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.

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 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.

Simplifying a ratio

Just as a fraction is usually written in lowest terms, a ratio is tidiest when its parts share no common factor. Picture the parts as dots and bundle them into equal groups: 4 dots and 6 dots bundle into 2 groups and 3 groups when the bundle size is 2, so 4 to 6 simplifies to 2 to 3. Dividing both parts by their common factor changes the numbers but not the balance, exactly the same move that reduces an equivalent fraction.

Sharing a quantity in a ratio

A common ratio task is dividing a total in a given ratio, such as splitting 20 lollies between two children 2 to 3. The trick is to count the parts first: 2 and 3 make 5 equal parts, so each part is 20 divided by 5, which is 4. The first child gets 2 parts, 8 lollies, and the second gets 3 parts, 12 lollies. The shares always add back to the original total, a quick way to check the answer is right.

Scaling to find a missing amount

Many real ratio problems give you one quantity and ask for its partner. A double number line handles this cleanly: put one part of the ratio on the top line and the other on the bottom, then stretch both by the same factor. With sugar to flour at 2 to 3, scaling up by 4 lines up 8 cups of sugar against 12 cups of flour. Reading straight across the two lines keeps the ratio fixed while the amounts grow, which is the heart of solving for a missing value.

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.