Ratio: Simplifying, Dividing Quantities, Solving Problems — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers ratio in the forms the specification requires. By the end of this guide you should be able to write and simplify a ratio, including ratios with units and ratios written in the form 1 : n; divide a quantity into a given ratio; and use a ratio together with one known part to find the other parts or the whole.

You should also be able to move between ratios and fractions, handle three-part ratios, combine two ratios that share a common quantity, and recognise when a problem gives you a difference between parts rather than a total.

The central idea throughout is the share: work out what one part of the ratio is worth, and every other quantity in the question follows from it by multiplication. Almost every ratio problem at GCSE reduces to finding the value of one share.

Key terms and definitions

Ratio — a comparison of two or more quantities, written with colons, as in 3 : 5.

Part (or share) — one unit of the ratio. A ratio of 3 : 5 has 8 parts in total, three of them in the first quantity and five in the second.

Simplifying — dividing every number in the ratio by the same value until no common factor remains, in the same way a fraction is cancelled.

Unitary form — a ratio written as 1 : n, obtained by dividing both sides by the first number. Useful for direct comparison and for scales.

Equivalent ratios — ratios describing the same relationship, such as 2 : 3, 4 : 6 and 10 : 15.

Total parts — the sum of the numbers in the ratio, and the number you divide the total quantity by when sharing.

Core concepts

Writing and simplifying a ratio

A ratio compares quantities of the same kind. If a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18.

Simplify exactly as you would a fraction, by dividing both sides by their highest common factor. Here the HCF of 12 and 18 is 6, so 12 : 18 simplifies to 2 : 3.

Order matters. The ratio of boys to girls is 2 : 3, but the ratio of girls to boys is 3 : 2. Always check which way round the question has asked.

Ratios with units

Before simplifying a ratio involving measurements, convert both quantities to the same unit. A ratio compares pure numbers, so units must match or the comparison is meaningless.

To simplify 40 cm : 2 m, convert the metres to centimetres: 40 cm : 200 cm. Dividing both by 40 gives 1 : 5. Simplifying 40 : 2 without converting would give 20 : 1, which is not merely wrong but inverted.

Once both quantities are in the same unit, the units are dropped and the ratio is written as bare numbers.

The form 1 : n

Dividing both sides by the first number produces a ratio of the form 1 : n. This makes two ratios easy to compare and is the standard way of writing a map or model scale.

To write 4 : 10 in the form 1 : n, divide both by 4: 1 : 2.5. Note that n need not be a whole number, and insisting that it is will lose the mark.

A scale of 1 : 50 000 on a map means one unit on the map represents 50 000 of the same unit on the ground.

Ratios and fractions

A ratio of 2 : 3 describes a whole made of 5 parts. The first quantity is therefore 2/5 of the total and the second is 3/5.

This is the single most common confusion in the topic. In the ratio 2 : 3, the first quantity is not 2/3 of the total — it is 2/3 of the second quantity, which is a different statement. When a question mentions a fraction of the total, add the parts first.

Dividing a quantity in a given ratio

This is the standard calculation, and it always follows the same three steps.

Add the parts to find the total number of shares. Divide the quantity by that total to find the value of one share. Multiply by each number in the ratio to find each portion.

To share £60 in the ratio 2 : 3: the total is 5 parts, so one part is 60 ÷ 5 = £12. The two portions are 2 × 12 = £24 and 3 × 12 = £36.

Always check that the portions add back to the original total. Here 24 + 36 = 60. ✓ That check takes a few seconds and catches almost every arithmetic slip.

Problems giving one part rather than the total

Many questions give the value of one share of the ratio instead of the whole. The method is the same, but the first step changes: divide the known quantity by its number of parts.

If money is shared in the ratio 3 : 7 and the smaller share is £21, then 3 parts are worth £21, so one part is £7. The larger share is 7 × 7 = £49, and the total is 10 × 7 = £70.

Problems giving a difference

A third type gives the difference between two shares. The difference corresponds to the difference in the parts.

