Ratio, proportion and rates of change
Choose your practice ↓Ratio, proportion and rates of change is 25% of the Foundation paper — joint heaviest with number — and 20% of Higher, and it is the area where GCSE maths looks most like the maths people actually use. It covers ratio notation and simplest form, dividing a quantity in a given ratio, expressing one quantity as a fraction or a percentage of another, and the whole of percentage change: increase and decrease as a multiplier, reverse percentages where you are given the new value and asked for the original, and simple interest. Proportion follows, direct and inverse, graphically and algebraically, alongside the compound units — speed, density, pressure, rates of pay and unit pricing — and the conversions between related units of length, area, volume, capacity and mass. Higher adds the equations that describe direct and inverse proportion, compound growth and decay, area and volume scale factors in similar shapes, and reading an instantaneous rate of change from the gradient of a tangent to a curve.
📚 What you learn here
- Ratio notation, simplest form, and dividing a quantity in a part:part or part:whole ratio
- Expressing one quantity as a fraction or percentage of another, and comparing quantities with percentages
- Percentage increase and decrease as a multiplier, original value (reverse percentage) problems, and simple interest
- Direct and inverse proportion, graphically and algebraically; (Higher) constructing the proportion equation
- Compound units: speed, density, pressure, rates of pay and unit pricing
- Changing between related standard units of length, area, volume, capacity, mass and time
- Scale factors, scale diagrams and maps
- Similar shapes: the length scale factor, and (Higher) the area and volume scale factors
- Compound interest, growth and decay, and (Higher) general iterative processes
- The gradient of a straight line as a rate of change; (Higher) gradients of chords and tangents as average and instantaneous rates
Frequently asked questions
How do I do a reverse percentage question?
Divide, do not subtract. If a price is £84 after a 20% increase, the £84 is 120% of the original, so the original is 84 ÷ 1.2 = £70. Taking 20% off £84 gives £67.20, which is a different number and the single most common error in the topic.
What is the difference between simple and compound interest?
Simple interest is calculated on the original amount every year, so it adds the same sum each time. Compound interest is calculated on the running total, so it grows: £1000 at 3% compound for 4 years is 1000 × 1.03⁴. Compound growth and decay sits in this content area and is normally a calculator-paper question.
If a shape is enlarged by scale factor 3, what happens to its area?
It is multiplied by 9, not by 3 — areas scale by the square of the length scale factor, and volumes by its cube. This link between lengths, areas and volumes in similar figures is the Higher part of the statement.
Do I need to memorise the speed, density and pressure formulae?
Yes. They are not on the exam formulae sheet. Speed is distance ÷ time, density is mass ÷ volume, pressure is force ÷ area — and the units in the question tell you which one you need.
Why do so many ratio questions turn into fraction questions?
Because a ratio 3:5 is the same statement as the fractions 3/8 and 5/8 of the whole. Recognising when to move between the two forms is exactly what the 'fractions in ratio problems' statement is asking for.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications