A Resource for Free-standing Mathematics Qualifications Working with %

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1 Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs ad its value falls by 1% per year. What will it be worth whe it is years old? What will it be worth whe it is 20 years old? A compay's sales of a product are fallig by 6% per aum. They sold 000 this year. If sales cotiue to fall at the same rate, estimate the aual sales 6 years from ow. A shareholder ows 2000 shares i a compay. She expects to get % more shares i a share issue ad the plas to sell 2% of her shareholdig. How may shares will she have after these trasactios? What percetage is this of her origial shareholdig? A shop marks up the goods it sells by 0%. I a sale it reduces its ormal prices by 2%. What is the overall percetage profit or loss o goods it sells i the sale? The price of a trai fare icreased by 2.% recetly. It ow costs 66.2 What was the cost before the price rise? After a 12.% discout, the isurace costs 2.90 What was the cost before the discout? The Nuffield Foudatio 1

2 Suppose you ivest 000 i a savigs accout at a fixed rate of % ad leave the moey i the accout over a umber of years. Assume iterest is added at the ed of each year. There are a umber of ways you ca work out how much will be i the accout at the ed of each year. Step-by-step Method Amout ivested 000 Iterest at ed of Year 1 = % of 000 = = 120 Amout at ed of Year 1 = 120 Iterest at ed of Year 2 = % of 120 = = Amout at ed of Year 2 = 2.80 ad so o. There are other methods. For example, you could divide by 100, the multiply by 0. Usig a multiplier Amout ivested 000 Amout at ed of Year 1 = 10% of 000 = = 120 Amout at ed of Year 2 = = 2.80 ad so o You ca do repetitive calculatios like this o your calculator. Complete this table to show the value of the ivestmet over the first years. Ed of year Amout ( ) Total = 100% of origial amout + % iterest Usig idices Amout ivested 000 Amout at ed of Year = eg Amout at ed of Year 2 = = 2.80 Amout at ed of Year = = 69.6 After years the amout will have bee multiplied by 1.0 times. What are the advatages ad disadvatages of each of these methods? The Nuffield Foudatio 2

3 Depreciatio Whe a quatity is reduced by a percetage, the multiplier is less tha 1. Suppose a car costs whe ew, but its value falls by 1% per year. I this case the car's value each year is 8% of its value the previous year ad the multiplier is 0.8. Use this multiplier to complete the table. You ca also fid the value of the car whe it is years old from the formula give below: Age of car years Value ( ) What assumptio is beig made by usig this method? Is this realistic. What will happe i the log ru? Value whe the car is years old = Use this formula to check the car's value whe it is years old. What will the car be worth whe it is 20 years old? Fallig Sales Suppose a compay's sales of a product are fallig by 6% per aum ad the umber sold this year was 000 Write dow a formula for aual sales years from ow. Use the formula to fid the estimated sales 6 years from ow. After years the umber will have bee multiplied by 0.9 times.. Check your aswer by usig repeated calculatios to complete the table. Sales after years Value ( ) The Nuffield Foudatio

4 Combiig % chages You ca combie percetage chages by usig oe multiplier followed by aother. Suppose a shareholder ows 2000 shares i a compay ad that she expects to get % more i a share issue. The she plas to sell 2% of her shareholdig. After receivig the extra % the umber she will have is = 2060 shares After sellig 2% she will have = 1 1 As a percetage of her origial shareholdig, this is 100 = 77.2% 2000 Multiplyig multipliers gives a quicker way to work this out. Check that = This method allows us to combie percetage chages without kowig the quatities. Combiig % chages without kowig the quatities Suppose a shop marks up the goods it sells by 0%, the i a sale it reduces its ormal prices by 2%. To fid the fial price as a % of the cost price work out: = 0.97 The shop is sellig goods at oly 97.% of cost price. They are losig 2.% o ay goods sold i the sale. Would they make ay profit if they reduced prices by 20% i the sale istead of 2%? Reversig % chages You ca reverse a percetage chage by dividig by the multiplier. Suppose that after a 2.% rise a trai fare costs 66.2 This is 102.% of the origial price (i.e times the origial price. So the origial price was = 6.80 You ca also reverse reductios. If isurace costs 2.90 after a 12.% discout, the this price is 87.% of the cost before the discout. The cost before the discout was = The Nuffield Foudatio

5 Teacher Notes Uit Advaced Level, Mathematical Priciples of Persoal Fiace Notes o Activity This activity ca be used at the start of the course to fid out what learers kow about percetage chages ad which methods they use. You ca ask studets to try the questios o page 1 idividually or discuss them i pairs or groups. The PowerPoit presetatio illustrates the differet methods ad shows how quickly solutios ca be foud if a multiplier is used. Pages 2 to give some of the worked solutios whilst leavig other parts for learers to complete. As well as discussig the solutios, you could also discuss how realistic assumptios such as a costat rate of depreciatio are. Oce they have completed pages 2 to, studets could keep them as a set of otes to provide a remider of the differet methods. Aswers Page 1 Amout i accout at the ed of years = Value of car after years = 7099, Value after 20 years = 620 Sales after 6 years = (earest 1000) Shareholder will have 1 shares. This is 77.2% of her origial shareholdig Shop loses 2.% o goods sold i the sale Price of trai fare before rise = 6.80 Price of isurace before discout = Page 2 Compoud Iterest Ed of year Amout ( ) Page Depreciatio Fallig Sales Formula for sales years from ow = Estimated sales 6 years from ow = (earest thousad) Age of car years Value ( ) Value after 20 years = 620 assumig the rate of depreciatio remais the same. Page Combiig % chages without kowig the quatities If the shop reduces prices i the sale by 20%, they make a overall profit of % Sales after years Value ( ) The Nuffield Foudatio

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