AC9M7N08 · Year 7 · Number

Ratios

ACARA v9 CONTENT DESCRIPTION “recognise, represent and solve problems involving ratios”

A ratio is a way of comparing quantities. Two parts squash to three parts cordial, three girls to two boys, one cup of sugar to four of flour: ratios appear in cooking, mixing, sharing and scaling all the time. They are the heart of proportional reasoning, one of the most useful threads running through Year 7 mathematics, and they build directly on the fractions and percentages you have already met.

This year you learn to recognise ratios, represent them clearly, and solve problems with them. The key is to see a ratio not as two separate numbers but as a fixed relationship between quantities, a relationship that holds even when the actual amounts grow or shrink.

Part to part and part to whole

A ratio compares amounts by counting parts. If a bar has 2 blue sections and 3 green, the ratio of blue to green is 2 to 3. This is a part-to-part comparison, matching one group against another. But you can also compare a part to the whole: blue to the entire bar is 2 to 5, since there are 5 sections in total. The single most common mistake with ratios is confusing these two, so it is worth pausing each time to ask whether a question wants part to part or part to whole.

A ratio splits into parts
Compare one part to another, or one part to the whole.
A ratio compares amounts by counting parts. With 2 blue and 3 green, the ratio of blue to green is 2 to 3, a part-to-part comparison. Blue to the whole bar is 2 to 5, a part-to-whole comparison. Knowing which one a problem asks for is half the work.

This distinction connects ratios to fractions. When the ratio of girls to boys is 3 to 2, the parts total 5, so girls make up 3 out of every 5 students, the fraction 3 fifths. Reading a part-to-whole ratio as a fraction is a powerful move, because it lets you bring everything you know about fractions and percentages to bear on a ratio problem.

Equivalent ratios

Ratios behave like fractions in another important way: you can scale them. Multiplying both parts by the same number gives an equivalent ratio that describes exactly the same relationship. 2 to 3 is the same as 4 to 6 and as 6 to 9, just as doubling or tripling a recipe keeps the flavour identical. The actual amounts change, but the balance between them does not.

Equivalent ratios
Scaling both parts by the same number gives the same ratio.
Multiplying both parts of a ratio by the same number gives an equivalent ratio. 2 to 3, 4 to 6 and 6 to 9 all describe the same split, just as a recipe doubled or tripled keeps the same balance of ingredients. The blue portion stays the same fraction of every bar.

Equivalent ratios are the engine that solves most ratio problems. If a recipe mixes sugar and flour in the ratio 1 to 4 and you use 3 cups of sugar, you scale the whole ratio up by 3 to get 3 to 12, so you need 12 cups of flour. The same scaling lets you simplify a ratio to its lowest terms, dividing both parts by a common factor, which is why 4 to 6 is usually written as 2 to 3. Recognising that a ratio can be scaled up or down without changing what it means is the central skill, and it is exactly the reasoning you will carry into rates, percentages and the modelling problems that follow.

Teaching tip: mixing drinks or paint makes ratios immediate. Mix one part cordial to four parts water, then make a bigger jug with two parts to eight, and taste that they are the same. Experiencing that the bigger batch tastes identical turns equivalent ratios from a rule into something the student has verified for themselves.

Keep watch for the part-to-part versus part-to-whole trap. When a class is 3 girls to 2 boys, a student may say girls are 3 fifths and then slip into calling the ratio of girls to boys 3 fifths as well. Naming clearly each time which two quantities are being compared heads this off before it takes root.

Builds on: Operations with rational numbers (AC9M7N06). That unit worked with fractions and percentages; this unit uses those forms to express and solve ratios.
Quick self-check
1. A fruit bowl has 4 apples and 6 oranges. What is the ratio of apples to oranges?
2. In a class the ratio of girls to boys is 3 : 2. What fraction of the class are girls?
3. Which ratio is equivalent to 2 : 3?
4. A recipe uses sugar and flour in the ratio 1 : 4. If you use 3 cups of sugar, how much flour?
5. A ratio compares blue to green as 2 : 5. This is best described as