WD and spends $40 per month on various items, how much can he save in a year? c If he reduces his monthly expenses by $15 and reduces reflection his savings by $200, how much extra would he have to Why is it important to have spend each year? Explain. savings goals? Discount, profit and loss Discount Stores offer discounts in order to sell slow-moving stock, unfashionable items and stock that is out of season. A discount can be quoted in terms of money, or a percentage of the original price of the item. Two common methods are used to determine the sale price of an item after a percentage discount. 1. Subtract the discount percentage from %. Multiply this new percentage (the % of the marked price to be paid) by the original price. % - discount % ì % of original price % of original price ì original price = discounted price 2. Find the actual discounted dollar price by multiplying the discount percentage by the original price, then subtract the discount amount from the original price. Original price ì % = discount ($) Original price - discount = discounted price Worked Example 6 Find the sale price on a hat marked $72 if a 10% discount is given. Think write 1 Find the percentage of the marked price that is paid by subtracting the percentage discount from %. % - 10% = 90% 2 Find the sale price of the hat. 90% of $72 = = 0.9 ì 72 = Write the answer in a sentence. The sale price of the hat is $ Maths Quest 9 for the Australian Curriculum

11 Worked Example 7 Peddles is a bicycle store that has offered a discount of 15% on all goods. Find: a the cash discount allowed on a bicycle costing $260 b the sale price of the bicycle. a THINK Find the discount, which is 15% of the marked price. Write a Discount = 15% of 260 = = 0.15 ì 260 = 39 The cash discount allowed is $39. b 1 To find the sale price, subtract the discount from the marked price. b Sale price = marked price - discount = = Answer the question in a sentence. The sale price of the bicycle is $221. To calculate the percentage discount, we write the monetary amount of discount as a percentage of the original price and multiply by %. cash discount Percentage discount = original price 1 %. Worked Example 8 At Peddles, the price of a bicycle is reduced from $260 to $200. Calculate the percentage discount. Think write 1 Calculate the amount of the discount. Discount = = $60 2 Write the discount as a percentage of the original price, correct to the nearest whole number. Percentage discount = ì % = ö 23% 3 Answer the question in a sentence. The percentage discount is about 23%. Finding the original price Sometimes it is important to know what the original price was, given the new, discounted one. There are two cases. For a fixed dollar amount discount: Original price = new price + discount amount For a percentage discount: new price Original price = (% - discount percentage) Chapter 11 Financial mathematics 357

12 Worked Example 9 Find the original prices for the following sale items. a A winter coat labelled reduced by $60 now $135 b A stereo amplifier labelled reduced by 15% now $239 THINK a 1 Identify the type of discount fixed amount or percentage. 2 Apply the appropriate formula: Original price = new price + discount amount a Write Based on the statement, this is a fixed amount discount. New price = $135 Discount amount = $60 Original price = = $195 3 Answer the question. The original price of the coat was $195. b 1 Identify the type of discount fixed amount or percentage. b Based on the statement, this is a percentage discount. 2 Apply the appropriate formula new price Original price = (% - discount percentage) Discount = 15% New price = $239 New percentage = % - 15% 239 Original price = ( % 15%) 3 Complete the calculation. Original price = % = = $ (rounded to the nearest 5 cents) 4 Answer the question. The amplifier s original price was $ Profit and loss When a retailer calculates the price to be marked on an article, many overhead costs must be taken into account: staff wages, rent, electricity, advertising and so on. These costs must be covered, and the business must be profitable. Profit or loss is the difference between the total of the retailer s costs (cost price) and the final selling price of the goods (selling price). elesson eles-0117 Small business Selling at a profit Profit: selling price - cost = positive $ amount Percentage profit = profit cost ì % Selling at a loss Loss: selling price - cost = negative $ amount Percentage loss = loss cost ì % 358 Maths Quest 9 for the Australian Curriculum

