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KS4 Mathematics

KS4 Mathematics. N5 Percentages. N5 Percentages. Contents. A. N5.2 Percentages of quantities. A. N5.3 Finding a percentage change. A. N5.1 Fractions, decimals and percentages. N5.4 Increasing and decreasing by a percentage. A. N5.5 Reverse percentages. A. N5.6 Compound percentages.

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KS4 Mathematics

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  1. KS4 Mathematics N5 Percentages

  2. N5 Percentages Contents • A N5.2 Percentages of quantities • A N5.3 Finding a percentage change • A N5.1 Fractions, decimals and percentages N5.4 Increasing and decreasing by a percentage • A N5.5 Reverse percentages • A N5.6 Compound percentages • A

  3. Writing percentages as fractions 180 46 46 23 = 100 100 100 50 180 = = 9 4 1 7.5 15 3 100 15 = = 5 5 100 200 40 200 ‘Per cent’ means ‘out of 100’. To write a percentage as a fraction we write it over a hundred. For example, 23 46% = Cancelling: 50 9 180% = Cancelling: 5 3 Cancelling: 7.5% = 40

  4. Writing percentages as decimals 46 0.2 7 100 100 100 130 100 We can write percentages as decimals by dividing by 100. For example, = 46 ÷ 100 = 0.46 46% = = 7 ÷ 100 = 0.07 7% = = 130 ÷ 100 = 1.3 130% = = 0.2 ÷ 100 = 0.002 0.2% =

  5. Percentages as fractions and decimals

  6. Writing fractions as percentages × 5 85 17 = = 100 20 × 5 × 4 7 32 128 1 = = = 25 25 100 × 4 To write a fraction as a percentage, we can find an equivalent fraction with a denominator of 100. For example, 85 and 85% 100 128 and 128% 100

  7. Writing fractions as percentages 1 3 3 = × 100% 2 8 8 3 × 100% = 8 75% = 2 37 % = To write a fraction as a percentage you can also multiply it by 100%. Remember, multiplying by 100% does not change the value of the number because it is equivalent to multiplying by 1. For example, 25 2

  8. Writing decimals as percentages Decimals can also be converted to percentages by multiplying them by 100%. For example, 0.08 = 0.08 × 100% 1.375 = 1.375 × 100% = 8% = 137.5%

  9. Using a calculator 5 = 16 4 = 7 5 13 1 = = 8 8 We can also convert fractions to decimals and percentages using a calculator. For example, = 31.25% 5 ÷ 16 × 100% = 57.14% (to 2 d.p.) 4 ÷ 7 × 100% 13 ÷ 8 × 100% = 162.5%

  10. Table of equivalences

  11. Ordering on a number line

  12. One number as a percentage of another 3 7 4 4 35 35 80% = 4 × 100% 11 % 7 = × 100% = 35 There are 35 sweets in a bag. Four of the sweets are orange flavour. What percentage of sweets are orange flavour? Start by writing the proportion of orange sweets as a fraction. 4 out of 35 = Then convert the fraction to a percentage. 20 7

  13. One number as a percentage of another = 80 1 1 3200 40 40 100% × 100% = 40 Petra put £32 into a bank account. After one year she received 80p interest. What percentage interest rate did she receive? To write 80p out of £32 as a fraction we must use the same units. In pence, Petra gained 80p out of 3200p. We then convert the fraction to a percentage. 5 = 2.5% 2

  14. N5 Percentages Contents N5.1 Fractions, decimals and percentages • A • A N5.3 Finding a percentage change • A N5.2 Percentages of quantities N5.4 Increasing and decreasing by a percentage • A N5.5 Reverse percentages • A N5.6 Compound percentages • A

  15. Calculating percentages using fractions × 90 15 100 15 × 90 = 100 27 = 2 1 13 = 2 Remember, a percentage is a fraction out of 100. Find 15% of 90 15% of 90, means “15 hundredths of 90” or 3 9 20 2

  16. Calculating percentages using decimals What is 4% of 9? We can also calculate percentages using an equivalent decimal operator. 4% of 9 = 0.04 × 9 = 4 × 9 ÷ 100 = 36 ÷ 100 = 0.36

  17. Estimating percentages We can find more difficult percentages using a calculator. It is always sensible when using a calculator to start by making an estimate. For example, estimate the value of: 19% of £82  20% of £80 = £16 27% of 38m  25% of 40m = 10m 73% of 159g  75% of 160g = 120g

