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KS3 Mathematics

KS3 Mathematics. N7 Percentages. N7 Percentages. Contents. N7.2 Calculating percentages mentally. N7.3 Calculating percentages on paper. N7.1 Equivalent fractions, decimals and percentages. N7.4 Calculating percentages with a calculator. N7.5 Comparing proportions. N7.6 Percentage change.

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KS3 Mathematics

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  1. KS3 Mathematics N7 Percentages

  2. N7 Percentages Contents N7.2 Calculating percentages mentally N7.3 Calculating percentages on paper N7.1 Equivalent fractions, decimals and percentages N7.4 Calculating percentages with a calculator N7.5 Comparing proportions N7.6 Percentage change

  3. Percentages 1900 - 2000 Many words begin with ‘cent’:

  4. Percentages Percent means . . .

  5. Percentages 1% 1 part per hundred 1 100 0.01 A percentage is just a special type of fraction. means or =

  6. Percentages 10% 10 parts per hundred 10 1 0.1 100 10 A percentage is just a special type of fraction. means or = =

  7. Percentages 25% 25 parts per hundred 25 1 0.25 100 4 A percentage is just a special type of fraction. means or = =

  8. Percentages 50% 50 parts per hundred 50 1 0.5 100 2 A percentage is just a special type of fraction. means or = =

  9. Percentages 100% 100 parts per hundred 100 1 100 A percentage is just a special type of fraction. means or =

  10. Percentages of shapes

  11. Estimating percentages

  12. Estimating percentages

  13. Equivalent fractions, decimals and percentages

  14. 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

  15. 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% =

  16. Percentages as fractions and decimals

  17. 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

  18. Writing fractions as percentages 3 3 1 = × 100% 2 8 8 3 × 100% = 8 75% = 2 37 % = To write a fraction as a percentage you can also multiply it by 100%. For example, 25 2

  19. Writing decimals as percentages To write a decimal as a percentage you can multiply it by 100%. For example, 0.08 = 0.08 × 100% 1.375 = 1.375 × 100% = 8% = 137.5%

  20. Using a calculator 5 = 5 4 = = 16 7 8 1 13 = 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%

  21. Table of equivalences

  22. Table of equivalences

  23. Ordering on a number line

  24. Dominoes

  25. N7 Percentages Contents N7.1 Equivalent fractions, decimals and percentages N7.3 Calculating percentages on paper N7.2 Calculating percentages mentally N7.4 Calculating percentages with a calculator N7.5 Comparing proportions N7.6 Percentage change

  26. Calculating percentages mentally Some percentages are easy to work out mentally: 1% Divide by 100 To find 10% Divide by 10 To find 25% Divide by 4 To find 50% Divide by 2 To find

  27. Calculating percentages mentally 20% 30% 60% 15% 75% 11% 2% 49% 150% 17.5% 0.5 % We can use percentages that we know to find other percentages. Suggest ways to work out:

  28. Spider diagram

  29. N7 Percentages Contents N7.1 Equivalent fractions, decimals and percentages N7.2 Calculating percentages mentally N7.3 Calculating percentages on paper N7.4 Calculating percentages with a calculator N7.5 Comparing proportions N7.6 Percentage change

  30. Calculating percentages using fractions × 90 16 100 16 × 90 = 100 72 = 5 2 14 = 5 Remember, a percentage is a fraction out of 100. 16% of 90, means “16 hundredths of 90” or 4 18 25 5

  31. Calculating percentages using fractions What is 23% of 57? × 57 23 11 100 100 23 × 57 = 100 1311 = 100 13 = We can use fractions: Working × 20 3 23% of 57 = 50 1000 150 1150 7 140 21 + 161 1 3 1 1 1

  32. Calculating percentages using fractions What is 87% of 28? × 28 87 100 87 × 28 = 100 609 = 25 9 24 = 25 Using fractions again: Working 87% of 28 = 87 7 × 7 60 9 25 4

