Ratio: simplifying ratios and sharing in a ratio

What is a ratio?

A ratio compares the sizes of two or more amounts. It's written with a colon: if a recipe uses 3 cups of flour for every 1 cup of sugar, the ratio of flour to sugar is 3:1. Children meet this in Year 6, where the national curriculum uses examples like "for every egg you need three spoonfuls of flour" and introduces the a:b notation.

The order matters. Flour to sugar is 3:1, but sugar to flour is 1:3. Always write the parts in the order the question names them.

A ratio doesn't tell you the actual amounts, only how they compare. 3 cups to 1 cup, 6 cups to 2 cups and 30 grams to 10 grams are all in the ratio 3:1.

How to calculate the ratio of two numbers

To find the ratio between two amounts, write them in order with a colon, then simplify. A class has 12 boys and 18 girls, so the ratio of boys to girls is 12:18, which simplifies to 2:3.

The amounts must be in the same units first. To compare 50p with £2, change £2 into 200p. The ratio is 50:200, which simplifies to 1:4. Don't write the units in the ratio itself.

How to simplify ratios

To simplify a ratio, divide every part by the same number, just like simplifying fractions. The ratio is in its simplest form when no number other than 1 divides into every part.

Worked example: simplify 24:36

1. Both numbers are even, so halve them: 24 ÷ 2 = 12 and 36 ÷ 2 = 18. That gives 12:18.
2. Both divide by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. That gives 2:3.
3. Nothing but 1 divides into both 2 and 3, so 2:3 is the simplest form.

Dividing by the highest common factor does it in one step. The HCF of 24 and 36 is 12, and 24 ÷ 12 = 2 and 36 ÷ 12 = 3. Prime factors show how to find the HCF of bigger numbers.

The same works for three parts. 6:9:15 divides by 3 to give 2:3:5.

If a ratio has decimals, multiply first to make whole numbers. For 1.5:2, multiply both parts by 2: 1.5 × 2 = 3 and 2 × 2 = 4, so the ratio is 3:4.

How to share in a ratio

To share an amount in a ratio, add the parts to find how many equal parts there are, divide to find one part, then multiply for each share. The national curriculum calls this unequal sharing in Year 6, and dividing a quantity in a given ratio at secondary school. A bar model shows it clearly.

Worked example: share £35 in the ratio 3:2

1. Add the parts: 3 + 2 = 5, so there are 5 equal parts.
2. Find one part: 35 ÷ 5 = 7, so each part is worth £7.

3. The first share has 3 parts: 3 × 7 = 21, so £21.
4. The second share has 2 parts: 2 × 7 = 14, so £14.

Check that the shares add back to the total: 21 + 14 = 35.

Worked example: 48 sweets in the ratio 5:3

1. Add the parts: 5 + 3 = 8.
2. Find one part: 48 ÷ 8 = 6.
3. Multiply: 5 × 6 = 30 and 3 × 6 = 18.

So the shares are 30 sweets and 18 sweets, and 30 + 18 = 48.

Worked example: when you know the difference

Sam and Ali share some cards in the ratio 4:1. Sam has 36 more cards than Ali. How many does each have?

The difference between the shares is 4 − 1 = 3 parts, and those 3 parts are 36 cards. So one part is 36 ÷ 3 = 12. Ali has 1 part, 12 cards. Sam has 4 parts: 4 × 12 = 48 cards.

Worked example: when you know one share

Squash is mixed with water in the ratio 1:4. How much water goes with 50 ml of squash? One part is 50 ml, so the water is 4 parts: 4 × 50 = 200, so 200 ml of water.

Ratio and fractions

A ratio compares part with part. A fraction compares part with the whole. To turn a ratio into fractions, add the parts to make the denominator. In the ratio 3:2 there are 5 parts, so the first share is ³⁄₅ of the total and the second is ²⁄₅.

This gives another way to share. For £35 in the ratio 3:2, ³⁄₅ of 35 = 21, the same as before. Fractions of amounts explains that step, and fractions, decimals and percentages shows how to turn a ratio into a percentage. The basics are on the fractions page.

Ratio and proportion

Ratio and proportion are closely linked. A ratio compares amounts; two amounts are in proportion when their ratio stays the same as they change. If a recipe for 4 people uses 300 g of pasta, a recipe for 8 people uses 600 g, because 4:300 and 8:600 are the same ratio. The GCSE content describes proportion as "equality of ratios". Direct and inverse proportion goes into this in detail.

Ratio questions at GCSE

At GCSE, ratio questions use the same skills in real settings. The GCSE subject content lists simplifying, sharing in part:part and part:whole ratios, and using ratio for conversion, comparison, scaling, mixing and concentrations, as well as fractions in ratio problems. Most questions come down to the steps above: find what one part is worth, then use it.

When do children learn ratio?

Ratio starts in Year 6 maths, with problems about relative sizes, recipes and unequal sharing. At key stage 3 children use ratio notation properly, simplify ratios and divide amounts in a given ratio. Key stage 3 covers Years 7 to 9 as one stage, and each school decides when to teach each topic, so ask the school for its scheme of work if you want to know when ratio comes up. The KS3 maths guide has the rest of the key stage. Ratio comes back at GCSE.

Common mistakes with ratio

• Getting the order wrong. Boys to girls is not the same as girls to boys.
• Dividing by one part instead of the total. For £35 in the ratio 3:2, divide by 5, not by 3 or 2.
• Writing a ratio as the wrong fraction. 3:2 doesn't mean ³⁄₂. The first share is ³⁄₅ of the total.
• Mixing units. Change everything into the same unit before writing the ratio.
• Not simplifying all the way. 12:18 simplifies to 6:9, but 2:3 is the simplest form.
• Not checking. The shares must add up to the total.

How to help at home

• Cook together. Doubling or halving a recipe is ratio in action. Ask how much of each ingredient you'd need for more people.
• Mix drinks. Squash and water, or paint colours, are ratios your child can see and taste.
• Share fairly and unfairly. Share sweets or coins in a ratio like 2:1 and ask your child to work out each share first.
• Keep times tables sharp. Simplifying and sharing are all multiplying and dividing. The times tables pages help.

Ratio questions to try

1. Simplify 15:25.
2. Simplify 8:12:20.
3. Share £40 in the ratio 3:5.
4. Share 60 in the ratio 7:5.
5. Red and blue counters are in the ratio 2:7. There are 14 red counters. How many blue counters are there?
6. Cats and dogs at a rescue centre are in the ratio 3:2. What fraction of the animals are dogs?

Answers

1. 3:5 (divide both by 5)
2. 2:3:5 (divide every part by 4)
3. £15 and £25 (40 ÷ 8 = 5, then 3 × 5 = 15 and 5 × 5 = 25)
4. 35 and 25 (60 ÷ 12 = 5, then 7 × 5 = 35 and 5 × 5 = 25)
5. 49 (one part is 14 ÷ 2 = 7, and 7 × 7 = 49)
6. ²⁄₅ (there are 3 + 2 = 5 parts, and dogs are 2 of them)