13 Worked Example 10 A music store buys CDs at $15 each and sells them for $28.95 each. What is the percentage profit made on the sale of a CD? Think write 1 Calculate the profit on each CD; that is, selling price - cost. 2 Calculate the percentage profit; that is, profit cost 3 Write the answer in a sentence, rounding to the nearest percentage if applicable. Profit = = $ ì %. Percentage profit = 15 = 93% ì % The profit is 93% of the cost price. Worked Example 11 Ronan operates a sports store at a fixed profit margin of 65%. For how much would he sell a pair of running shoes that cost him $40? Think write 1 Find the selling price by first adding the percentage Selling price = 165% of 40 profit to % then multiply the percentage by the cost price. = = 1.65 ì 40 = 66 2 Write the answer in a sentence. The running shoes would sell for $66. Worked Example 12 In the same sports store, Ronan sells a tracksuit for $ What would Ronan have paid for the tracksuit? Think write 1 Write the selling price as 165% of the cost price. 165% of cost price = $ Write the percentage as a fraction and change the of to a ì. 165 ì cost price = Write the percentage as a decimal ì cost price = Divide both sides of the equation by Give your cost price = answer to the nearest cent = Write the answer in a sentence. Ronan would have paid $60.58 for the tracksuit. Chapter 11 Financial mathematics 359

14 remember 1. A discount is usually a percentage of the marked price. 2. To find the discounted price of an item use either of the following formulas: % - discount % = % of original price % of original price ì original price = discounted price or Original price ì % = discount ($) Original price - discount = discounted price 3. To find the percentage discount: cash discount Percentage discount = original price 1 %. 4. The original price can be calculated from the new price and the discount offered: For fixed dollar amount discounts: original price = new price + discount amount new price For percentage discounts: original price = (% - discount percentage) 5. (a) Profit: selling price - cost = positive $ value (b) Loss = selling price - cost = negative $ value 6. (a) Percentage profit = profit ì % cost (b) Percentage loss = loss cost ì % 7. (a) Selling price = (% + percentage profit) of cost price = (% + percentage profit) ì cost price (b) Selling price = (% - percentage loss) of cost price = (% - percentage loss) ì cost price Exercise 11B Individual pathways Activity 11-B-1 Reviewing discount, profit and loss doc-4129 Activity 11-B-2 Using discount, profit and loss doc-4130 Activity 11-B-3 Calculating discount, profit and loss doc-4131 Discount, profit and loss fluency 1 We 6 Find the sale price of each article when the marked price and discount are shown as in this table. Marked price (RRP) Discount a $0 15% b $250 20% c $95 12% d $ % e $ % 2 We 7 A sale discount of 20% off was offered by the music store Solid Sound. Find: a the cash discount allowed on a $350 sound system b the sale price of the system. 360 Maths Quest 9 for the Australian Curriculum

15 Digital doc SkillSHEET doc-6244 Digital doc SkillSHEET doc MC A calculator wristwatch is advertised at $69.95, less 10% discount. Find the sale price. A $7.00 B $59.95 C $62.96 D $49.95 E $ A store-wide clearance sale advertised 15% off everything. a What would be the selling price of a pair of jeans marked at $49? b If a camera marked at $189 was sold for $160.65, was the correct percentage deducted? 5 T-shirts are advertised at $15.95 less 5% discount. How much would Jim pay for five T-shirts? 6 We 8 Find the percentage discount given on the items shown in the table. Round to the nearest percent. Original price Selling price a $25 $15 b $ $72 c $69 $50 d $89.95 $70 7 MC Calculators were advertised at $20, discounted from $25. What percentage discount was given? a 20% b 25% c 5% d 0.2% e 2% 8 A tennis racquet marked at $79.95 sells for $60. What percentage discount is this, to the nearest whole number percentage? 9 CDs normally selling for $28.95 were cleared for $ What percentage discount was given? 10 All sports cleared their stock of soccer balls for $ They had been priced at $ What percentage discount was given? 11 We 9 Calculate the original price of these following sale items. a A shirt selling for $31.40 at a 15% discount b A digital watch selling for $42.40 at a 20% discount c A dishwashing machine selling for $ at a discount of 16% 12 MC A mobile phone is offered at 82% of its regular price. If the new price is $134.50, the original price was: A $164 B $146 C $747 D $447 E None of the above. 13 We 10 A supermarket buys frozen chickens for $3.50 each and sells them for $5.60. What is the percentage profit made on the sale of each chicken? 14 A restored motorbike was bought for $350 and later sold for $895. a How much profit was made? b What percentage was profit? Give your answer correct to the nearest whole number. 15 For each of the following items, find the percentage profit or loss. Cost price Selling price a $15 $20 b $40 $50 c $40 $30 d $75 $85 e $38.50 $29.95 Chapter 11 Financial mathematics 361

