GCSE Higher — Ratio, proportion and rates of change
On GCSE Higher, ratio, proportion and rates of change is worth 20% of the marks — enough that a weak patch here shows up in the final grade. 16 statements make up ratio, proportion and rates of change here. Each has its own page with the specification wording, the common mistakes, and questions. The list includes converting between standard and compound units, scale factors, scale diagrams and maps, one quantity as a fraction of another, ratio notation and simplest form, dividing in a ratio and multiplicative relationships as ratios or fractions. The Higher-only part of this area is instantaneous rate of change: gradients of curves. The assessment objectives split 40% AO1 against 60% AO2 and AO3 combined, which is why so many questions ask you to explain or to justify rather than simply to calculate. Most of this area is natural Paper 1 material, so practise it without a calculator before you practise it with one.
- R1 — Converting between standard and compound units
- R2 — Scale factors, scale diagrams and maps
- R3 — One quantity as a fraction of another
- R4 — Ratio notation and simplest form
- R5 — Dividing in a ratio
- R6 — Multiplicative relationships as ratios or fractions
- R7 — Proportion as equality of ratios
- R8 — Ratios, fractions and linear functions
- R9 — Percentages and percentage change
- R10 — Direct and inverse proportion
- …and 6 more ratio, proportion and rates of change statements at this tier
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Ratio, proportion and rates of change at GCSE Higher: the 16 DfE statements
Every question on this site is filed against one of these statements. Pick one to practise it on its own.
Sample ratio, proportion and rates of change questions for GCSE Higher
- The price of a share falls by 10% on Monday and then rises by 10% on Tuesday. Work out the overall percentage change from Monday's starting price.(a)0%(b)−1%(c)−2%(d)+1%
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−1% — Method: write each change as a multiplier and multiply them. A 10% fall is × 0.9 and a 10% rise is × 1.1. Working: 0.9 × 1.1 = 0.99, so the final price is 99% of the original, which is 1% less. Answer: an overall change of −1%. The distractors: 0% comes from assuming a 10% fall and a 10% rise cancel — they do not, because the rise is 10% of a smaller amount; +1% has the size right but the sign wrong, from reading the multiplier 0.99 as 1% above 1 instead of 1% below it; −2% comes from finding the 1% fall and then counting it once for each of the two changes. - The price of a jacket increases by 50% and then decreases by 50%. Describe the overall change from the original price.(a)no change(b)an increase of 25%(c)a decrease of 25%(d)a decrease of 50%
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a decrease of 25% — Method: use multipliers. An increase of 50% is × 1.5 and a decrease of 50% is × 0.5. Working: 1.5 × 0.5 = 0.75, so the final price is 75% of the original. Answer: a decrease of 25%. The distractors: no change comes from assuming +50% and −50% cancel; a decrease of 50% comes from applying only the second change; an increase of 25% has the direction wrong. - The number of members of a running club increases from 45 to 54. Work out the percentage increase.(a)9%(b)120%(c)20%(d)16.7%
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20% — Method: percentage increase = (increase ÷ original) × 100. Working: the increase is 54 − 45 = 9, and 9 ÷ 45 = 0.2, so the percentage increase is 0.2 × 100 = 20. Answer: 20%. The distractors: 9% comes from writing the actual increase as a percentage; 16.7% comes from dividing by the new value 54 instead of the original 45; 120% is the multiplier 1.2 written as a change rather than the change itself. - The rent on a flat increases by 10% one year and by a further 10% the following year. Work out the overall percentage increase over the two years.(a)121%(b)11%(c)21%(d)20%
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21% — Method: an increase of 10% is a multiplier of 1.1, and two successive increases are found by multiplying the multipliers. Working: 1.1 × 1.1 = 1.21, so the rent is 121% of the original, which is an increase of 21%. Answer: 21%. The distractors: 20% comes from adding the two percentages, which ignores that the second 10% is taken of a larger amount; 121% is the multiplier written as the change rather than the change itself; 11% comes from slipping in the multiplication and getting 1.11 instead of 1.21. - 40% of a number is 12 more than 25% of the same number. Work out the number.(a)48(b)80(c)15(d)30
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80 — Method: the difference between 40% and 25% of the number is 15% of the number, and that difference is 12. Working: 15% of the number is 12, so 1% of the number is 12 ÷ 15 = 0.8, and the number is 0.8 × 100 = 80. Check: 40% of 80 is 32, 25% of 80 is 20, and 32 − 20 = 12. Answer: 80. The distractors: 30 comes from solving 40% of the number = 12; 48 comes from solving 25% of the number = 12; 15 is the percentage difference written as the answer.
