Ratio, explained with examples
A ratio compares the sizes of two or more quantities. Ratio questions appear on every GCSE Maths paper, and almost all of them can be solved with one idea: work out the value of one part.
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What a ratio means
A ratio such as 3 : 2 means that for every 3 of the first thing there are 2 of the second. If a fruit bowl has apples and oranges in the ratio 3 : 2, there might be 3 apples and 2 oranges, or 6 and 4, or 30 and 20. The ratio tells you the relationship, not the actual amounts. The order matters: 3 : 2 is not the same as 2 : 3.
Simplifying ratios
Divide every part by their highest common factor, just like simplifying a fraction. Make sure all parts are in the same units first, and remove decimals or fractions by multiplying.
Example 1: simplifying
Simplify 24 : 36, 1.5 : 2.5, and 40 cm : 2 m.
- 24 : 36: the HCF is 12, so 2 : 3.
- 1.5 : 2.5: multiply by 2 to get 3 : 5, which is 3 : 5.
- 40 cm : 2 m = 40 cm : 200 cm = 1 : 5. Converting units first is essential.
Ratios and fractions
A ratio can be turned into fractions of the whole. In the ratio 3 : 2, there are 3 + 2 = 5 parts in total, so the first quantity is 3/5 of the whole and the second is 2/5. Notice that the first quantity is not 3/2 of the whole; the denominator is the total number of parts. The first quantity is, however, 3/2 times the second quantity.
Ratios in the form 1 : n
Some questions ask for a ratio in the form 1 : n, which makes comparisons easy. Divide both parts by the first number. For example, 4 : 10 becomes 1 : 2.5. Map scales and gear ratios are often written this way, and n can be a decimal.
Sharing in a ratio
To share an amount in a given ratio:
- Add the parts of the ratio to find the total number of parts.
- Divide the amount by the total to find the value of one part.
- Multiply one part by each number in the ratio.
- Check that the shares add up to the original amount.
Example 2: sharing money
Share £120 in the ratio 3 : 5.
- Total parts: 8. One part: 120 ÷ 8 = £15.
- Shares: 3 × 15 = £45 and 5 × 15 = £75. Check: 45 + 75 = 120.
Example 3: three parts
Share 360 sweets between Ali, Ben and Cara in the ratio 2 : 3 : 7.
- Total parts: 12. One part: 30.
- Ali 60, Ben 90, Cara 210.
When you know one part, not the total
Many exam questions give the size of one share, or the difference between two shares, rather than the total. Use the given information to find one part first.
Example 4: from one share
Red and blue beads are in the ratio 4 : 7. There are 28 red beads. How many beads are there altogether?
- 4 parts = 28, so one part = 7.
- Total = 11 parts = 77 beads.
Example 5: from a difference
Sam and Tia share money in the ratio 2 : 5. Tia gets £45 more than Sam. How much does Sam get?
- The difference is 5 − 2 = 3 parts = £45, so one part = £15.
- Sam gets 2 × 15 = £30.
Ratios that change (Higher)
Harder questions describe a ratio that changes after something is added or removed. Use algebra: write each quantity in terms of an unknown, apply the change, and form an equation from the new ratio.
Example 6: a changing ratio
A box has red and green counters in the ratio 3 : 4. After 6 red counters are added, the ratio becomes 1 : 1. How many counters were there at the start?
- Let red = 3x and green = 4x. After adding 6 red: 3x + 6 = 4x, so x = 6.
- Start: 18 red and 24 green, so 42 counters.
Combining ratios (Higher)
If a : b and b : c are given separately, make the common quantity b the same in both, then combine.
Example 7: combining
a : b = 2 : 3 and b : c = 4 : 5. Find a : b : c.
- Make b the same: multiply the first by 4 and the second by 3. a : b = 8 : 12 and b : c = 12 : 15.
- a : b : c = 8 : 12 : 15.
Ratio in real life
- Recipes: a recipe for 4 people scaled up for 10 multiplies every amount by 10/4 = 2.5.
- Maps and scale drawings: a scale of 1 : 50,000 means 1 cm on the map is 50,000 cm = 500 m in real life.
- Mixing: paint or concrete mixed in a fixed ratio.
- Best buys: compare the cost per unit, such as price per 100 g, to decide which pack is better value.
Example 8: a map scale
On a map with scale 1 : 25,000, two villages are 8 cm apart. How far apart are they in real life?
- 8 × 25,000 = 200,000 cm.
- 200,000 cm = 2000 m = 2 km.
Common mistakes
- Dividing the total by one number of the ratio instead of by the total number of parts.
- Simplifying before converting to the same units.
- Writing the ratio in the wrong order.
- Giving the fraction as 3/2 instead of 3/5 for the ratio 3 : 2.
Practice questions
Try each one before opening the answer.
Q1. Simplify 45 : 60.
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Q2. Share £84 in the ratio 1 : 6.
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Q3. Boys and girls are in the ratio 5 : 4. There are 20 girls. How many students are there?
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Q4. Write 750 g : 1.5 kg in its simplest form.
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Frequently asked questions
Are ratio questions on the Foundation paper?
Yes. Simplifying, sharing and simple ratio problems are on both tiers. Changing ratios and combining ratios are mainly Higher.
What is the difference between ratio and proportion?
A ratio compares parts with each other, such as 3 : 2. A proportion compares a part with the whole, such as 3/5. Direct and inverse proportion describe how two quantities change together.
Key takeaways
- A ratio compares quantities; order matters.
- Simplify by dividing by the HCF, in the same units.
- Sharing: total parts, then one part, then each share.
- If you know one share or a difference, use it to find one part.
- For changing ratios, use algebra and form an equation.