Eduqas GCSE Mathematics Ratio, proportion and rates of change: a complete overview of ratio, proportion, percentages, compound measures and growth

What the Ratio content demands

Ratio, proportion and rates of change is where Number becomes useful. Eduqas dresses these topics in real contexts (recipes, best buys, maps, savings, speed and depreciation), so they are among the most common questions on the paper, and because problem solving (AO3) is worth a quarter of the marks, many appear as multi-step worded problems. The content runs from ratio sharing through proportion and percentages to compound measures and, at Higher tier, exponential growth and decay and graph interpretation. Component 2 (calculator) is where most of these are tested, but the methods must be secure on both components.

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

Ratio and scale

Simplify a ratio by dividing both parts by their highest common factor, after converting to the same units. To share a quantity in a ratio, add the parts for the number of shares, find one share, then multiply out each part. A scale such as $1 : 25000$ relates map distance to real distance: multiply to go from map to real, divide to go back. Similar shapes have lengths in a fixed ratio (the scale factor), with areas scaling by its square and volumes by its cube at Higher tier.

Direct and inverse proportion

Direct proportion, $y = kx$, means two quantities rise together; the unitary method (find one unit, then scale) solves it without algebra. Inverse proportion, $y = \dfrac{k}{x}$, means one rises as the other falls, with the product $xy$ constant. At Higher tier you form a proportion equation, find the constant of proportionality from a known pair, and use it, including proportion to a square, cube or square root.

Percentage change and interest

Convert every percentage change to a multiplier: $+15\%$ is $\times 1.15$, $-15\%$ is $\times 0.85$. Percentage change between values is $\dfrac{\text{change}}{\text{original}} \times 100$. A reverse percentage divides by the multiplier to recover the original. Compound interest raises the multiplier to the power of the number of periods, so 2000 pounds at $3\%$ for $4$ years is $2000 \times 1.03^4$.

Compound measures

Speed $= \dfrac{\text{distance}}{\text{time}}$, density $= \dfrac{\text{mass}}{\text{volume}}$, pressure $= \dfrac{\text{force}}{\text{area}}$, each rearranging to find any of the three. The marks-sensitive skill is units: convert time to hours for km/h, and multiply m/s by $3.6$ to get km/h (divide to reverse).

Growth, decay and rates of change

At Higher tier, exponential growth and decay apply a multiplier once per period: an amount after $n$ periods is (start) $\times (\text{multiplier})^n$. On a real-life graph the gradient is a rate of change (distance-time gradient is speed; speed-time gradient is acceleration), found at an instant by a tangent. The area under a speed-time graph is the distance, estimated by splitting into triangles and trapezia.

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 $18 : 24$. (1 mark)
2. Share 300 pounds in the ratio $2 : 3$. (2 marks)
3. Increase 80 by $25\%$. (2 marks)
4. A price falls from 50 to 42. Find the percentage decrease. (2 marks)
5. After a $20\%$ discount a coat costs 64 pounds. Find the original price. (3 marks)
6. A car travels 120 km in 1 hour 30 minutes. Find its average speed in km/h. (2 marks)
7. 2000 pounds is invested at $4\%$ compound interest per year. Find its value after 2 years. (2 marks)
8. $y$ is inversely proportional to $x$, and $y = 12$ when $x = 3$. Find $y$ when $x = 4$. (2 marks)