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KS3 Mathematics

KS3 Mathematics. N8 Ratio and proportion. N8 Ratio and proportion. Contents. N8.2 Dividing in a given ratio. N8.1 Ratio. N8.3 Direct proportion. N8.4 Using scale factors. N8.5 Ratio and proportion problems. Stacking blocks. What is the ratio of red counters to blue counters?.

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KS3 Mathematics

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  1. KS3 Mathematics N8 Ratio and proportion

  2. N8 Ratio and proportion Contents N8.2 Dividing in a given ratio N8.1 Ratio N8.3 Direct proportion N8.4 Using scale factors N8.5 Ratio and proportion problems

  3. Stacking blocks

  4. What is the ratio of red counters to blue counters? A ratio compares the sizes of parts or quantities to each other. Ratio For example, red : blue = 9 : 3 = 3 : 1 For every three red counters there is one blue counter.

  5. For example, What is the ratio of blue counters to red counters? A ratio compares the sizes of parts or quantities to each other. Ratio The ratio of blue counters to red counters is not the same as the ratio of red counters to blue counters. blue : red = 3 : 9 = 1 : 3 For every blue counter there are three red counters.

  6. What is the ratio of red counters to yellow counters to blue counters? Ratio red : yellow : blue = 12 : 4 : 8 = 3 : 1 : 2 For every three red counters there is one yellow counter and two blue counters.

  7. ÷ 7 ÷ 7 ÷ 3 ÷ 3 Simplifying ratios Ratios can be simplified like fractions by dividing each part by the highest common factor. For example, 21 : 35 = 3 : 5 For a three-part ratio all three parts must be divided by the same number. For example, 6 : 12 : 9 = 2 : 4 : 3

  8. Equivalent ratio spider diagrams

  9. Simplifying ratios with units Simplifythe ratio 90p : £3 ÷ 30 ÷ 30 When a ratio is expressed in different units, we must write the ratio in the same units before simplifying. First, write the ratio using the same units. 90p : 300p When the units are the same we don’t need to write them in the ratio. 90 : 300 = 3 : 10

  10. Simplifying ratios with units Simplify the ratio 0.6 m : 30 cm : 450 mm ÷ 15 ÷ 15 First, write the ratio using the same units. 60 cm : 30 cm : 45 cm 60 : 30 : 45 = 4 : 2 : 3

  11. Simplifying ratios containing decimals Simplify the ratio 0.8 : 2 × 10 × 10 ÷ 4 ÷ 4 When a ratio is expressed using fractions or decimals we can simplify it by writing it in whole-number form. We can write this ratio in whole-number form by multiplying both parts by 10. 0.8 : 2 = 8 : 20 = 2 : 5

  12. Simplifying ratios containing fractions Simplify the ratio : 4 2 : 4 3 × 3 × 3 ÷ 2 ÷ 2 2 3 We can write this ratio in whole-number form by multiplying both parts by 3. = 2 : 12 = 1 : 6

  13. ÷ 5 ÷ 5 ÷ 8 ÷ 8 We can compare ratios by writing them in the form 1 : morm: 1, wheremis any number. Comparing ratios For example, the ratio 5 : 8 can be written in the form 1 : m by dividing both parts of the ratio by 5. 5 : 8 = 1 : 1.6 The ratio 5 : 8 can be written in the form m : 1 by dividing both parts of the ratio by 8. 5 : 8 = 0.625 : 1

  14. The ratio of boys to girls in class 9P is 4:5. The ratio of boys to girls in class 9G is 5:7. Which class has the higher proportion of girls? The ratio of boys to girls in 9P is 4 : 5 ÷ 4 ÷ 5 ÷ 4 ÷ 5 The ratio of boys to girls in 9G is 5 : 7 Comparing ratios = 1 : 1.25 = 1 : 1.4 9G has a higher proportion of girls.

  15. N8 Ratio and proportion Contents N8.1 Ratio N8.2 Dividing in a given ratio N8.3 Direct proportion N8.4 Using scale factors N8.5 Ratio and proportion problems

  16. Mixing paint

  17. Ratios and proportions on a metre rule

  18. Ratios and proportions on a metre rule

  19. Divide £40 in the ratio 2 : 3. Dividing in a given ratio A ratio is made up of parts. We can write the ratio 2 : 3 as 2 parts : 3 parts The total number of parts is 2 parts + 3 parts = 5 parts We need to divide £40 by the total number of parts. £40 ÷ 5 = £8

  20. Divide £40 in the ratio 2 : 3. and 3 parts = Dividing in a given ratio Each part is worth £8 so 2 parts = 2 × £8 = £16 3 × £8 = £24 £40 divided in the ratio 2 : 3 is £16 : £24 Always check that the parts add up to the original amount. £16 + £24 = £40

  21. A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1. How much of each type of juice is contained in 750 ml of the cocktail? Dividing in a given ratio First, find the total number of parts in the ratio. 6 parts + 3 parts + 1 part = 10 parts Next, divide 750 ml by the total number of parts. 750 ml ÷ 10 = 75 ml

