4.03 Ratios, proportion and rates

A ratio compares the relationship between two or more quantities of the same type. It shows how much there is of one thing compared to another.

Suppose a train carriage has 45 people in it, of which 25 are male and 20 are female. We can express the ratio of men to women as 25:20.

A ratio can express a part-to-part relationship, as in the example above. But it can also describe a part-to-whole relationship.

For example, the ratio of males to all passengers on the train is 25:45 because there are 25 males out of the 45 people in total.

The order that the words are written in the question typically corresponds to the order of the values in the ratio.

Ratios are simplified in the same way as a fraction . A ratio is said to be in its simplest form when all terms in the ratio have a highest common factor 1.

Consider the train carriage with the ratio of men to woman as 25:20. Simplified, this would be 5:4. In other words, for every 5 males there are 4 females.

When simplifying ratios ensure any units are are written in the same unit of measure. To compare lengths 65 cm and 3 musing a ratio, it is important to convert one of the units first. In this case, write both in centimetres (65:300) before simplifying to 13:60.

Examples

A rate is more general than a ratio since it compares different units. A common example of a rate is speed, which is often written in kilometres per hour or km/h. You can see that this describes a relationship between two measurements-kilometres and hours. We can write this relationship as:

\text{Speed}=\dfrac{\text{Distance}}{\text{Time}} or S=\dfrac{D}{T}.

A rate describes the relationship between different units like distance to time, or cents to grams.

Examples

When 2 rates or ratio are equal (equivalent) they are in proportion. The following ratios are in proportion:

3:5=15:25=\dfrac{1}{5}:\dfrac{1}{3}

You can use this concept of proportion to find a missing value by making equivalent ratios. If a recipe calls for 2 eggs for every 3 cups of flour, how many eggs are needed for 15 cups of flour? This can be written as:

You would need 10 eggs.

Alternatively, you could use equivalent fractions and make the denominators the same. In this case, multiplying the top and bottom of the left-hand-side fraction by 5 would give the same answer.

Examples

You may wish to divide a quantity by a given ratio.

If we were dividing a quantity of4 items using a ratio of 1:3, it could be represented using these blue and green dots.

Here there are 4 parts in the ratio and the quantity being divided is 4. What happens if we have 40 items and we want to divide them in the ratio 1:3?

First, calculate the total number of parts in the ratio, then use it to divide the quantity into a given ratio.

The total number of parts in the (part-part) ratio is found by adding all the parts. In this case 1+3=4. Then we can divide the total quantity, which is 40 in this case by the total number of parts, which is 4, to give 10. Then using the ratio, you have a blue group of 1 \times 10 = 0 and a green group of 3 \times 10 = 30. Here we've multiple each term in the ratio by 10.

To divide a quantity by a ratio you first identify the number of parts in the ratio.

A ratio of 3:8:1 would have 12 parts in total.

Examples

This is a method of carrying out a calculation to find the value of a number of items by first finding the cost of one of them. This method of solving problems is often handy for solving word problems.

Examples