How to Simplify and Divide Ratios in GCSE Maths

📋 Spec coverage: GCSE AQA Mathematics (8300) This article covers:

• AQA R4 - Ratio notation (Foundation): using ratio notation and reducing ratios to their simplest form.
• AQA R5 - Dividing quantities in a ratio (Foundation): part-to-part and part-to-whole ratios, expressing a division as a ratio, and applying ratios in contexts.

To simplify a ratio, divide every part by the same common factor. To divide an amount in a ratio, add the ratio parts, find the value of one part, then multiply.

Simplifying a ratio

A ratio is in its simplest form when its parts have no common factor greater than one.

Simplify $18:30$:

The highest common factor (HCF) of $18$ and $30$ is $6$.

$$18÷6:30÷6=3:5$$

Therefore, the simplified ratio is $3:5$.

If quantities have different units, convert them first. For example:

$$2m:50cm=200cm:50cm=4:1$$

Dividing an amount in a ratio

Divide $84$ in the ratio $3:4$.

First, find the total number of parts:

$$3+4=7$$

Find the value of one part:

$$84÷7=12$$

Calculate each share:

$$3\times 12=36$$ $$4\times 12=48$$

So the two shares are $36$ and $48$. Check that $36+48=84$.

For a part-to-whole ratio, such as $3:7$, the first quantity is $\frac{3}{7}$ of the total. The remaining quantity is $\frac{4}{7}$. These methods also apply to scaling, mixing, concentrations and comparisons.

If an amount is divided into $24$ and $40$, express the division as a ratio and simplify:

$$24:40=3:5$$

Common exam mistake

Do not divide the total by just one ratio number. Always divide by the sum of the parts. Also, simplify every part of a ratio by the same number.

Completeness check: AQA R4 is covered through ratio notation and simplification. AQA R5 is covered through part-to-part and part-to-whole division, reversing shares into a ratio, and applying the method to ratio contexts.