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

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

3. © Nuffield Foundation 2012 Amount invested = £3000 Interest rate = 4% Amount at end of Year 1= 104% of £3000 = 1.04 x £3000 = £3120 and so on 2 Repeating calculations using a multiplier Amount at end of Year 2 = 1.04 x £3120= £3244.80 Try repeated calculations like this one on your calculator A Compound interest

4. © Nuffield Foundation 2012 £3000 invested at 4% interest End of year n Amount £ A 03000.00 1 2 3 4 5 3120.00 3244.80 3374.59 3509.58 3649.96 How much is in the account after 5 years? Repeated calculations A Compound interest

5. © Nuffield Foundation 2012 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 Amount at end of Year 5 = 1.04 2 x £3000 = 1.04 5 x £3000 = £3244.80 = £3649.96 Think about What are the advantages and disadvantages of each method? 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. A Compound interest

6. © Nuffield Foundation 2012 A new car costs £16 000. Age of car ( n years)Value (£ A ) 016 000 1 2 3 4 5 13 600 11 560 9826 8352 7099 What will it be worth when it is 5 years old? What will the car be worth when it is 20 years old? In this case the multiplier is0.85 Think about What assumption is being made? Is it realistic? B Depreciation Its value falls by 15% per year

7. © Nuffield Foundation 2012 Formula for annual sales n years from now Try this B A companys sales of a product are falling by 6% per annum. Estimate the annual sales 6 years from now. They sold 45 000 this year. = 0.94 n x 45 000 Estimate of annual sales 6 years from now = 0.94 6 x 45 000 about 31 000 Check this by repeated calculations. In this case the multiplier is 0.94 B Falling sales

8. © Nuffield Foundation 2012 C Combining percentage changes Number after receiving 3% extra = 103% of 2000 = 1.03 x 2000 A shareholder owns 2000 shares. How many shares will she have after these transactions? She expects to get 3% more shares then plans to sell 25% of her shareholding. = 2060 Number after selling 25%= 75% of 2060 = 0.75 x 2060 = 1545 What % is this of her original shareholding? = 77.25% or 1.03 x 0.75= 0.7725 1545 2000 100

9. © Nuffield Foundation 2012 Sale price = 75% of normal price = 75% of 130% of cost price Try this C A shop marks up the goods it sells by 30% What is the overall % profit or loss on goods sold in the sale? In a sale it reduces its normal prices by 25% The shop makes a 2.5% loss on goods it sells in the sale. = 0.975 of cost price = 0.75 x 1.3 x cost price C Combining percentage changes

10. © Nuffield Foundation 2012 D Reversing percentage changes 1.025 x previous price = £66.42 Previous price The price of a train fare increased by 2.5% recently. How much did it cost before the rise in price? It now costs £66.42 Previous price= £64.80 = £66.42 1.025

11. © Nuffield Foundation 2012 0.875 x full price = £25.90 Full price Try this D After a 12.5% discount, insurance costs £25.90 Full price = £29.60 = £25.90 0.875 What was the cost before the discount? D Reversing percentage changes

12. © 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?