Ratio word problems — mixing ingredients, sharing money, comparing distances — ask you to find an unknown quantity from a given ratio. Write the ratio in simplest form, then choose the parts method (share a total) or the scaling method (find one part first) based on what the question gives you.
How do you identify a ratio problem?
A ratio problem usually contains key words or phrases:
- "shared in the ratio…"
- "for every x of A there are y of B…"
- "the mixture contains x parts A and y parts B…"
- "A and B are in the ratio x : y…"
Once you spot the ratio statement, write it out clearly before doing any calculations.
Method 1: the parts method (sharing a total)
Use this method when you are given the total and asked to find each share.
Steps:
- Add the parts of the ratio to find the total number of parts.
- Divide the given total by the total number of parts to find the value of one part.
- Multiply to find each share.
Worked example: £240 is shared between Anna and Ben in the ratio 3 : 5. How much does each person receive?
| Step | Working |
|---|---|
| Total parts | 3 + 5 = 8 |
| Value of 1 part | £240 ÷ 8 = £30 |
| Anna's share | 3 × £30 = £90 |
| Ben's share | 5 × £30 = £150 |
| Check | £90 + £150 = £240 ✓ |
Method 2: the scaling method (using one quantity to find another)
Use this method when you know one of the quantities and need to find the other.
Worked example: a recipe uses flour and sugar in the ratio 4 : 1. If you use 200 g of flour, how much sugar do you need?
- Flour is 4 parts; 200 g = 4 parts, so 1 part = 200 ÷ 4 = 50 g.
- Sugar is 1 part: 1 × 50 = 50 g.
Worked example: in a class the ratio of boys to girls is 3 : 4. There are 15 boys. How many girls are there?
- Boys = 3 parts; 15 boys = 3 parts, so 1 part = 15 ÷ 3 = 5.
- Girls = 4 parts: 4 × 5 = 20 girls.
Multi-step ratio problems
Some questions combine ratio with other operations.
Worked example: Priya and Quinn share sweets in the ratio 5 : 3. Quinn gives 6 of his sweets to Priya, and they now have sweets in the ratio 3 : 1. How many sweets did they start with?
| Before transfer | After transfer |
|---|---|
| Priya: 5k | Priya: 5k + 6 |
| Quinn: 3k | Quinn: 3k − 6 |
After transfer, ratio = 3 : 1: (5k + 6) / (3k − 6) = 3/1 5k + 6 = 9k − 18 24 = 4k k = 6
Total sweets = 5(6) + 3(6) = 30 + 18 = 48 sweets.
Check: after the transfer, Priya has 36 and Quinn has 12. Ratio = 36 : 12 = 3 : 1 ✓
Common problem structures
| Problem type | Key information | Method |
|---|---|---|
| Share a total in a ratio | Total given, ratio given | Parts method |
| Find one quantity from another | One quantity and the ratio given | Scaling method |
| Find the total from one share | One share and the ratio given | Find 1 part, multiply by total parts |
| Ratio with algebra (harder) | Expressions in ratio | Form and solve an equation |
Frequently asked questions
What if there are three quantities in the ratio?
The parts method works identically for three-part ratios. Add all three ratio numbers to get the total parts, find the value of one part, then multiply each ratio number by that value. For example, sharing 360 in the ratio 2 : 3 : 4 gives total parts = 9, one part = 40, and shares of 80, 120 and 160.
How do I know which method to use?
If the question gives you the total, use the parts method. If it gives you just one of the quantities (but not the total), use the scaling method to find the value of one part. In some harder questions you will need to set up an equation — this links to the algebraic ratio work at GCSE.
What if the total does not divide neatly into the required parts?
This usually means the question involves a fractional answer, which is fine. Alternatively, check whether you have simplified the ratio too far or added the parts incorrectly. If the answer should be a whole number and it isn't, re-read the problem.
How do I check a ratio answer?
For the parts method: add your individual shares together and confirm they equal the original total. For the scaling method: confirm the ratio of your two answers simplifies to the given ratio. Both checks take under a minute.
For step-by-step ratio help from Professor Pi at KS3, visit aitutors.me.