16 Digital doc SkillSHEET doc Running shoes bought for $ were sold after six months for $60. a How much was the loss? b What was the percentage loss? Give your answer to the nearest whole number. 17 A sports card collection costing $80 was sold for $65. What was the percentage loss? 18 Running shoes bought by a sports store for $30 per pair were sold at $ What percentage profit was made? 19 We 11 Kyle runs a jewellery business that uses a fixed profit margin of 98%. For how much would he sell a necklace that cost him $830? 20 Find the selling price for each of the following items. a Jeans costing $20 are sold with a profit margin of 95%. b A soccer ball costing $15 is sold with a profit margin of 80%. c A sound system costing $499 is sold at a loss of 45%. d A skateboard costing $30 is sold with a profit margin of 120%. 21 WE12 MC A camping goods shop operates on a profit margin of 85%. How much would the shop have paid for a sleeping bag that sells for $89.95? A $ B $13.49 C $48.62 D $76.46 E $85.00 Understanding 22 A discount of 15% reduced the price of a CD by $3.20. a What was the original price of the CD? b What was its selling price? 23 A discount of 22% reduced the price of an outfit by $48. a What was its original price? b At what reduced price was it selling? 24 MC I pay $1290 for 10 identical articles, being allowed a total discount of $130. The marked price of each article is: A $142 B $116 C $1420 D $2590 E $ MC I am allowed a discount of 10% off the total price of 6 articles which cost $x each. The final price paid is: A $60x B $6x C $0.06x D $0.6x E $5.4x 26 A major department store marks up each item using a profit margin of 120%. a How much did it cost the store to buy a shirt if it is sold for $55? b How much did it cost the store to buy perfume that is sold for $106? c What was the cost to the store of a kitchen appliance sold for $89.95? 27 Sonja bought an old bike for $20. She spent $47 on parts and paint and renovated it. She then sold it for $115 through her local newspaper. The advertisement cost $10. a What were her total costs? b What percentage profit did she make on costs? c What percentage profit was made on the selling price? 28 MC A fruit and vegetable retailer buys potatoes by the tonne for $180 and sells them in 5-kg bags for $2.45. What percentage profit is made? A 58% B 310% C 172% D 272% E 245% 362 Maths Quest 9 for the Australian Curriculum

17 29 During a sale, a retailer allows a discount of 15% off the marked price. His sale price of $60 still gives him a profit of 10%. a What did the article cost him? b What was the marked price? 30 MC A clothing store operates on a profit margin of 150%. The selling price of an article bought for $p is: A $151p B $150p C $2.5p D $1.5p E None of these 31 Chris calculated the costs of producing a new diving mask would be $20, but he needed to have a 60% profit margin. What price will Chris have to charge in order to retain this profit margin? Digital doc WorkSHEET 11.1 doc-6243 Reasoning 32 MC Dale saw an ad in the local paper for a rental tuxedo. The tuxedo cost $120 to rent for the evening. When Dale arrived at the store to pay for his tuxedo, the owner offered him a 10% discount on the $120 advertised cost. After Dale paid the discounted price, the owner realised that Dale was a member of the local surf club and offered him a 5% discount on the price Dale had already paid. What was the final price of the tuxedo? A $108 B $102 C $ D $105 E $ A student received a 10% discount on mathematical books during the bookstore s sale. He was then offered another 5% frequent customer discount to be applied to the discounted price. If the customer paid $220 for the books, what was the original price of the books? Justify your answer. 34 Glen s surf shop increased their profit margin on wetsuits to 200%, while Melissa s surf shop increased their profit margin on wetsuits to 50%. After the changes in profit margin, both surf shops prices were equal. If Glen s original cost was $120 for a wetsuit, then what was Melissa s original cost for a wetsuit? Justify your answer. 35 Chilee s dress shop is going out of business and selling 20 dresses at a loss in order to pay outstanding debts. If Chilee s cost for each of these dresses is $120, by what percentage should she decrease the price of her dresses in order to obtain $1680 in cash? 36 The cost of producing a litre of milk is: milk payment to farmers: $40 per litres packaging $0.12 per litre transportation from farm to dairy $45 per 200 litres dairy processing costs $10.50 per 25 litres. a What is the cost of producing a litre of milk? b If the dairy wishes to make a profit of $0 per 4000 litres, what is the selling price of a litre of milk? c The government decides to force dairies to reduce their selling price by $0.10 per litre. What is the profit per litre now? Justify your answer. 37 A set of reference books originally costs $780 and is offered at a 20% discount in March. In April the discounted price is increased by 20%. a Does the price in April equal the price before March? b If the answer to part a is no, explain why. reflection Why is profit an important part of all buying and selling transactions? Chapter 11 Financial mathematics 363

