FreeStanding Mathematics Activity Working with percentages Nuffield Foundation

Free-Standing Mathematics Activity Working with percentages © Nuffield Foundation 2012

A Compound interest Amount invested = £ 3000 Interest rate = 4% 1 Step-by-step method Interest at end of Year 1 = 4% of £ 3000 Think about Is the answer the same if you divide by 100, then multiply by 4? = 0. 04 x £ 3000 = £ 120 Amount at end of Year 1 = £ 3120 Interest at end of Year 2 = 4% of £ 3120 = 0. 04 x £ 3120 = £ 124. 80 Amount at end of Year 2 = £ 3120 + £ 124. 80 = £ 3244. 80 © Nuffield Foundation 2012 and so on

A Compound interest Amount invested = £ 3000 Interest rate = 4% 2 Repeating calculations using a multiplier Amount at end of Year 1 = 104% of £ 3000 = 1. 04 x £ 3000 = £ 3120 Amount at end of Year 2 = 1. 04 x £ 3120 = £ 3244. 80 and so on Try repeated calculations like this one on your calculator © Nuffield Foundation 2012

A Compound interest £ 3000 invested at 4% interest Repeated calculations How much is in the account after 5 years? End of year n Amount £ A 0 3000. 00 1 3120. 00 2 3244. 80 3 3374. 59 4 3509. 58 5 3649. 96 © Nuffield Foundation 2012

A Compound interest Amount invested = £ 3000 Interest rate = 4% 3 Using indices Amount at end of Year n = 1. 04 n x £ 3000 Amount at end of Year 2 = 1. 042 x £ 3000 = £ 3244. 80 Amount at end of Year 5 = 1. 045 x £ 3000 = £ 3649. 96 Try this A An account gives 3% interest per annum. £ 5000 is invested. How much will be in the account after 6 years? Use each method. Think about What are the advantages and disadvantages of each method? © Nuffield Foundation 2012

B Depreciation A new car costs £ 16 000. Its value falls by 15% per year What will it be worth when it is 5 years old? In this case the multiplier is 0. 85 Age of car (n years) Value (£ A) 0 16 000 1 13 600 2 11 560 3 9826 4 8352 5 7099 © Nuffield Foundation 2012 What will the car be worth when it is 20 years old? Think about What assumption is being made? Is it realistic?

B Falling sales Try this B A company’s sales of a product are falling by 6% per annum. They sold 45 000 this year. Estimate the annual sales 6 years from now. In this case the multiplier is 0. 94 Formula for annual sales n years from now = 0. 94 n x 45 000 Estimate of annual sales 6 years from now = 0. 946 x 45 000 about 31 000 Check this by repeated calculations. © Nuffield Foundation 2012

C Combining percentage changes A shareholder owns 2000 shares. She expects to get 3% more shares then plans to sell 25% of her shareholding. How many shares will she have after these transactions? Number after receiving 3% extra = 103% of 2000 = 1. 03 x 2000 = 2060 Number after selling 25% = 75% of 2060 = 0. 75 x 2060 = 1545 What % is this of her original shareholding? 1545 100 = 77. 25% or 1. 03 x 0. 75 = 0. 7725 2000 © Nuffield Foundation 2012

C Combining percentage changes Try this C A shop marks up the goods it sells by 30% In a sale it reduces its normal prices by 25% What is the overall % profit or loss on goods sold in the sale? Sale price = 75% of normal price = 75% of 130% of cost price = 0. 75 x 1. 3 x cost price = 0. 975 of cost price The shop makes a 2. 5% loss on goods it sells in the sale. © Nuffield Foundation 2012

D Reversing percentage changes The price of a train fare increased by 2. 5% recently. It now costs £ 66. 42 How much did it cost before the rise in price? 1. 025 x previous price = £ 66. 42 Previous price = £ 66. 42 1. 025 Previous price © Nuffield Foundation 2012 = £ 64. 80

D Reversing percentage changes Try this D After a 12. 5% discount, insurance costs £ 25. 90 What was the cost before the discount? 0. 875 x full price = £ 25. 90 Full price = £ 25. 90 0. 875 Full price = £ 29. 60 © Nuffield Foundation 2012

Reflect on your work • Which of the methods do you think is most efficient? • How can a graphic calculator or spreadsheet help? © Nuffield Foundation 2012