A ratio compares quantities: 3 : 4 means 3 parts of the first for every 4 parts of the second. Simplify a ratio by dividing every part by the highest common factor. To share an amount in a ratio, add the parts, find the value of one part, then multiply. A ratio also gives fractions: in 3 : 7, the first part is 3/10 of the total.
Ratio methods
- Simplify: divide every part by the HCF
- Share: total ÷ number of parts = one part; then multiply
- Units: convert to the same unit before simplifying
- Fractions: in a : b, the first part is a/(a + b) of the whole
- Difference problems: difference in parts = difference in amounts
Writing and simplifying ratios
A ratio such as 18 : 24 compares two quantities. Divide both parts by their highest common factor, 6, to get the simplest form 3 : 4. A simplified ratio has whole-number parts with no common factor. If the quantities have different units, convert them first: 75 cm : 2 m becomes 75 : 200, which simplifies to 3 : 8.
Sharing in a ratio
Add the parts to find the total number of parts, divide the amount by this to find one part, then multiply for each share. To share £63 in the ratio 4 : 5, there are 9 parts, one part is £7, and the shares are £28 and £35. Check that the shares add up to the original amount.
Three-part ratios
The same method works with three or more parts. To share 600 in the ratio 2 : 3 : 7, there are 12 parts, so one part is 50, giving shares of 100, 150 and 350.
Ratios and fractions
A ratio tells you what fraction each part is of the whole. In 3 : 7 there are 10 parts, so the first quantity is 3/10 of the total and the second is 7/10. This is why a ratio question can be answered with fractions, and vice versa.
Problems using the difference
Sometimes you are told the difference between two shares instead of the total. If Ali and Beth share money in the ratio 7 : 4 and Ali gets £36 more, the difference is 3 parts, so one part is £12 and the total of 11 parts is £132.
Worked examples
Example 1. A box contains 18 red pens and 24 blue pens. Write the ratio of red to blue in its simplest form.
- 18 : 24.
- Divide both by 6.
Answer: 3 : 4.
Example 2. Share £63 in the ratio 4 : 5.
- Total parts: 9. One part: £63 ÷ 9 = £7.
- 4 × 7 = 28 and 5 × 7 = 35.
Answer: £28 and £35.
Example 3. Write 75 cm : 2 m in its simplest form.
- Same units: 75 cm : 200 cm.
- Divide by 25.
Answer: 3 : 8.
Example 4. Share 600 in the ratio 2 : 3 : 7.
- Total parts: 12. One part: 50.
- 2 × 50, 3 × 50 and 7 × 50.
Answer: 100, 150 and 350.
Example 5. Ali and Beth share money in the ratio 7 : 4. Ali receives £36 more than Beth. How much money is shared?
- The difference is 7 − 4 = 3 parts, worth £36, so one part is £12.
- Total: 11 parts × £12.
Answer: £132.
Common mistakes
- Dividing by the ratio parts instead of the total. For 4 : 5, divide by 9, not by 4 or 5.
- Not converting units first. 75 cm : 2 m is not 75 : 2.
- Writing the ratio in the wrong order. Keep the order given in the question.
- Not simplifying fully. Divide by the highest common factor.
- Using the total when the difference is given. Match the difference in parts to the difference in amounts.
How ratio is examined
Ratio is one of the most common topics on Edexcel GCSE (1MA1) and International GCSE (4MA1) papers, at both tiers, and it often appears inside problem-solving questions. You may be asked to:
- simplify a ratio, including with different units
- share an amount in a given ratio
- find a total from one share or from the difference between shares
- link ratios to fractions
Work through the sharing in a ratio worksheet, which has answers, and use the GCSE proportion revision engine for more.
Ratio leads to direct and inverse proportion and, at Higher tier, ratio with algebra.
Frequently asked questions
How do you simplify a ratio?
Divide every part by the highest common factor.
How do you share an amount in a ratio?
Add the parts, divide the amount by the total to find one part, then multiply each part by that value.
How is a ratio related to a fraction?
In a ratio a : b, the first part is a/(a + b) of the whole.