18 11c Interactivities Simple interest int-2770 Effects of P, r, i and t int-0745 Simple interest The time-value of money If you borrow $10 from a friend (or a bank), she expects to get paid back, in the future, more than $10. Should it be $10.01, $10.10, $11 or some other value? This is known as the time-value of money the fact that over time a sum of money will increase in value if invested wisely. If you borrow money it is called a loan; the person (or bank) you borrow from makes an investment. In the above example the $10 is the principal and the amount over $10 is the interest. The length of time between the borrowing and the paying back is called the time period. Worked Example 13 A young apprentice butcher agrees to borrow $ and pay $110 back in 4 months time. a Identify the amount of principal and interest in this transaction. b Based on the rate of interest above, how much interest would be charged if the apprentice borrowed $500 for 3 months? THINK a 1 Calculate the difference between what was borrowed and what was paid back. This is the interest. Write a Interest = = 10 The interest was $10. 2 The amount borrowed is the principal. The principal was $. b 1 Determine the rate of interest, that is, how much interest was charged per month. 2 Use that rate of interest (2.5% per month) to calculate the interest on $500 for 1 month. 3 Multiply the monthly interest charge ($12.50) by the number of months (3). b Interest = $10 Principal = $ Rate of interest = 10 Percentage interest rate = 10 ì 1 = 10% per 4 months or 2.5% per month. Interest on $500 for 1 month = 2.5% of 500 = 2. 5 ì 500 = $12.50 Interest for 3 months = ì 3 = $ Answer the question. The amount of interest on $500 for 3 months is $37.50 Simple interest Worked example 13 is an example of a loan (or investment) using simple interest calculations. The amount of interest is proportional to the principal, the time period and the rate of interest. 364 Maths Quest 9 for the Australian Curriculum

19 Let P = the amount of the principal, T = the time period and r = the interest rate per unit of time. Then the amount of interest (I) is given by the formula: P r T I =. Note that r and T must be in the same units; that is, if the interest rate (r) is in months, then the length of time (T) must be in months also. The amount (A) of the loan (or investment) is the sum of the principal and interest. Amount = principal + interest A = P + I Worked Example 14 Using the simple interest formula, calculate the interest, and then the amount of the following investments. a Gerry invests $400 at 7% per year for 6 years. b Hetty invests $500 at 6% per year for 10 months. THINK a 1 Identify the principal, interest rate and time period. Ensure that the interest rate and time are in the same units. In this case they are both in years. 2 Use the simple interest formula with the principal as given ($400). 3 Calculate the amount of the investment using the formula: Amount = principal + interest. Write a Principal = $400 Interest rate (r) = 7% per year Time period (T) = 6 years. Interest = PrT = = 168 Interest = $168 Principal = $400 Interest = $168 Amount = = $568 4 Answer the question. The investment will amount to $568. b 1 Identify the principal, the interest rate and time period, ensuring that they are in the same units. In this case interest rate is in years while time period is months. Convert interest rate into a monthly rate by dividing by Use the simple interest formula with the principal as given ($500). b Principal = $500 annual interest rate Monthly interest rate = 12 = 6 12 = 0.5% Interest rate (r) = 0.5% (per month) Time period = 10 months Interest = PrT = = 25 Interest = $25 Chapter 11 Financial mathematics 365

