Spot whether two quantities rise together or one falls as the other rises, then use y=kx or y=xk. It is Higher only, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Direct proportion: as one doubles, so does the other, and y=kx for some constant k. Inverse: as one doubles, the other halves, and y=xk. A graph through the origin that is a straight line is direct; a graph that is a rectangular hyperbola (never hitting the axes) is inverse.
Find k from one pair of values, write the formula, then find the missing value. The sentence 'y is inversely proportional to the square of x' is y=x2k — include the power. A table that does not have a constant y/x is not direct; a table that does not have a constant xy is not inverse.
Direct: more of one means more of the other (y=kx). Inverse: more of one means less of the other (y=xk), often 'tap-hours' or 'people-days' staying constant. Find k from the pair you are given, then use it.
Objectives
Scale a job when more workers share the same work.
Find y when y is directly proportional to x.
Find y, or k, when y is inversely proportional to x.
Key points
Recognise direct proportion (double one, double the other) and inverse proportion (double one, halve the other).
Use the unitary method: find the value for one, then scale up.
Find k from one pair of values and write the formula y=kx or y=xk.
Solve worker/time and tap/fill problems using a fixed total (worker-days, litres).
Worked examples
Example 1
4 workers take 6 hours to finish a job. How many hours do 8 workers take?
2 marks
Find total worker-hours needed: 4 × 6 = 24.
Divide total work by 8 workers: 24 / 8 = 3 hours.
The team of 8 takes 3 hours.
Answer: 3
Common slips
Slip: Divide 6 by 8 directly. Total work is workers multiplied by time.
Slip: Sam doubled the time to 12. More workers share the job, so it takes less time.
Slip: Conclude 12 hours. Doubling workers halves the time required.
Example 2
y is directly proportional to x. y = 10 when x = 2. Find y when x = 5.
2 marks
Find constant of proportionality: k = y / x = 10 / 2 = 5.
Substitute x = 5 into y = kx: y = 5 × 5 = 25.
The resulting value of y is 25.
Answer: 25
Common slips
Slip: Add 5 to 10 directly. Direct proportion uses a constant multiplier k.
Slip: Tara multiplied 10 by 5 without finding k. You must scale x = 5 by the constant k = 5.
Slip: Conclude y = 50. Substituting x = 5 gives y = 25.
Example 3
y is inversely proportional to x. y = 12 when x = 3. Find y when x = 4.
3 marks
Find the inverse constant: k = xy = 3 × 12 = 36.
Use y = k / x with x = 4: y = 36 / 4 = 9.
The resulting value of y is 9.
Answer: 9
Common slips
Slip: Divide 12 by 3 directly. In inverse proportion, the product xy is constant.
Slip: Umar wrote 16. Inverse proportion divides the product by the new x.
Slip: Conclude y = 16. As x increases from 3 to 4, y decreases to 9.
Questions and answers
GCSE Maths direct & inverse proportion questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
6 workers take 4 hours to finish a job. How many hours do 3 workers take?
Show answer
Answer: 8
Find total worker-hours: 6 × 4 = 24.
Divide by 3 workers: 24 / 3 = 8.
M1 — Find total worker-hours: 6 × 4 = 24.
A1 — Divide by 3 workers: 24 / 3 = 8.
Question 2 Easy2 marks
5 workers take 6 hours to finish a job. How many hours do 10 workers take?
Show answer
Answer: 3
Find total worker-hours: 5 × 6 = 30.
Divide by 10 workers: 30 / 10 = 3.
M1 — Find total worker-hours: 5 × 6 = 30.
A1 — Divide by 10 workers: 30 / 10 = 3.
Question 3 Medium2 marks
y is directly proportional to x. y = 6 when x = 3. Find y when x = 7.
Show answer
Answer: 14
Find constant: k = 6 / 3 = 2.
Calculate y = 2 × 7 = 14.
M1 — Find constant: k = 6 / 3 = 2.
A1 — Calculate y = 2 × 7 = 14.
Question 4 Medium2 marks
y is directly proportional to x. y = 8 when x = 4. Find y when x = 6.
Show answer
Answer: 12
Find constant: k = 8 / 4 = 2.
Calculate y = 2 × 6 = 12.
M1 — Find constant: k = 8 / 4 = 2.
A1 — Calculate y = 2 × 6 = 12.
Question 5 Hard2 marks
y is inversely proportional to x. y = 20 when x = 2. Find y when x = 5.
Show answer
Answer: 8
Find constant: k = 20 × 2 = 40.
Calculate y = 40 / 5 = 8.
M1 — Find constant: k = 20 × 2 = 40.
A1 — Calculate y = 40 / 5 = 8.
Question 6 Hard2 marks
y is inversely proportional to x. y = 8 when x = 5. Find y when x = 4.
Show answer
Answer: 10
Find constant: k = 8 × 5 = 40.
Calculate y = 40 / 4 = 10.
M1 — Find constant: k = 8 × 5 = 40.
A1 — Calculate y = 40 / 4 = 10.
Question 7 Exam-style3 marks
y is directly proportional to x. y = 12 when x = 4. Find y when x = 9.
Show answer
Answer: 27
Find the constant: k = 12 / 4 = 3.
Multiply by x = 9: y = 3 × 9 = 27.
M1 — Writes y = kx, or 12 = 4k
M1 — k = 3
A1 — 27
Common mistakes
Most lost marks on this topic are one slip.
Doubling the time when the workers double
Halve the time
Multiplying y by the new x in an inverse question
Divide the constant product by the new x
Using xy as k for direct proportion
Direct k is y divided by x
Spot the slip
4 workers take 6 hours to do a job. Sue says 8 workers would take 12 hours. How long do 8 workers really take?
3
12
6
Show the slip
3 Twice the workers take half the time: 4×6 = 24 worker-hours, 24 ÷ 8 = 3 hours.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the general formula (y=kx or y=xk) as your first line — it is usually worth a mark on its own.
In a worded problem, say which type of proportion it is before you start: 'more builders, less time — inverse'.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.