  18. Using a calculator 38% = 0.38 0 . 3 8 × 6 5 = One way to work out a percentage using a calculator is by writing the percentage as a decimal. For example, What is 38% of £65? So we key in: The calculator will display the answer as 24.7. We write the answer as £24.70

  19. Using a calculator 57 100 5 7 ÷ 1 0 0 × 8 = 0 We can also work out a percentage using a calculator by converting the percentage to a fraction. For example, What is 57% of £80? 57% = = 57 ÷ 100 So we key in: The calculator will display the answer as 45.6 We write the answer as £45.60

  20. Using a calculator 0 . 5 9 × 3 7 . = 5 We can also work out percentages on a calculator by finding 1% first and then multiplying by the required percentage. What is 37.5% of £59? 1% of £59 is £0.59 so, 37.5% of £59 is £0.59 × 37.5. We key in: And get an answer of 22.125 We write the answer as £22.13 (to the nearest penny).

  21. Calculating percentages

  22. N5 Percentages Contents N5.1 Fractions, decimals and percentages • A N5.2 Percentages of quantities • A • A N5.3 Finding a percentage change N5.4 Increasing and decreasing by a percentage • A N5.5 Reverse percentages • A N5.6 Compound percentages • A

  23. Finding a percentage increase or decrease actual increase Percentage increase = × 100% original amount actual decrease Percentage decrease = × 100% original amount Sometimes, we are given an original value and a new value and we are asked to find the percentage increase or decrease. We can do this using the following formulae:

  24. Finding a percentage increase 0.7 The percentage increase = × 100% 3.5 A baby weighs 3.5 kg at birth. After 6 weeks the baby’s weight has increased to 4.2 kg. What is the baby’s percentage increase in weight? The actual increase = 4.2 kg – 3.5 kg = 0.7 kg = 20%

  25. Finding a percentage decrease All t-shirts were £25 now only £17! What is the percentage decrease? 8 The percentage decrease = × 100% = 25 The actual decrease = £25 – £17 = £8 32%

  26. Finding a percentage profit 18 Her percentage profit = × 100% 32 A shopkeeper buys chocolate bars wholesale at a price of 32p per bar. She then sells the chocolate bar in her shop at 50p each. What is her percentage profit? Her actual profit = 50p – 32p = 18p = 56.25%

  27. Finding a percentage loss 0.46 Her percentage loss = × 100% 3.68 A share dealer buys a number of shares at £3.68 each. After a week the price of the shares has dropped to £3.22. What is her percentage loss? Her actual loss = £3.68 – £3.22 = 46p Make sure the units are the same. = 12.5%

  28. Finding a percentage change

  29. N5 Percentages Contents N5.1 Fractions, decimals and percentages • A N5.2 Percentages of quantities • A N5.3 Finding a percentage change • A N5.4 Increasing and decreasing by a percentage • A N5.5 Reverse percentages • A N5.6 Compound percentages • A

  30. Percentage increase Method 1 We can work out 20% of £150 000 and then add this to the original amount. The value of Frank’s house has gone up by 20% in three years. If the house was worth £150 000 three years ago, how much is it worth now? There are two methods to increase an amount by a given percentage. The amount of the increase = 20% of £150 000 = 0.2 × £150 000 = £30 000 The new value = £150 000 + £30 000 = £180 000

  31. Percentage increase Method 2 If we don’t need to know the actual value of the increase we can find the result in a single calculation. We can represent the original amount as 100% like this: 100% 20% When we add on 20%, we have 120% of the original amount. Finding 120% of the original amount is equivalent to finding 20% and adding it on.

  32. Percentage increase So, to increase £150 000 by 20% we need to find 120% of £150 000. 120% of £150 000 = 1.2 × £150 000 = £180 000 In general, if you start with a given amount (100%) and you increase it by x%, then you will end up with (100 + x)% of the original amount. To convert (100 + x)% to a decimal multiplier we have to divide (100 + x) by 100. This is usually done mentally.