If a ratio is 3 : 7 and one person receives £28 more than the other, then the difference of 7 − 3 = 4 parts is worth £28, so one part is £7, exactly as above.

Recognising which of the three — total, one part, or difference — the question has given is what decides the first line of working.

Three-part ratios and combining ratios

Three-part ratios work identically: 2 : 3 : 5 has ten parts altogether.

Where two ratios share a quantity, they can be combined by scaling until the shared quantity matches. If A : B = 2 : 3 and B : C = 6 : 7, scale the first ratio so that B is 6 in both: multiplying 2 : 3 by 2 gives 4 : 6. Now A : B : C = 4 : 6 : 7.

Worked examples

Example 1: Dividing in a ratio

Share £84 between two people in the ratio 5 : 7.

Total parts: 5 + 7 = 12.

One part: 84 ÷ 12 = £7.

The shares: 5 × 7 = £35 and 7 × 7 = £49.

Check: 35 + 49 = 84. ✓ The larger number in the ratio has produced the larger share, which is a further sense check worth making.

Example 2: Given one part, find the total

Sand and cement are mixed in the ratio 4 : 1. A builder uses 3 kg of cement. How much sand is needed, and what is the total mass of the mixture?

Cement is 1 part, and that part is 3 kg. So one part = 3 kg.

Sand is 4 parts: 4 × 3 = 12 kg.

Total is 5 parts: 5 × 3 = 15 kg.

The wording matters here. A student who treated 3 kg as the total would divide rather than multiply and reach 2.4 kg of sand, which is wrong. Identify which quantity the given figure belongs to before calculating.

Example 3: Simplifying with units, and the 1 : n form

Write the ratio 250 g : 2 kg in its simplest form, and then in the form 1 : n.

Convert to a common unit first: 2 kg = 2000 g, so the ratio is 250 : 2000.

The HCF of 250 and 2000 is 250, so dividing gives 1 : 8.

That is already in the form 1 : n, with n = 8. Had the simplified ratio been, say, 2 : 5, the 1 : n form would require dividing both by 2 to give 1 : 2.5.

Common mistakes and how to avoid them

Forgetting to convert units. Always bring both quantities to the same unit before simplifying, or the ratio will be wrong by whatever the conversion factor was.

Confusing ratio with fraction. In 2 : 3, the first share is 2/5 of the total, not 2/3. Add the parts whenever a fraction of the whole is wanted.

Reversing the order. "The ratio of cats to dogs" must be written cats first. Re-read the question before writing the ratio down.

Dividing the total when only one part was given. Check whether the figure in the question is the whole amount, one share, or the difference between shares. The first line of working depends entirely on this.

Multiplying a percentage-style shortcut onto a ratio. Ratios are shared out in parts; there is no shortcut that avoids finding the value of one part.

Leaving the answer as a number of parts. The question asks for money, mass or length, so convert back and include the units.

Exam technique for "Ratio"

Write the total number of parts as an explicit first line. It is the step examiners most often award a method mark for, and it commits you to the correct denominator.

State clearly what one part is worth before finding any share. Every remaining quantity in the question follows from that one number, so making it visible turns a multi-mark question into a short sequence of multiplications.

Check that the shares add back to the total. It costs almost nothing and detects nearly every arithmetic error.

Read carefully to identify whether you have been given the total, one share, or the difference between shares — and underline that figure in the question.

Give the answer in the units asked for. A ratio problem that ends with "7" rather than "£7" or "7 kg" is incomplete.

Quick revision summary

Simplify a ratio by dividing all parts by their highest common factor, exactly as with a fraction. Convert to a common unit first whenever measurements are involved.

The form 1 : n comes from dividing both sides by the first number, and n does not have to be a whole number.

In the ratio a : b, the total is a + b parts, and the first quantity is a/(a + b) of the whole — not a/b.

To divide a quantity: add the parts, divide to find one part, then multiply for each share. Check that the shares sum to the original.

If the question gives one share, divide that figure by its own number of parts. If it gives the difference between shares, divide by the difference in the parts.

Combine two ratios sharing a quantity by scaling until that shared quantity matches in both.