Ratio and proportion

Share quantities in a ratio, use unit rates and scales, and solve direct and inverse proportion problems.

Also known as: direct proportion, inverse proportion, sharing in a ratio, scale.

SAT · Problem-Solving and Data Analysis GCSE Maths · Ratio, proportion and rates of change IGCSE Maths · Number

Key ideas

To share in a ratio, find the value of one part first.
Direct proportion: $y = kx$ (doubling $x$ doubles $y$ ).
Inverse proportion: $y = \frac{k}{x}$ , so $xy$ is constant.
Convert units carefully when using scales.

Sharing in a ratio

To share $45$ in the ratio $2 : 3$ , there are $2 + 3 = 5$ parts, so one part is $45 \div 5 = 9$ . The shares are $18$ and $27$ .

If you know a difference instead of a total (e.g. "12 more girls than boys" in a ratio $3 : 5$ ), the difference is $5 - 3 = 2$ parts, so one part is $6$ .

Direct and inverse proportion

If $y \propto x$ , write $y = kx$ . With $y = 12$ when $x = 4$ , $k = 3$ , so when $x = 10$ , $y = 30$ .

If $y \propto \frac{1}{x}$ , write $y = \frac{k}{x}$ . With $y = 8$ when $x = 3$ , $k = 24$ , so when $y = 6$ , $x = 4$ .

Scales and unit rates

A map scale of $1 : 25\,000$ means $1$ cm represents $25\,000$ cm $= 250$ m. So $6$ cm represents $1.5$ km.

For "best value" or recipe problems, find the cost or amount for one unit first.

Worked examples

Example 1 · Ratio with a difference

The ratio of boys to girls is 3 : 5. There are 12 more girls than boys. How many students are there in total?

Step 1.Difference = 5 − 3 = 2 parts = 12, so one part = 6.
Step 2.Total = 8 parts = 48.

Answer: 48

Example 2 · Inverse proportion

$y$ is inversely proportional to $x$ , and $y = 8$ when $x = 3$ . Find $x$ when $y = 6$ .

Step 1. $xy = k = 24$ .
Step 2. $x = \frac{24}{6} = 4$ .

Answer: $x = 4$

Common mistakes

Dividing the total by one side of the ratio instead of the total number of parts.
Using direct proportion when quantities are inversely related (e.g. workers and time).
Forgetting to convert centimetres to kilometres (100 000 cm in 1 km).

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Ratio and proportionEasier

$45$ sweets are shared in the ratio $2 : 3$ . How many sweets are in the larger share?

Question 2Ratio and proportionEasier

$4$ pens cost $6$ . At the same price per pen, how much do $10$ pens cost?