  33. Percentage increase Here are some more examples using this method: Increase £50 by 60%. Increase £86 by 17.5%. 160% × £50 = 1.6 × £50 117.5% × £86 = 1.175 × £86 = £80 = £101.05 Increase £24 by 35% Increase £300 by 2.5%. 135% × £24 = 1.35 × £24 102.5% × £300 = 1.025 × £300 = £32.40 = £307.50

  34. Percentage decrease Method 1 We can work out 30% of £75 and then subtract this from the original amount. The amount taken off = 30% of £75 A CD walkman originally costing £75 is reduced by 30% in a sale. What is the sale price? There are two methods to decrease an amount by a given percentage. = 0.3 × £75 = £22.50 The sale price = £75 – £22.50 = £52.50

  35. Percentage decrease Method 2 We can use this method to find the result of a percentage decrease in a single calculation. We can represent the original amount as 100% like this: 100% 70% 30% When we subtract 30% we have 70% of the original amount. Finding 70% of the original amount is equivalent to finding 30% and subtracting it.

  36. Percentage decrease So, to decrease £75 by 30% we need to find 70% of £75. 70% of £75 = 0.7 × £75 = £52.50 In general, if you start with a given amount (100%) and you decrease it by x%, then you will end up with (100 – x)% of the original amount. To convert (100 – x)% to a decimal multiplier we have to divide (100 – x) by 100. This is usually done mentally.

  37. Percentage decrease Here are some more examples using this method: Decrease £65 by 20%. Decrease £320 by 3.5%. 80% × £65 = 0.8 × £65 96.5% × £320 = 0.965 × £320 = £52 = £308.80 Decrease £56 by 34% Decrease £1570 by 95%. 66% × £56 = 0.66 × £56 5% × £1570 = 0.05 × £1570 = £36.96 = £78.50

  38. Percentage increase and decrease

  39. N5 Percentages Contents N5.1 Fractions, decimals and percentages • A N5.2 Percentages of quantities • A N5.3 Finding a percentage change • A N5.5 Reverse percentages N5.4 Increasing and decreasing by a percentage • A • A N5.6 Compound percentages • A

  40. Reverse percentages I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them. What is the original price of the jeans? Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount. We can solve this using inverse operations. Let p be the original price of the jeans. £25.50 ÷ 0.85 = £30 p× 0.85 = £25.50 so p =

  41. Reverse percentages What is the original price of the jeans? × 0.85% Price before discount. ÷ 0.85% Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount. I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them. We can show this using a diagram: Price after discount.

  42. Reverse percentages

  43. Reverse percentages We can also use a unitary method to solve these type of percentage problems. For example, Christopher’s monthly salary after a 5% pay rise is £1312.50. What was his original salary? The new salary represents 105% of the original salary. 105% of the original salary = £1312.50 1% of the original salary = £1312.50 ÷ 105 100% of the original salary = £1312.50 ÷ 105 × 100 = £1250 This method has more steps involved but may be easier to remember.

  44. N5 Percentages Contents N5.1 Fractions, decimals and percentages • A N5.2 Percentages of quantities • A N5.3 Finding a percentage change • A N5.6 Compound percentages N5.4 Increasing and decreasing by a percentage • A N5.5 Reverse percentages • A • A

  45. Compound percentages A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? It is not 30%! When a percentage change is followed by another percentage change do not add the percentages together to find the total percentage change. The second percentage change is found on a new amount and not on the original amount.

  46. Compound percentages A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? To find a 20% decrease we multiply by 80% or 0.8. To find a 10% decrease we multiply by 90% or 0.9. A 20% discount followed by a 10% discount is equivalent to multiplying the original price by 0.8 and then by 0.9. original price × 0.8 × 0.9 = original price × 0.72

  47. Compound percentages A 20% discount followed by a 10% discount A 28% discount A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? The sale price is 72% of the original price. This is equivalent to a 28% discount.

  48. Compound percentages A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? Suppose the original price of the jacket is £100. After a 20% discount it costs 0.8 × £100 = £80 After an other 10% discount it costs 0.9 × £80 = £72 £72 is 72% of £100. 72% of £100 is equivalent to a 28% discount altogether.

  49. Compound percentages Jenna invests in some shares. After one week the value goes up by 10%. The following week they go down by 10%. Has Jenna made a loss, a gain or is she back to her original investment? To find a 10% increase we multiply by 110% or 1.1. To find a 10% decrease we multiply by 90% or 0.9. original amount × 1.1 × 0.9 = original amount × 0.99 Fiona has 99% of her original investment and has therefore made a 1% loss.

  50. Compound percentages

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