Ratio: Simplifying and Equivalent Ratios

What you will learn

• How to read and write ratio notation using :.
• How a ratio compares quantities in the same order as the words in the question.
• How to simplify a ratio by dividing every part by a common factor.
• How to make equivalent ratios by multiplying or dividing every part by the same number.
• How to tell part-to-part language from part-to-whole language.
• Why it is risky to treat a ratio as a fraction before checking what the parts mean.

Key facts before you start

Simplifying and equivalent ratios infographic

Why this matters

Ratios appear in recipes, map scales, paint mixtures, currency, speed, density, probability and similar shapes.

If you can simplify and scale ratios accurately, later GCSE questions about sharing, direct proportion and rates become much more manageable.

Prior knowledge

You should already be comfortable with:

• multiplication and division facts,
• factors and highest common factor,
• equivalent fractions and simplifying fractions,
• reading worded questions carefully,
• checking whether an answer fits the context.

Clear explanation

Ratio notation

A ratio compares quantities. The colon symbol : is read as to.

This means there are 3 red parts for every 2 blue parts. The order matters: 3 : 2 is not the same as 2 : 3.

Simplifying ratios

To simplify a ratio, divide every part by the same common factor. Keep dividing until the parts have no common factor bigger than 1.

You must divide every part. For a three-part ratio, all three parts are treated together.

Equivalent ratios

Equivalent ratios compare quantities in the same way. Multiply or divide every part by the same number.

Part-to-part and part-to-whole language

In the ratio boys : girls = 2 : 3, the 2 compares boys with girls and the 3 compares girls with boys. That is part-to-part language.

The whole group has 2 + 3 = 5 equal parts. So boys are 25 of the whole group, not 23 of the whole group.

Do not treat ratios as fractions too early

A ratio such as 2 : 3 is not automatically the fraction 23 of the whole. First ask: what are the parts, and what is the whole?

Once the meaning is clear, fractions can be useful. Before that, they can lead to the wrong denominator.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For simplifying and equivalent ratios, marks usually come from scaling every part by the same factor, matching units before simplifying, dividing by the highest common factor where possible, keeping the order of the ratio parts exactly as the question gives them, and using the total number of parts only when the question asks for a fraction or share of the whole.

Common mistakes

• Changing only one part: equivalent ratios must multiply or divide every part by the same number.
• Stopping too early: 12 : 18 can become 6 : 9, but it is not fully simplified until 2 : 3.
• Reversing the order: if the question says red : blue, keep red first and blue second.
• Using the wrong whole: in 2 : 3, the total is 5 parts, so the first part is 25 of the whole.
• Treating ratios as fractions too early: decide whether the question is part-to-part or part-to-whole before writing a fraction.

Extension challenge

The ratio of adult tickets to child tickets sold is 5 : 8. A total of 78 tickets are sold. How many child tickets are sold?

Exam-board guidance

Ratio notation, simplifying ratios and equivalent ratios are shared GCSE Maths skills. They are often tested directly, then used inside longer proportion and problem-solving questions.

Next lesson

Next, use ratios to divide a total into unequal shares.