Ratio and Proportion
Number
The method
- Add the parts so the whole is expressed as a number of parts
- One part is then the quantity divided by that total
- Each share follows by multiplying a single part by its own number in the ratio
- Check by adding the shares back to the original amount, which catches an arithmetic slip immediately
Exam tip
One part earns the method mark
Sharing in a ratio is set in 12 of the 22 papers, and every scheme gives a method mark for the value of one part: the amount divided by the number of parts it stands for. That value, or any share worked out from it, earns the mark even when the final answer goes wrong. So write one part on its own line before working out what the question asks for.
Worked example
Dividing a quantity in a given ratio
£540 is divided between two people in the ratio 4 : 5. Work out each share.
Solution:
- A ratio of 4 : 5 means the money is cut into 4 + 5 = 9 equal parts, not into 4 or 5
- Finding the value of one part is the step every ratio-sharing question turns on
- One part = 540 ÷ 9 = £60
- The first share takes 4 of those parts: 4 × 60 = £240
- The second takes 5 of them: 5 × 60 = £300
- Check by adding the shares back: 240 + 300 = 540, which is the amount divided
- A second check is the ratio itself: 240 : 300 cancels by 60 to 4 : 5
- Keep the shares in the order the question set them, since giving £300 to the first person is a different answer
Worked example
Dividing into three parts
96 counters are divided in the ratio 3 : 4 : 5. Work out how many counters are in each group.
Solution:
- The method does not change when there are three parts instead of two
- Total parts = 3 + 4 + 5 = 12
- One part = 96 ÷ 12 = 8 counters
- The three groups take 3, 4 and 5 of those parts
- 3 × 8 = 24, 4 × 8 = 32 and 5 × 8 = 40
- The groups hold 24, 32 and 40 counters
- Check by adding: 24 + 32 + 40 = 96
- The division came out exactly, which it must: a whole number of counters cannot be split into fractions, so a total that does not divide by the number of parts is a signal to re-read the question
When the question gives a difference instead of a total
- The difference between two shares is worth the difference between their two numbers of parts
- Divide to find one part, then continue exactly as before
- In the ratio 7 : 4 one share exceeds the other by 33, and 7 − 4 = 3 parts, so one part is 11
- The shares are 77 and 44, and the total is 11 parts, or 121
When the question gives one share instead of a total
- Match the known quantity against the number of parts it stands for, then divide to find one part
- Paint is mixed red to white in the ratio 5 : 2, and 45 litres of red are available
- 5 parts is 45 litres, so one part is 9 litres, giving 18 litres of white and 63 litres of mixture
Worked example
When the amount given is not the total
(a) Some money is shared between three people in the ratio 2 : 5 : 9. The third person receives £117. Work out the total amount shared.
(b) Two people share some money in the ratio 4 : 9. The second person receives £40 more than the first. Work out the total amount they share.
Solution:
- In both parts the amount given is not the total, so first decide how many parts it stands for
- (a) The £117 is the third person's own share, which is 9 parts
- One part = 117 ÷ 9 = £13
- The total is 2 + 5 + 9 = 16 parts, so it is 16 × 13 = £208
- Check: the three shares are £26, £65 and £117, which add to £208
- (b) The £40 is the gap between the two shares, which is 9 − 4 = 5 parts
- One part = 40 ÷ 5 = £8
- The total is 4 + 9 = 13 parts, so it is 13 × 8 = £104
- Check: the shares are £32 and £72, which differ by £40
- Dividing by 16 in (a), or by 13 in (b), would treat the given amount as the total, which it is not
- Both questions ask for the total, so stopping at one person's share leaves the answer unfinished
Combining two ratios
- Two two-part ratios that share a quantity join into a single three-part ratio
- Scale each ratio so that the shared quantity is represented by the same number in both, using the lowest common multiple of its two values
- Write the three parts out in one line once they agree
- With A : B = 3 : 4 and B : C = 6 : 5, the shared B is 4 and 6, whose lowest common multiple is 12
- Scaling gives A : B = 9 : 12 and B : C = 12 : 10, so A : B : C = 9 : 12 : 10