KS3 ratio: notation, simplifying and sharing in a ratio

Ratio and sharing in a ratio, explained

A ratio such as 3 : 5 compares two quantities, here saying that for every 3 of one there are 5 of the other. The order matters: a class with 12 boys and 18 girls has a boys to girls ratio of 12 : 18 but a girls to boys ratio of 18 : 12. Like a fraction, a ratio is simplified by dividing both parts by a common factor, so 12 : 18 becomes 2 : 3, the simplest form, where the parts have no common factor other than 1. If the parts are measured in different units, they must be converted first, so 50 cm to 2 m is 50 : 200, which simplifies to 1 : 4.

Sharing in a ratio is the main skill. In a part:part ratio like 3 : 5, add the parts to find the total number of equal shares (3 + 5 = 8), divide the amount by that number to find one share, and multiply to find each person’s part. Sometimes the information is part:whole instead, such as “3 out of every 8 tickets sold were for children”, which can be written as children : all tickets = 3 : 8. Then the 8 is already the whole, so there is no adding: divide the total by 8, multiply by 3 for the children’s tickets, and the other 5 shares make up the rest. The reverse is also taught: when a quantity of 36 is split into 20 and 16, children write the split as the ratio 20 : 16, which simplifies to 5 : 4.

Ratios are connected to fractions. If two people share money in the ratio 3 : 5, the first gets 3/8 of the total and the second 5/8, because the whole is 8 parts. Ratio also describes a multiplicative relationship, meaning one quantity is a number of times the other. If A : B is 3 : 5, then A is 3/5 of B and B is 5/3, or 1⅔, times A. Children learn that the same relationship can be expressed as a ratio or as a fraction, and which is best depends on the question.

The last link is to linear functions. A recipe that needs 2 cups of rice for every 5 cups of water gives the rule water = 2.5 × rice, which is a linear function. On a graph of water against rice it is a straight line through the origin whose gradient is the 2.5, so doubling one quantity always doubles the other. Seeing ratio, fraction, equation and graph as different views of the same relationship is the heart of Key Stage 3 proportional reasoning.

Words your child will hear

Ratio

A comparison of quantities, written with a colon, such as 3 : 5.

Simplest form

A ratio whose parts have no common factor except 1, such as 2 : 3 instead of 12 : 18.

Equivalent ratios

Ratios that describe the same relationship, such as 2 : 3 and 6 : 9.

Part:part ratio

A ratio comparing one part with another, such as 2 red to 3 blue.

Part:whole ratio

A ratio comparing one part with the total, such as 3 out of every 8.

Share

One of the equal portions into which a total is split when sharing in a ratio.

Proportion

A part compared with the whole, often written as a fraction (3/8 of the class). Two quantities are “in proportion” when they change by the same factor.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

• Use ratio notation, including reduction to simplest form
• Divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio
• Understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction
• Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Common mistakes and how to help

How to help at home

• Cook with ratios. Recipes are ratios: rice and water, flour and butter, squash and water. Scale a recipe up for guests or down for one, and ask your child to explain how they kept the ratio the same.
• Share something fairly. Share a bag of 40 sweets or a sum of pocket money in a ratio such as 3 : 2 and check the answer adds up. Swap the order and see what changes.
• Spot ratios in the world. Look for the ratio of boys to girls in a team, of red cars to the rest, or of screen time to outdoor time. Write each as a ratio, simplify it, and then turn it into a fraction.
• Mix a drink. Mix a drink with 1 part squash to 4 parts water and ask what fraction of the drink is squash (1/5). Then ask how much water goes with 150 ml of squash (600 ml). A question with a purpose is easier to remember.

Related topics

Everything in Key Stage 3 maths