  22. A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1. How much of each type of juice is contained in 750 ml of the cocktail? Dividing in a given ratio Each part is worth 75 ml so, 6 parts of orange juice = 6 × 75 ml = 450 ml 3 parts of lemon juice = 3 × 75 ml = 225 ml 1 part of lime juice = 75 ml Check that the parts add up to 750 ml. 450 ml + 225 ml + 75 ml = 750 ml

  23. Dividing in a given ratio spider diagram

  24. N8 Ratio and proportion Contents N8.1 Ratio N8.2 Dividing in a given ratio N8.3 Direct proportion N8.4 Using scale factors N8.5 Ratio and proportion problems

  25. What proportion of these counters are red? 3 4 Proportion compares the size of a part to the size of a whole. Proportion There are many ways to express a proportion. For example, We can express this proportion as: 12 out of 16 3 in every 4 0.75 or 75%

  26. Proportional sets

  27. 3 packets of crisps weigh 90 g. How much do 6 packets weigh? Direct proportion problems 3 packets weigh 90 g. × 2 × 2 6 packets weigh 120 g. If we double the number of packets then we double the weight. The number of packets and the weights are in direct proportion.

  28. 3 packets of crisps weigh 90 g. How much do 6 packets weigh? Direct proportion problems 3 packets weigh 90 g. ÷ 3 ÷ 3 1 packet weighs 30 g. If we divide the number of packets by three then divide the weight by three. Once we know the weight of one packet we can work out the weight of any number of packets.

  29. 3 packets of crisps weigh 90 g. How much do 7 packets weigh? × 7 × 7 Direct proportion problems 3 packets weigh 90 g. ÷ 3 ÷ 3 1 packet weighs 30 g. 7 packets weigh 210 g. This is called using a unitary method.

  30. N8 Ratio and proportion Contents N8.1 Ratio N8.2 Dividing in a given ratio N8.4 Using scale factors N8.3 Direct proportion N8.5 Ratio and proportion problems

  31. How can we get from 4 to 5 using only multiplication and division? Using scale factors We could divide 4 by 4 to get 1 and then multiply by 5. (4 ÷ 4) × 5 = 5 We could also multiply 4 by 5 to get 20 and then divide by 4. (4 × 5) ÷ 4 = 5

  32. We call the a multiplier or scale factor. 5 5 = = 4 4 × × 5 5 5 4 5 = 4 4 4 How can we divide by 4 and multiply by 5 in a single step? Using scale factors Dividing by 4 and multiplying by 5 is equivalent to 1.25 We can write the scale factor as a decimal, 125% We can also write it as a percentage,

  33. 5 4 ÷ 4 or 1 × × 1 5 4 4 Using a diagram to represent scale factors We can represent the scaling from 4 to 5 using a diagram: × 5

  34. How can we get from 5 to 4 using only multiplication and division? Using scale factors We could divide 5 by 5 to get 1 and then multiply by 4. (5 ÷ 5) × 4 = 4 We could also multiply 5 by 4 to get 20 and then divide by 5. (5 × 4) ÷ 5 = 4

  35. We call the a multiplier or scale factor. 4 4 = = 5 5 × × 4 4 4 5 4 = 5 5 5 How can we divide by 5 and multiply by 4 in a single step? Using scale factors Dividing by 5 and multiplying by 4 is equivalent to 0.8 We can write the scale factor as a decimal, 80% We can also write it as a percentage,

  36. 5 4 ÷ 5 or 1 × × 1 4 5 5 Using a diagram to represent scale factors We can represent the scaling from 5 to 4 using a diagram: × 4

  37. 5 4 To scale from 4 to 5 we multiply by To scale from 5 to 4 we multiply by 4 5 × × 4 5 5 4 Inverse scale factors When we scale from a smaller number to a larger number the scale factor must be more than 1. 4 5 When we scale from a larger number to a number smaller the scale factor must be less than 1.

  38. Scale factor diagrams

  39. Scale factor diagrams

  40. × × × × 9 3 4 7 b a 4 3 9 7 b a Using scale factors To scale atob we multiply by To scale btoa we multiply by For example, 4 9 7 3

  41. £8 is worth 13 euros. How much is £2 worth? 1 To scale from £8 to £2 we or × 0.25 × 4 1 1 × × 4 4 or × 0.25 or × 0.25 We can use scale factors to solve problems involving direct proportion. Using scale factors and direct proportion £8 is worth 13€ £2 is worth (13 ÷ 4)€ = 3.25€

  42. £8 is worth 13 euros. How much is £2 worth? 13 Alternatively, to scale from 8 to 13 we or × 1.625 × 8 13 13 or × 1.625 or × 1.625 × × 8 8 We can use scale factors to solve problems involving direct proportion. Using scale factors and direct proportion £8 is worth 13€ £2 is worth (2 × 1.625)€ = 3.25€

  43. £8 is worth 13 euros. How much is £2 worth? 13 or × 1.625 × 8 8 × or × 0.615 (to 3 dp) 13 We can use scale factors to solve problems involving direct proportion. Using scale factors and direct proportion We can convert between any number of pounds or euros using pounds euros

  44. Using scale factors and direct proportion

  45. N8 Ratio and proportion Contents N8.1 Ratio N8.2 Dividing in a given ratio N8.5 Ratio and proportion problems N8.3 Direct proportion N8.4 Using scale factors

  46. Direct proportion spider diagrams

  47. Cog wheels

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