Ratios and direct proportion
You already know fractions, decimals and percentages. Ratios are one more way to compare two quantities — how much of one corresponds to the other. You meet them most often in cooking, mixing paint, on maps, and in sports statistics.
What a ratio is
A ratio of two numbers a and b is written a : b and read "a to b". It tells you how many times more (or less) of the first quantity you have compared to the second.
In the class there are 12 girls and 18 boys. The ratio of girls to boys is 12 : 18.
Ratio in simplest form
Just like fractions, a ratio can be simplified. Divide both terms by a common factor — best of all by their greatest common factor (GCF).
12 : 18 = (12 ÷ 6) : (18 ÷ 6) = 2 : 3
The ratio 2 : 3 says the same as 12 : 18 — for every 2 girls there are 3 boys. It's the same ratio in its simplest form.
Equivalent ratios
You can also scale up a ratio — multiply both terms by the same number. The result is an equivalent ratio.
| Ratio | × 2 | × 3 | × 4 |
| 2 : 3 | 4 : 6 | 6 : 9 | 8 : 12 |
All five ratios say the same thing: for every 2 parts of the first, there are 3 parts of the second.
Direct proportion
When as one quantity grows, the other grows in the same ratio, we call it direct proportion.
If 1 bread roll costs 40 p, then 2 rolls cost 80 p, 5 rolls cost £2, 10 rolls cost £4.
The ratio of price to count stays the same: 40 p per roll. This is the unit rate.
The rule of three
Given three known values of a direct proportion, the fourth is found by the rule of three:
- Find what corresponds to one unit (divide).
- Multiply by the new number of units.
5 notebooks cost £7.50. How much do 8 notebooks cost?
1 notebook: £7.50 ÷ 5 = £1.50.
8 notebooks: £1.50 × 8 = £12.
What you will learn next
- Ratio in simplest form — simplify a ratio using the GCF.
- Equivalent ratio — fill in the missing term.
- Unit rate — find the price per item.
- Rule of three — from 3 values find the 4th.
- Recipe scaling — adjusting a recipe for more people.