Frequently asked questions
How much of the GCSE Higher paper is ratio, proportion and rates of change?
20% of the total marks, and the three papers each carry an equal share of the qualification, so it is spread across all of them rather than concentrated in one.
How many topics are there in ratio, proportion and rates of change at GCSE Higher?
16 DfE content statements, coded R1 to R16 within this area. Every question in the bank is tagged to one of them.
Which ratio, proportion and rates of change topics are Higher only?
Instantaneous rate of change: gradients of curves (R15). A further 2 statements have a Higher-only part inside otherwise shared content.
Can I use a calculator for ratio, proportion and rates of change questions?
Paper 1 is non-calculator; Papers 2 and 3 allow one. Of the 16 statements in this area, 6 are naturally non-calculator and 1 naturally calculator, with the rest workable either way — so practise both.
What kind of marks does ratio, proportion and rates of change carry?
The GCSE Higher split is 40% AO1 (use and apply standard techniques), 30% AO2 (reason, interpret and communicate) and 30% AO3 (solve problems in and out of context). Questions in this area are set across all three.
Are these past paper questions?
No. Every question at /gcse-higher is original, written to the DfE content statements and checked before it is published. We do not host past papers or mark schemes.
Do I need an account?
No — you can start practising straight away. Progress is saved automatically in your browser.
Is it free?
Yes. The questions, worked answers and printable worksheets are all free, with no adverts.
Tips for ratio, proportion and rates of change at GCSE Higher
- A percentage increase of p% is a multiplier of (1 + p/100); a decrease is (1 − p/100). Repeated growth or decay over n periods is multiplied by the multiplier n times: amount = P × (1 + r)ⁿ.
- Reverse percentages go backwards through the multiplier: if a price after a 20% rise is £96, the original was 96 ÷ 1.2 = £80. Never subtract the same percentage.
- For direct proportion, y = kx: find k from one pair of values, then use it. For inverse proportion, y = k/x, so xy is constant. 'Proportional to the square' means y = kx².
- Compound measures are ratios in disguise: speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area. Keep the units consistent — km/h needs hours, m/s needs seconds.
- For similar shapes the length scale factor is k, the area scale factor is k², and the volume scale factor is k³.
Common mistakes (and how to avoid them)
- Adding a percentage back on to reverse it — a 20% discount is undone by dividing by 0.8, not by adding 20%.
- Using simple growth P × (1 + rn) for compound growth P × (1 + r)ⁿ.
- Mixing units in a speed calculation, such as minutes with km/h, without converting first.
- Multiplying an area by the length scale factor instead of by its square, or a volume by anything other than the cube.
Worked examples
- After a 40% discount, 60% of the original price remains, so the multiplier is 0.6.
- 0.6 × original = 90, so original = 90 ÷ 0.6.
- 90 ÷ 0.6 = 150.
- A 15% fall is a multiplier of 0.85 per year, applied three times: 12 000 × 0.85³.
- 0.85³ = 0.614125.
- 12 000 × 0.614125 = 7369.5, which rounds to £7370.
- Inverse proportion means y = k/x, so k = xy = 4 × 6 = 24.
- When x = 3, y = 24 ÷ 3 = 8.
Ratio, proportion and rates of change is a fifth of the Higher paper, and its questions almost always hide two steps inside one sentence: a reverse percentage, a growth factor applied several times, a compound measure with a unit conversion. Learn to see a percentage as a multiplier — every 'increase', 'decrease' and 'go back to the original' becomes a multiplication or a division by the same number. Keep units honest, and check that a reversed price or a decayed value is on the right side of where you started.
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