  33. 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

  34. N7 Percentages Contents N7.1 Equivalent fractions, decimals and percentages N7.2 Calculating percentages mentally N7.3 Calculating percentages on paper N7.4 Calculating percentages with a calculator N7.5 Comparing proportions N7.6 Percentage change

  35. 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

  36. Using a calculator 0 . 3 8 × 6 5 = By writing a percentage as a decimal, we can work out a percentage using a calculator. Suppose we want to work out 38% of £65. 38% = 0.38 So we key in: And get an answer of 24.7 We write the answer as £24.70

  37. 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. Suppose we want to work out 57% of £80. 57% = = 57 ÷ 100 So we key in: And get an answer of 45.6 We write the answer as £45.60

  38. Using a calculator 0 . 5 9 × 3 7 . = 5 We can also work out percentage on a calculator by finding 1% first and multiplying by the required percentage. Suppose we want to work out 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).

  39. N7 Percentages Contents N7.1 Equivalent fractions, decimals and percentages N7.2 Calculating percentages mentally N7.3 Calculating percentages on paper N7.5 Comparing proportions N7.4 Calculating percentages with a calculator N7.6 Percentage change

  40. 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

  41. One number as a percentage of another = 80 1 1 3200 40 40 100% 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 × 100% = = 2.5% 2

  42. Using percentages to compare proportions Matthew sat tests in English, Maths and Science. His results were: English Maths Science 74 17 66 80 20 70 Which test did he do best in? To compare the marks we can write each fraction as a percentage.

  43. Using percentages to compare proportions 74 66 74 66 17 17 70 70 20 20 80 80 English = × 100% = 74 ÷ 80 × 100% = 92.5% Maths = × 100% = 17 ÷ 20 × 100% = 85% Science = × 100% = 66 ÷ 70 × 100% = 94.3% (to 1 d.p.) We can see that Matthew did best in his Science test.

  44. Using percentages to compare proportions Chocolate Cookies Cheesy Crisps Nutrition Information Nutrition Information Typical Value Per 10g biscuit Typical Value Per 23 g bag Energy Protein Carbohydrate Fat Fibre Sodium 233kj 0.6g 6.7g 2.2g 0.2g <0.05g Energy Protein Carbohydrate Fat Fibre Sodium 504kj 1.6g 13g 7g 0.3g 0.2g Which product contains the smallest percentage of carbohydrate?

  45. Using percentages to compare proportions 6.7 10 13 23 The chocolate cookies contain 6.7g of carbohydrate for every 10g of biscuits. 6.7g out of 10g = × 100% = 6.7 ÷ 10 × 100% = 67% The cheesy crisps contain 13g of carbohydrate for every 23g of crisps. 13g out of 23g = × 100% = 13 ÷ 23 × 100% = 56.5% (to 1 d.p) The cheesy crisps contain a smaller percentage of carbohydrate.

  46. N7 Percentages Contents N7.1 Equivalent fractions, decimals and percentages N7.2 Calculating percentages mentally N7.3 Calculating percentages on paper N7.6 Percentage change N7.4 Calculating percentages with a calculator N7.5 Comparing proportions

  47. Percentage increase and decrease House prices predicted to fall by 2% next year SALE 20% off all marked prices! PC now only £568 Plus 17 % VAT Orange Shampoo 25% extra free! 1 2 Factory workers demand 15% pay increase Bus fares set to rise by 30%

  48. Percentage increase To increase an amount by a 20%, for example, we can find 20% of the amount and then add it on to the original amount. 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.

  49. Percentage increase For example, Increase £50 by 60%. 160% × £50 = 1.6 × £50 = £80 Increase £20 by 35% 135% × £20 = 1.35 × £20 = £27

  50. Percentage increase What happens if we increase an amount by 100%? We take the original amount and we add on 100%. 100% 100% We now have 200% of the original amount. This is equivalent to 2 times the original amount.

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