20 3 Calculate the mount of the investment using the formula: Amount = principal + interest. Principal = $500 Interest = $25 Amount = = $525 4 Answer the question. The investment will amount to $525. An application: term deposits A term deposit is a form of savings account, where the money is left in the bank for a long time (term) and (generally) cannot be accessed until the term is completed. Term deposits usually pay a higher interest rate than regular savings accounts. In Australia, taxation law requires that any term deposit where the term is 1 year or longer can only pay simple interest. The interest paid each year is given to the depositor in cash or deposited into a regular savings account. Worked Example 15 Calculate the interest paid each year and over the length of the term deposit for the following. a Ina invests $2530 for 5 years at 6.5% per year. b Jose invests $4280 for 3.3 years at 9.5% per year. THINK Write a 1 Identify the principal, interest rate and time. 2 Use the simple interest formula with the principal as given ($2530) for 1 year (T = 1). 3 Multiply the value of the 1st year s interest by the length of the term (5 years). a Principal = $2530 Interest rate (r) = 6.5% per year Time = 1 year Interest = PrT = = $ Total interest = 5 ì = $ Answer the question. Total interest paid is $ b 1 Identify the principal and interest rate. b Principal = $4280 Interest rate (r) = 7.5% (per year) 2 Use the simple interest formula with the principal as given ($4280) for 1 year (T = 1). 3 Multiply the value of the 1st year s interest by the length of the term (3.3 years). Interest = PrT = = $321 Total interest = 3.3 ì 321 = $ Answer the question. Total interest paid is $ Maths Quest 9 for the Australian Curriculum

21 Developing a simple interest spreadsheet The spreadsheet below calculates the total amount of simple interest for a given number of years A Year B C D E F Principal Interest rate (per year) Time (years) Principal Interest New value Inputs (yellow cells) Cell D2: the amount of principal. Above, the principal is $0. Cell D3: the interest rate, as a percentage. Above, the interest rate is 5%. Cell D4: the term. Above, the term is 6 years. Outputs (Row 7 and beyond) Column B: shows the years: 1, 2, 3,... 6 Column C: shows the principal each year. Set C7 = $D$2 and fill down. Column D: shows the interest calculation. Set D7 = C7*$D$3/ and fill down. Cell E7: Shows the new value after year 1. Set E7 = C7 + D7. Cell E8: Shows the new value after year 2. Set E8 = E7 + D8 and fill down. For time periods greater than 6 years, highlight Row 12 s cells and fill down. remember 1. If one borrows or lends money, its value increases over time. This is called the time-value of money. 2. The amount of money borrowed or loaned is called the principal (P). 3. The increase in value is called the interest (I). 4. The length of the loan or investment is called the time period (T). 5. With simple interest, the amount of interest is proportional to P, r and T: I = PrT 6. The amount of the loan or investment is the sum of principal and interest: Amount = principal + interest 7. A term deposit is a form of savings account where money is left in the bank for a long time (term) and cannot be accessed until the term is completed. Chapter 11 Financial mathematics 367

22 Exercise 11C Individual pathways Activity 11-C-1 Calculating simple interest doc-4135 Activity 11-C-2 Rearranging the simple interest formula doc-4136 Activity 11-C-3 Comparing investments doc-4137 Simple interest Fluency 1 WE13 A Year 9 girl agrees to borrow $200 and pay $225 back in 5 months time. a Identify the amount of principal and interest in this transaction. b Based on the rate of interest above, how much interest would be charged if the girl borrowed $700 for 4 months? 2 Jan borrowed $0 for 4 months and paid back $1050. How much would she have to pay back if she borrowed for 9 months instead? 3 WE14 Using the simple interest formula calculate the interest, and then the amount of the following investments. a Huey invests $800 at 4% per year for 3 years. b Dewey invests $1200 at 7% per year for 13 months. c Louie invests $1600 at 0.5% per month for 2 years. 4 Use the simple interest formula to calculate the interest on the following loans. a Larry borrows $6200 at 8.5% per year for 5 years. b Moe borrows $7200 at 9.5% per year for 10 months. c Curly borrows $7200 at 10.5% per year for 13 years. 5 A construction company borrows money to finance its projects. It offers investors 10% per year simple interest for amounts over $ Calculate the interest and total value for the following investors. a Stanley invests $ for 2.5 years. b Oliver invests $ for 11 months. 6 WE15 Calculate the interest paid each year and over the length of the term deposit for the following: a George invests $6520 for 2.6 years at 6.2% per year. b Gracie invests $5250 for 2.4 years at 6.8% per year. 7 MC If a $4000 investment returns $550 in interest over a 1.5 year term then the annual interest rate is: A 13.75% B 9.17% C 5% D 10% E 4.58% 8 For each of the five loans in the table, calculate: i the simple interest ii the amount repaid. 368 Maths Quest 9 for the Australian Curriculum Principal ($) Interest rate per annum Time a % 2 years b % 3 years c % 48 months d % 2 years 6 months e % 42 months 9 Find the final value of each of the following investments: a $3000 for 2 years at 5% p.a. b $5000 for 3 months at 4.3% p.a.

