OCR GCSE Mathematics Ratio, proportion and rates of change: a complete overview of ratio, proportion, percentages and rates

What the Ratio content demands

Ratio, proportion and rates of change applies number skills to real-life problems, and it is one of the most heavily examined areas of OCR GCSE Mathematics. The content runs from ratio and scale through proportion and percentages to compound measures and rates of change. Because the questions are usually contextual and multi-step, this area carries a large share of the AO3 problem-solving marks that OCR weights at $30$ percent.

This guide walks through the five areas of the Ratio content and ties together the matching dot-point pages, each of which has its own practice questions.

Ratio and scale

Simplify a ratio by dividing every part by the highest common factor, and write it as $1 : n$ to compare ratios or use scales. Divide a quantity in a ratio by adding the parts, finding one part, then multiplying out each share. A map or model scale such as $1 : 50000$ means each unit on the drawing represents that many units in reality, so convert between centimetres, metres and kilometres carefully. Similar shapes share a length scale factor.

Direct and inverse proportion

In direct proportion ($y = kx$) two quantities increase together; in inverse proportion ($y = \tfrac{k}{x}$) one increases as the other decreases. The unitary method finds the value of one unit first. At Higher tier, form a proportion equation with a constant of proportionality $k$, find $k$ from a given pair, then use the equation; proportion can be to $x$, $x^2$, $x^3$ or $\sqrt{x}$.

Percentage change and interest

Use a multiplier: a $15\%$ increase multiplies by $1.15$, a $15\%$ decrease by $0.85$. The percentage change between two values divides the change by the original. A reverse percentage divides the final value by the multiplier to recover the original. Compound interest applies the multiplier each period, so $n$ years of $r\%$ growth multiplies by $\left(1 + \tfrac{r}{100}\right)^n$.

Compound measures

Speed $= \tfrac{\text{distance}}{\text{time}}$, density $= \tfrac{\text{mass}}{\text{volume}}$, and pressure $= \tfrac{\text{force}}{\text{area}}$. Each rearranges to find any quantity, and a formula triangle helps. Units are the main pitfall: $30$ minutes is $0.5$ hours, and converting m/s to km/h multiplies by $3.6$. The same material has the same density, which links multi-part questions.

Growth, decay and rates of change

Exponential growth or decay applies the same multiplier each period: $P \times \left(1 \pm \tfrac{r}{100}\right)^n$. On a real-life graph the gradient is a rate of change (speed on a distance-time graph, acceleration on a speed-time graph), and the area under a speed-time graph is the distance travelled. These ideas underpin the later study of calculus.

Check your knowledge

A mix of ratio, proportion, percentage and compound-measure questions. Attempt them under timed conditions, then check against the solutions.

1. Simplify the ratio $15 : 25$. (1 mark)
2. Share £$360$ in the ratio $5 : 4$. (3 marks)
3. Increase £$240$ by $12\%$. (2 marks)
4. A coat costs £$63$ after a $10\%$ reduction. Find the original price. (3 marks)
5. $y$ is directly proportional to $x$. When $x = 4$, $y = 10$. Find $y$ when $x = 14$. (3 marks)
6. A car travels $180$ km in $3$ hours. Find its average speed. (2 marks)
7. £$1000$ is invested at $5\%$ compound interest. Find its value after $2$ years. (2 marks)
8. Convert $20$ m/s to km/h. (2 marks)