What Is Ratio and Proportion? KS3 Maths Explained

A ratio compares two or more quantities in the same units, written as a:b. Proportion states that two ratios are equal, or describes how one quantity changes relative to another. Both topics appear throughout KS3 and GCSE and underpin everyday problems from cooking and maps to mixing concrete.

What is a ratio?

A ratio tells you how much of one thing there is compared to another. If a fruit bowl contains 3 apples and 5 oranges, the ratio of apples to oranges is 3:5.

Notice two things:

The order matters. 3:5 (apples to oranges) is not the same as 5:3 (oranges to apples).

Ratios compare parts. Neither 3 nor 5 is a total — the total number of fruits is 3 + 5 = 8.

Key vocabulary

Term | Meaning
Ratio | A comparison of two or more quantities: a:b
Simplest form | A ratio written using the smallest possible whole numbers
Part | Each section a ratio is divided into
Proportion | A statement that two ratios are equal, or the fraction one part is of the whole

How do you simplify a ratio?

To simplify a ratio, divide every part by their highest common factor (HCF).

Worked example 1 — simplifying a ratio

Simplify the ratio 12:18.

Step 1: Find the HCF of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. HCF = 6.

Step 2: Divide each part by 6.

12 ÷ 6 = 2 and 18 ÷ 6 = 3

Simplified ratio: 2:3

Check: Can 2 and 3 be divided by the same number? No — so the ratio is fully simplified.

Worked example 2 — simplifying a three-part ratio

Simplify 8:12:20.

HCF of 8, 12 and 20 is 4.

8 ÷ 4 = 2, 12 ÷ 4 = 3, 20 ÷ 4 = 5

Simplified ratio: 2:3:5

How do you share an amount in a ratio?

This is one of the most common ratio questions at KS3 and GCSE.

Method: total parts → value of one part → multiply

Worked example 3 — sharing in a two-part ratio

Share £56 between Aisha and Ben in the ratio 3:5.

Step 1: Count the total number of parts. 3 + 5 = 8 parts.

Step 2: Find the value of one part. £56 ÷ 8 = £7 per part.

Step 3: Multiply each share.

Aisha: 3 × £7 = £21

Ben: 5 × £7 = £35

Check: £21 + £35 = £56 ✓

Worked example 4 — sharing in a three-part ratio

Concrete mix requires sand, cement, and gravel in the ratio 3:1:2. A builder uses 24 kg of sand. How much cement and gravel does he need?

Step 1: The sand part is 3. If 3 parts = 24 kg, then 1 part = 24 ÷ 3 = 8 kg.

Step 2: Cement = 1 × 8 = 8 kg. Gravel = 2 × 8 = 16 kg.

Total mix: 24 + 8 + 16 = 48 kg ✓

What is proportion?

Proportion describes a relationship where two quantities change together at the same rate. There are two types you need at KS3.

Direct proportion

Two quantities are in direct proportion if, when one doubles, the other doubles too. Their ratio stays constant.

Notation: y ∝ x means y is directly proportional to x, so y = kx where k is the constant of proportionality.

Worked example 5 — direct proportion

5 pens cost £3.50. How much do 8 pens cost?

Method 1 — unitary method (find the cost of 1 first):

Cost of 1 pen = £3.50 ÷ 5 = £0.70

Cost of 8 pens = 8 × £0.70 = £5.60

Answer: £5.60

Method 2 — scaling factor:

Scale factor = 8 ÷ 5 = 1.6

£3.50 × 1.6 = £5.60 ✓

Both methods give the same answer; at KS3, the unitary method is the safest to use.

Worked example 6 — proportion with a table

A car travels at a constant speed. Complete the table.

Distance (km) | 60 | 90 | 150 | ?
Time (min) | 40 | 60 | 100 | 80

Because the speed is constant, distance and time are in direct proportion. Find the constant:

k = distance ÷ time = 60 ÷ 40 = 1.5 km per minute

For time = 80 min: distance = 1.5 × 80 = 120 km

Missing value: 120 km

What is inverse proportion?

In inverse proportion, when one quantity doubles the other halves. For example, if 3 workers take 12 days to build a wall, 6 workers would take 6 days (assuming everyone works at the same rate).

workers × days = constant → 3 × 12 = 36, so 6 × ? = 36 → ? = 6 days

Inverse proportion is introduced in Year 9 and extends to GCSE.

How do ratios appear on the national curriculum?

The DfE's KS3 mathematics programme of study (published by the Department for Education) requires students to use ratio notation, simplify ratios, divide a quantity in a given ratio, and understand and use proportion, including percentage. The AQA GCSE Mathematics specification confirms that ratio and proportion questions appear on every GCSE maths paper and reward students who can move fluently between ratio, fraction, and percentage representations.

Common mistakes to avoid

Mistake | Example | Correct approach
Adding instead of multiplying when finding a share | Aisha gets 3 + 7 instead of 3 × 7 | Find value of 1 part first, then multiply
Forgetting that order matters | Writing 5:3 when the question says 3:5 | Read the question carefully; label each part
Not simplifying fully | Leaving 6:9 instead of 2:3 | Divide by the HCF, not just any common factor
Confusing ratio with fraction | Writing ratio 3:5 as the fraction 3/5 | The fraction of the whole is 3/(3+5) = 3/8

Frequently asked questions

What is the difference between a ratio and a fraction?

A ratio compares parts to each other (e.g. 3:5 means 3 of one thing for every 5 of another). A fraction compares one part to the whole. In the ratio 3:5, the fraction of the total that the first part represents is 3/(3+5) = 3/8. Knowing how to convert between the two forms is a key KS3 skill.

How do you check your answer when sharing in a ratio?

Add all the shares together. The total must equal the original amount. For example, if you share £56 in the ratio 3:5 and get £21 and £35, check: £21 + £35 = £56 ✓. If it does not add up, recheck your value of one part.

What is the unitary method in proportion?

The unitary method means finding the value of one unit first, then scaling. For example, if 4 books cost £10, the cost of 1 book is £10 ÷ 4 = £2.50. Then multiply by the new number of books. It works for any direct proportion problem and is reliable under exam pressure.

How do I know if two quantities are in direct or inverse proportion?

If one quantity increases and the other also increases at the same rate, they are in direct proportion (e.g. more items → higher cost). If one increases while the other decreases, they may be in inverse proportion (e.g. more workers → fewer days). Drawing a table of values and checking whether one quantity divided by the other is constant (direct) or whether their product is constant (inverse) will confirm which applies.

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