23 Understanding 10 A woman purchases a term deposit for 3 years at 6.5% per annum. What percentage of the principal is paid out in interest over the 3 year period? 11 One million dollars was invested for 5 days at 6.2% p.a. How much interest was earned on this investment? 12 Carla borrows $3762 for an overseas trip at 8.9% p.a. simple interest over 20 months. If repayment is made later in equal monthly instalments over the same time period, how much is each instalment? 13 Jodie wants to earn $200 in interest over the next two years. If she can invest her money at 8% p.a., how much does she need to invest to earn the $200? 14 Find the missing quantity in each row of the table. Principal Rate of interest p.a. Time Interest earned a $ % $ b $ % $ c 7% 3 years $ d 4.9% 1 year 9 months $ e $ years $ f $ months $ MC If an investment of $400 pays 8% simple interest per year for 3 years, then the amount of the investment is: A $32 B $96 C $432 D $496 E $1296 Reasoning 16 A $ business is purchased on $ deposit and the balance payable over 5 years at 8.95% p.a. flat rate. a How much money is borrowed to purchase this business? b How much interest is charged? c What total amount must be repaid? d Find the size of each of the equal monthly repayments. 17 A construction worker agrees to the following contract. She starts out earning $0 per week, then every 5 weeks her wages increase by a certain amount. If over a 25-week period she earns a total of $38 750, what was the increase every 5 weeks? (Hint: Try to set up a spreadsheet for this problem.) 18 Olivia and Anita go to the same bank, and Olivia invests $5000 more than Anita. Olivia receives an annual interest rate of 6%, while Anita receives 5%. After one year Olivia received $1560 simple interest. Determine: a how much Olivia invested b how much Anita invested c how much simple interest Anita earned. Chapter 11 Financial mathematics 369

24 Digital doc Worksheet 11.2 doc You have been given $ from your parents to place in two different investments (Silicon World and Solar Energy). Silicon World offers an annual interest rate of 10% and Solar Energy offers a 6% annual interest rate. Together the investments return a total of $1800 at the end of the year. Determine the amount of money placed in each investment. Justify your answer. reflection What is the easiest way to memorise the simple interest formula? 370 Maths Quest 9 for the Australian Curriculum

25 Summary Spending wisely number AND algebra Money and financial mathematics It is always better to spend less than you earn. Students can save money from part-time jobs in order to buy things they want. Savings goals can be considered as short, medium or long term. Use unit pricing to compare product prices. Discount, profit and loss A discount is usually a percentage of the marked price. To find the discounted price of an item use either of the following formulas: % - discount % = % of original price % of original price ì original price = discounted price or Original price ì % = discount ($) Original price - discount = discounted price To find the percentage discount: cash discount Percentage discount = original price 1 %. The original price can be calculated from the new price and the discount offered: For fixed dollar amount discounts: original price = new price + discount amount new price For percentage discounts: original price = (% - discount percentage) (a) Profit: selling price - cost = positive $ value (b) Loss = selling price - cost = negative $ value (a) Percentage profit = profit ì % cost (b) Percentage loss = loss cost ì % (a) Selling price = (% + percentage profit) of cost price = (% + percentage profit) ì cost price (b) Selling price = (% - percentage loss) of cost price = (% - percentage loss) ì cost price Simple interest If one borrows or lends money, its value increases over time. This is called the time-value of money. The amount of money borrowed or loaned is called the principal (P). The increase in value is called the interest (I). The length of the loan or investment is called the time period (T). With simple interest, the amount of interest is proportional to P, r and T : I = PrT Chapter 11 Financial mathematics 371

