10/02/2016 Percentages A percentage is simply a fraction out of 100: 24 100 78 100 97 100 24% 78% 97% The symbol % is used to stand for βout of 100β. So, all of the fractions above can be written: To convert any fraction into a percentage, simply multiply it by 100: For example: 1 4 1 × 100 = 25% 4 Example A student scored 41 out of 60 possible marks, express their score as a percentage. The fraction of marks obtained is: 41 60 To convert it to a percentage, multiply it by 100: 41 × 100 = 68.3 % 60 Percentages Values, such as prices, can change; we need to be able to express these changes as percentages. The basic equation which affects these value changes is: πππ€ π£πππ’π = πππ π£πππ’π × π 100 π=π× π 100 which we will shorten to: Notice: If π = 100, there would be no change in value: π = π. If π < 100, there would be a decrease in value: π < π. If π > 100, there would be an increase in value: π > π. Example A PC costing £750 is reduced in price by 15%, what is its new price? To reduce the price by 15%, use: π = 100 β 15 = 85 π=π× π 100 π = 750 × 85 100 = £637.50 1 10/02/2016 Percentages Values, such as prices, can change; we need to be able to express these changes as percentages. The basic equation which affects these value changes is: πππ€ π£πππ’π = πππ π£πππ’π × π 100 π=π× π 100 which we will shorten to: Notice: If π = 100, there would be no change in value: π = π. If π < 100, there would be a decrease in value: π < π. If π > 100, there would be an increase in value: π > π. Example An employee earning £27,000 gets a 5% pay rise; what is his new salary? To raise the salary by 5%, use: π=π× π = 100 + 5 = 105 π 100 π = 27000 × 105 = £28,350 100 Percentages Example A house was bought for £75,000 and then resold for £87,000; what is the percentage increase in the value of the house? π=π× π 100 Here: π = 75,000 π = 87,000 Substitute these values into the equation and re-arrange for π: 87,000 = 75,000 × π= π 100 87,000 × 100 75,000 π = 116 The increase in percentage is therefore 16% 2 10/02/2016 Percentages Past Exam Question A language school is trying to encourage its clients to use their website. They offer a 12% discount to anyone who pays for their courses online. a) A 4 week summer course usually costs £429, how much would it cost if paid for online? b) An 8 week course cost £648. It was bought online, how much did it cost before the discount? a) π = π × π Here: 100 π = 429 π = 429 × b) π = π × π Here: 100 π =? π =? 88 = 377.52 100 π = 648 648 = π × π= π = 100 β 12 = 88 π = 100 β 12 = 88 88 100 648 × 100 88 π = 736.36 Percentage Profit Companies produce products and services in the hope of making a profit. ππππππ‘ = π ππ£πππ’π β πππ π‘ Where revenue is the money received by the company The cost is what the company pays out in order to produce its products and services. There are two ways of measuring the profit as a percentage: Profit over cost: ππππππ‘ × 100 πΆππ π‘ Profit margin: ππππππ‘ × 100 π ππ£πππ’π 3 10/02/2016 Percentage Profit Past Exam Question a) ππππππ‘ = π ππ£πππ’π β πππ π‘ Let π = ππ’ππππ ππ πππππ’πππ‘πππ π πππ. ππππππ‘ = 4500 π ππ£πππ’π = 5.50 × π πππ π‘ = 12,000 4500 = 5.5π β 12000 π= 4500 + 12000 5.5 π = 3000. b) Profit over cost: 4500 × 100 = 37.5% 12000 Percentage Profit Past Exam Question c) ππππππ‘ = π ππ£πππ’π β πππ π‘ π ππ£πππ’π = ππ’ππππ π πππ × πππππ = 2500π ππππππ‘ = 2500π β 12000 Profit over cost = 24% ππππππ‘ × 100 = 24 πΆππ π‘ 2500π β 12000 × 100 = 24 12000 2500π = 24 × 12000 + 12000 100 π = £ 5.95 4
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