Mathematics 0580 · syllabus E2.8
Cambridge IGCSE Mathematics 0580 — Direct and inverse proportion topical past papers
Writing direct and inverse proportion with ∝ and a constant k, for powers and roots such as y ∝ x², y ∝ √x and y ∝ 1/x³, and using it to find values.
- Questions
- 74
- Years
- 2009–2026
- With mark scheme
- 86%
- Questions 2022–2026
- 20
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Direct and inverse proportion covers
Proportion links two variables by a constant multiplier. If y is directly proportional to x, written y ∝ x, then y = kx for a constant k, and doubling x doubles y. If y is inversely proportional to x, then y = k/x, and doubling x halves y. The relationship can also be with a square, square root, cube or cube root, so y may be proportional to x², √x or ∛x, or inversely proportional to x² or x³, for example.
Every question follows the same three steps: write the relationship as an equation with k, use the pair of values given to find k, then use the completed formula to find the value asked for. Questions also ask what happens to one variable when the other is multiplied by a factor, such as the effect on y of doubling x when y ∝ x², and some describe the relationship in words, such as the time for a job being inversely proportional to the number of workers.
Proportion in algebraic form is Extended content only, examined on Papers 2 and 4; Core candidates meet proportion only as ratio and unit-price reasoning. It draws on substitution and changing the subject, and the graphs of y = kx, y = kx² and y = k/x are among the graphs of functions.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- E2.8 Proportion
What you should be able to do
- Write a statement of direct or inverse proportion using ∝, then as an equation with k
- Find the constant of proportionality from a given pair of values
- Use the formula to find an unknown value of either variable
- Work with proportion to squares, cubes, square roots and cube roots
- Describe the effect on one variable of multiplying the other by a factor
Key ideas
- y ∝ x means y = kx; y ∝ 1/x means y = k/x
- y ∝ x² means y = kx², so multiplying x by 3 multiplies y by 9
- y ∝ 1/√x means y = k/√x, so multiplying x by 4 divides y by 2
- The constant k is the same for every pair of values in the relationship
How Direct and inverse proportion has been examined
From 2022 to 2026, Direct and inverse proportion was on Paper 2 · Non-calculator (Extended) in 10 of 14 exam series, worth about 1% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 1 | 1% | 1 of 14 |
| Paper 2 · Non-calculator (Extended) | 13 | 1% | 10 of 14 |
| Paper 3 · Calculator (Core) | 1 | 1% | 1 of 14 |
| Paper 4 · Calculator (Extended) | 5 | 1% | 4 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Variation.
A worked example
y is inversely proportional to the square of x. When x = 3, y = 8. Find y when x = 6, and find x when y = 0.5, given that x > 0.
- Write the relationship: y = k/x².
- Substitute x = 3 and y = 8: 8 = k/9, so k = 72 and y = 72/x².
- When x = 6: y = 72/36 = 2.
- When y = 0.5: 0.5 = 72/x², so x² = 72 ÷ 0.5 = 144 and x = 12, taking the positive root because x > 0.
Answer: y = 2 when x = 6; x = 12 when y = 0.5
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Writing y = x² + k instead of y = kx²: proportion means multiplying by a constant, not adding one.
- Confusing 'inversely proportional to the square' with 'proportional to the square root': y ∝ 1/x² is y = k/x², but y ∝ √x is y = k√x.
- Rounding k before using it: keep k exact, as a fraction if necessary, so that the final answer is accurate.
- Stopping after finding k: the remaining marks need the completed formula used with the new value.
Exam technique
- Always start by writing the equation with k; that line earns the first mark.
- Write the formula with the value of k in it, such as y = 5x², before finding the new value.
- Check the direction: in direct proportion both variables rise together; in inverse proportion one rises as the other falls.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.