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This page sets out the method for ratio in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Ratio topic page.
When you see 'share £180 in the ratio 4 : 5', add the parts first.
When you are told what one person gets, or how much more one gets than another, find one part from that.
Add the ratio numbers. That is how many equal parts the whole makes.
2:3 has 5 parts. £50 is £10 per part.
Multiply one part by each number in the ratio.
The shares are £20 and £30.
The shares add back to the original amount.
20 + 30 = 50.
Sometimes you are told how much more one person gets.
The difference in parts is worth that amount. Find one part from the difference.
Add all three ratio numbers before you divide.
Then multiply back for each person.
The first share in 2:3 is 2/5 of the total, not 2/3.
The denominator is the total number of parts.
The ratio 2:5 does not mean £2 and £5. It means 2 pieces and 5 pieces of the same size. There are 7 pieces altogether. If the total is £84, one piece is £84 ÷ 7 = £12. The shares are then 2 × 12 and 5 × 12.
Dividing by 2 or by 5 on their own skips the rest of the pieces. That is why the shares would not add back to £84. The add-back is the reason the method is trusted. Two correct shares are a partition of the original amount, so their sum is a check, not a new calculation to invent.
Method: 1. Add the numbers in the ratio to get the total number of parts. 2. Divide the amount by that total to find one part. 3. Multiply one part by each ratio number. 4. Add the shares and check they return the original amount.
If Bea receives more than Amir, the extra money is the extra pieces, not the whole share. In 3:7 Bea has 4 more pieces than Amir. A gap of £36 means each piece is £36 ÷ 4 = £9. Amir has 3 pieces, so Amir has £27.
Students sometimes divide the gap by one of the ratio numbers. That treats the gap as if it were a full share. The gap only covers the pieces that one person has and the other does not. Subtract the ratio numbers first. Then the division matches the story.
Method: 1. Subtract the smaller ratio number from the larger one. 2. Divide the money gap by that difference of parts. 3. Multiply one part by the ratio number you were asked for. 4. Subtract the two shares and check the gap returns.
If you know one person’s money, you know the value of their pieces. Dev has 3 pieces and receives £45, so one piece is £45 ÷ 3 = £15. Anyone else’s share is their number of pieces times £15. The whole prize is every piece times £15.
This is the same one-part idea run backwards. You are not dividing the unknown total. You are using a known bundle of parts to price a single part. Once that price is known, every other bundle is a multiplication. The ratio still decides how many pieces each person holds.
Method: 1. Read which ratio number matches the known share. 2. Divide the known share by that number to find one part. 3. Multiply one part by the ratio number you need. 4. Build the whole by multiplying one part by the sum of the parts.
Three names do not change the method. Add all three numbers. For 1:3:4 there are 8 pieces. A prize of £240 gives £30 a piece. The shares are £30, £90 and £120. The smallest ratio number still gets the smallest share.
Leave nobody out of the addition. Using only the first two numbers throws the size of one piece off, and then every share is wrong. After you multiply, add all three shares. They should return the prize exactly, because every piece has been given to someone.
Method: 1. Add all three ratio numbers. 2. Divide the total by that sum, or use a known share to price one part. 3. Multiply one part by each of the three numbers. 4. Add the three shares and check the total.
In a multi-step exam question, two people share money in a ratio, and then each of them gives away a fraction of their share. The order matters. First find the value of one part, using any amount you are given. Then find each person share. Only after that do you take the fraction of each share. A fraction of the whole amount would be wrong, because each person gives a fraction of their own money.
Work with one clear line at a time, such as "Priya share = 72" and "she gives 1/3 of 72 = 24". Labelling each line makes it easy to follow and protects the method marks. At the end, check that the totals add up. What was given away plus what is left must equal what each person had before.
Method: 1. Add the ratio parts and find the value of one part. 2. Work out each share by multiplying one part. 3. Take the stated fraction of each share, not of the total. 4. Combine the amounts as the question asks and check the totals.
Tip: Write every share on its own line. A fraction of the wrong amount is the usual slip in this type of question.
Some ratios are not in whole numbers, such as 2.6 : 1. The method does not change. Add the parts, 2.6 + 1 = 3.6, then divide the total by 3.6 to find one part. Work with a calculator and keep the full value. The final shares are often whole numbers, which is a good check. If you can, multiply the ratio by 10 to remove the decimal first, so that 2.6 : 1 becomes 26 : 10 or 13 : 5.
Another style gives a ratio with an unknown, such as 4 : n, and says one person gets 40% of the total. A percentage tells you the fraction of the total: 40% = 2/5. The first number in the ratio over the total of the parts is 4/(4 + n). Set the two fractions equal and solve, which gives 4 + n = 10 and n = 6.
Method: 1. Clear the decimals by multiplying the ratio, or work with the decimal total. 2. For an unknown part, write dfracparttotal parts as a fraction or percentage. 3. Put it equal to the given fraction or percentage and solve. 4. Check by substituting your value back into the ratio.
Sometimes you are given a : b and b : c and need a : b : c. The letter b appears in both, so make its value the same in each ratio. If a : b = 2 : 3 and b : c = 4 : 5, the number for b is 3 in one and 4 in the other. Use the lowest common multiple, 12, as the common value. Multiply the first ratio by 4 to get 8 : 12 and the second by 3 to get 12 : 15.
Then the combined ratio is 8 : 12 : 15. It can now be used to share an amount in the usual way, so the three parts add to 35. Copying the ratios side by side as 2 : 3 : 5 is a very common slip, since the middle term is not the same size in the two ratios. Check the final ratio by dividing the first two terms and comparing with the original.
Method: 1. Find the letter that appears in both ratios. 2. Use the lowest common multiple of its two values. 3. Multiply each ratio so that the common letter has that value. 4. Write the combined ratio, then share as normal.
Some questions say that one person gives money to another so that they end up with equal amounts or a new ratio. The total does not change. Find the total, then use the new ratio to work out each new share. The difference between old and new share is what moved. If both people gain or lose the same fixed amount, the ratio changes but the difference between their amounts stays the same.
When the same amount is added to both parts of a ratio, set up an equation. Let one part be k, so the amounts are 7k and 5k. After adding 2 to each, the new ratio gives an equation in k, which you cross-multiply. Solve for k, then substitute to get the amounts. Always check by adding the amount to your answers and dividing to see whether you reproduce the new ratio.
Method: 1. Work out the total, and whether it stays the same. 2. Use the starting ratio to find each starting amount. 3. Use the new ratio on the same total to find each new amount. 4. The difference between old and new is the amount that moved. Check with the final ratio.
Tip: If the total does not change, work with the total. If the same amount is added to both parts, work with the difference.
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