A ratio is a comparison between two quantities.
Example
If there are 20 girls and 10 boys in a class, the ratio of girls to boys is 20 to 10. This is written as 20:10. Ratios, like fractions, can be simplified, so 20:10 could be written as 2:1.
In this example, the two 'quantities' being compared are girls and boys. 20 girls are compared with 10 boys. This comparison is summarised by using the colon symbol (:).
The quantities are always compared in the order of the statement. In the example, girls are compared with boys. So the ratio of girls to boys is 20:10. If the boys are compared with the girls, the ratio is 10:20, or 1:2.
The ratio 20:10 may be simplified to 2:1. The ratio 2:1 expresses the fact that there are 2 girls for every 1 boy in the class. The original ratio 20:10 not only expresses the fact that there are 20 girls for every 10 boys in the class, but also the fact that the class contains 30 pupils (see the page on proportion).
In the example above, like quantities are being compared: both girls and boys are pupils. Ratios must always compare two values of the same type of data. So if six hours are to be compared with one day, the ratio is 6:24 because there are 24 hours in a day.
Worked examples
Example one
A school has 360 key stage 3 (KS3) pupils and 200 key stage 4 (KS4) pupils. Show this information as a ratio.
The ratio which compares the number of KS3 pupils with the number of KS4 pupils is:
360:200
This can be simplified by dividing by 10 to:
36:20
This ratio shows that for every 36 KS3 pupils there are 20 KS4 pupils.
Simplifying this further by dividing each number by 4 gives:
9:5
The ratio 9:5 states that for every 9 KS3 pupils there are 5 KS4 pupils.
The ratio of KS3 pupils to KS4 pupils is 9:5.
Example two
In a 25-hour week of lessons a pupil has 5 hours of science and 2.5 hours of design and technology (D&T). Find the ratio of D&T hours to science hours and the ratio of science hours to the total lesson hours. Express both as ratios in their lowest form.
The ratio of D&T hours to science hours is:
2.5 to 5 hours or 2.5:5, which is 1:2 in its simplest form. So for every 2.5 hours of D&T, there were 5 hours of science.
The ratio of science hours to total lesson hours is: 5 to 25 or 5:25, which is 1:5 in its lowest form.
Avoiding common errors
Most common errors can be avoided by simplifying the ratio and ensuring the numbers in the ratio are in the same order as the required comparison.