4.03 Ratios, proportion and rates

Expressing and simplifying ratios

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$45 people in it, of which $25$25 are male and $20$20 are female. We can express the ratio of men to women as $25:20$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$25:45 because there are $25$25 males out of the $45$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.

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

Practice questions

Question 1

Rates

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:

$\text{Speed }=\frac{\text{Distance }}{\text{Time }}$Speed =Distance Time ​

or

$S=\frac{D}{T}$S=DT​

Practice questions

Question 2

Proportion

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

$3:5=$3:5=$15:25=$15:25=$\frac{1}{5}:\frac{1}{3}$15​:13​

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

You would need $10$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$5 would give the same answer.

Practice question

Question 3

Dividing quantities using ratio

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

If we were dividing a quantity of $4$4 items using a ratio of $1:3$1:3, it could be represented using these blue and green dots.

Here there are $4$4 parts in the ratio and the quantity being divided is $4$4. What happens if we have $40$40 items and we want to divide them in the ratio $1:3$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$1+3=4. Then we can divide the total quantity, which is $40$40 in this case by the total number of parts, which is $4$4, to give $10$10. Then using the ratio, you have a blue group of $1\times10=10$1×10=10 and a green group of $3\times10=30$3×10=30. Here we've multiple each term in the ratio by $10$10.

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

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

Practice questions

Question 4

The unitary method

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.

Practice questions

Question 5