26 The amount of the loan or investment is the sum of principal and interest: amount = principal + interest. A term deposit is a form of savings account, where money is left in the bank for a long time (term) and cannot be accessed until the term is completed. Homework Book Mapping your understanding Use the terms in the summary, and other terms if you wish, to construct a concept map that illustrates your understanding of the key concepts covered in this chapter. Compare this concept map with the one that you created in What do you know? on page 347. Have you completed the two Homework sheets, the Rich task and two Code puzzles in your Maths Quest 9 Homework Book? 372 Maths Quest 9 for the Australian Curriculum

27 Chapter review number AND algebra Money and financial mathematics Fluency 1 Claire wishes to save $0 for a holiday in Hawaii. She works part-time earning $120 per week and can save 40% each week. How long will it take to reach her savings goal? A 17 weeks B 7 weeks C 3 weeks D 20 weeks E 21 weeks 2 If a trendy sports coat is sold for 3 times what it cost to make, then the profit as a percentage of the selling price is: A 66.7% B 75% C % D 200% E Cannot be determined from the given information. 3 If a person invests $400 and it earns $300 simple interest in 6 years, then the interest rate per year is: A 16.7% B 20% C 8% D 12.5% E 75% 4 Marie-Louise, a Year 9 girl, earns $13.40 per hour for an 18-hour fortnight. After her expenses of $40 per fortnight, she saves one-third of the rest to put towards her college education. She decides that she needs to save $8000 in total. After 50 weeks, how much more does she need to save? 5 A digital wristwatch has $5.70 worth of material, $1.80 worth of packaging and $0.22 worth of advertising. In addition it costs $42 to transport 200 watches. If the watch sells for $30 how much profit is made on 200 watches? 6 Calculate the original prices on the following sale prices. a A cot on sale for $78 at a discount of 20% b A single bed on sale for $420 at a discount of 30% c Last year s model mp3 player on sale for $ at a discount of 40%. 7 Mary wishes to purchase a very expensive pair of dress shoes. They cost $160 but she only has $30 to spend. The store agrees to a lay-by where she must pay an equal amount each week for 7 weeks. How much is her weekly payment? 8 Calculate the simple interest on the following savings accounts. a $520 at 4.5% per year for 2 years b $1230 at 5.5% per year for 3.5 years c $3457 at 6.5% per year for 14 months 9 To increase sales, the manager of a clothing store offers the following discount on a $400 overcoat. The 1st week it is sold at the regular price, but each week thereafter the price is reduced by 10% over the previous week s price. Determine the price in weeks 2, 3, 4 and 5. problem solving 1 Bargain Barry loves to shop. After receiving a 15% discount on his chainsaw and 35% off his lawnmower, he tells his friend Cheap Chris about his bargain. Chris told Barry that he was cheated because he would have received a better deal if the total prices were added together and discounted by 25%. The cost of the lawnmower was twice the amount of the chainsaw. a Explain why Chris is not correct in his advice. b When would Chris advice be correct? 2 Sara borrows $ from one bank, and Jose borrows $2000 from another bank. Sara s account pays 3% more than Jose s account. If together the accounts earn $800 simple interest in one year, what is the annual interest rate for: a Sara s account? b Jose s account? 3 Cory is offered two different deals on a $ motorcycle: Deal A: an initial discount of 15%, and an 8% discount, applied to the discounted price, for being the th customer of the day Deal B: a 22% discount on the original price. a What is the overall discount amount for Deal A? b What is the overall discount percentage for Deal A? c Which deal would save you the most money? 4 Catherine increased the cost of dresses by 20% in her boutique for one week, then reduced the new price by 30% the following week to $60. What was the original price of the dresses? Interactivities Test yourself Chapter 11 int-2701 Word search Chapter 11 int-2699 Crossword Chapter 11 int-2700 Chapter 11 Financial mathematics 373