What Is a Ratio and How to Simplify Ratios

Ratios are one of those topics that seem simple on the surface but can get surprisingly tricky. This guide starts from the basics and builds up to the types of problems you will see in Year 7 and Year 8 maths.

What is a ratio?

A ratio compares two or more quantities of the same kind. It tells you how much of one thing there is relative to another.

If a class has 12 boys and 18 girls, the ratio of boys to girls is 12:18. This does not tell you the total number of students. It tells you the relationship between the two groups.

Ratios can also compare more than two quantities. A recipe might call for flour, sugar and butter in the ratio 3:2:1, meaning for every 3 parts flour, you use 2 parts sugar and 1 part butter.

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How to simplify ratios

Simplifying a ratio works just like simplifying a fraction. Divide all parts of the ratio by their highest common factor (HCF).

Example 1

Simplify 12:18.

The HCF of 12 and 18 is 6.

12 / 6 = 2

18 / 6 = 3

So 12:18 simplifies to 2:3.

Example 2

Simplify 45:30.

The HCF of 45 and 30 is 15.

45 / 15 = 3

30 / 15 = 2

So 45:30 simplifies to 3:2.

Example 3

Simplify 24:36:12.

The HCF of 24, 36 and 12 is 12.

24 / 12 = 2

36 / 12 = 3

12 / 12 = 1

So 24:36:12 simplifies to 2:3:1.

If you are not sure of the HCF, you can simplify in stages. For 24:36:12, you could first divide everything by 2 to get 12:18:6, then by 2 again to get 6:9:3, then by 3 to get 2:3:1. You reach the same answer.

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Ratios with decimals or fractions

Sometimes ratios involve decimals. The trick is to multiply all parts by the same number to get rid of the decimals, then simplify.

Example

Simplify 1.5:2.5.

Multiply both parts by 2 to remove the decimals: 3:5.

3 and 5 have no common factors, so 3:5 is the simplest form.

For fractions, multiply all parts by the lowest common denominator to clear the fractions first.

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Dividing a quantity in a given ratio

This is one of the most common ratio problems in Australian maths courses. You are given a total and a ratio, and asked to split the total accordingly.

The method

1. Add up the parts of the ratio to find the total number of parts.
2. Divide the total quantity by the total number of parts to find the value of one part.
3. Multiply each ratio number by the value of one part.

Example 1

Divide $120 in the ratio 3:5.

Total parts = 3 + 5 = 8

Value of one part = $120 / 8 = $15

First share = 3 x $15 = $45

Second share = 5 x $15 = $75

Check: $45 + $75 = $120. It adds up, so you know it is correct.

Example 2

Three friends share 240 lollies in the ratio 2:3:7. How many does each person get?

Total parts = 2 + 3 + 7 = 12

Value of one part = 240 / 12 = 20

First person = 2 x 20 = 40 lollies

Second person = 3 x 20 = 60 lollies

Third person = 7 x 20 = 140 lollies

Check: 40 + 60 + 140 = 240.

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Equivalent ratios

Just like equivalent fractions, you can multiply or divide all parts of a ratio by the same number to create an equivalent ratio.

2:3 is equivalent to 4:6, 6:9, 8:12 and so on. They all describe the same relationship.

This is useful when comparing ratios. To check if 6:9 and 10:15 are equivalent, simplify both: 6:9 = 2:3 and 10:15 = 2:3. They are the same.

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Common mistakes to avoid

• Mixing up the order: The ratio of boys to girls (12:18) is different from the ratio of girls to boys (18:12). Order matters.
• Comparing different units: If you are finding the ratio of 2 metres to 50 centimetres, convert to the same unit first. That is 200 cm to 50 cm, which simplifies to 4:1.
• Thinking a ratio is a fraction: The ratio 3:5 does not mean 3/5. It means 3 parts out of a total of 8 parts. The fraction of the total represented by the first part is 3/8, not 3/5.
• Not checking your answer: When dividing a quantity by a ratio, always add your answers together to make sure they equal the original total.

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Why ratios matter

Ratios are everywhere. Cooking recipes, mixing cordial, sharing costs, map scales, and enlarging or reducing images all use ratios. In later years, ratios lead into rates, proportional reasoning and similar figures in geometry.

If you want to build confidence with ratios through guided practice, imSteyn covers ratios and rates as part of the Year 7 and Year 8 Australian Curriculum. Instead of just marking answers, it asks you questions that help you work through each step and understand the reasoning behind it.

The most important thing to remember about ratios is that they describe a relationship, not a fixed amount. Once you see them that way, simplifying and dividing quantities becomes a clear, logical process rather than a set of rules to memorise.