What is inverse proportion?
Two quantities are in inverse proportion if, as one increases, the other decreases at a rate that keeps their product constant. Doubling one quantity always halves the other; trebling one always divides the other by three.
Worked Example 1
It takes 4 workers 15 hours to build a wall. How long would it take 6 workers, assuming the same total amount of work is shared out?
1
Find the total "worker-hours" needed for the job: 4 × 15 = 60
2
Divide the total by the new number of workers: 60 ÷ 6 = 10
Answer10 hours
The constant of proportionality
Inverse proportion can be written as an equation: y = k⁄x, where k is the constant of proportionality. Since xy = k, multiplying the two known values together gives k directly.
Worked Example 2
y is inversely proportional to x. When x = 4, y = 9. Find the equation connecting y and x, then find y when x = 6.
1
Write the proportion equation: y = k⁄x, which means xy = k
2
Substitute the known values to find k: 4 × 9 = 36, so k = 36
3
Write the full equation, then substitute x = 6: y = 36⁄x = 36⁄6 = 6
Answery = 36⁄x, and y = 6 when x = 6
Inverse proportion graphs
A graph of two inversely proportional quantities is a curve, not a straight line. It gets closer and closer to both axes without ever touching them, since neither quantity can reach zero.
As speed doubles (30 → 60mph), time halves (4hr → 2hr) — speed × time stays constant at 120 miles. The curve never touches either axis.
Inverse proportion with squares and cubes
Some quantities are inversely proportional to a power of another quantity, such as y ∝ 1⁄x² or y ∝ 1⁄x³. The method is the same — just include the power when finding and using k.
Worked Example 3
y is inversely proportional to x². When x = 2, y = 9. Find y when x = 3.
1
Write the proportion equation: y = k⁄x², which means yx² = k
2
Substitute the known values to find k: 9 × 2² = 9 × 4 = 36, so k = 36
3
Substitute x = 3 into y = 36⁄x²: y = 36 ÷ 9 = 4
Answer4
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1It takes 3 painters 8 hours to paint a fence. How long would it take 4 painters, assuming inverse proportion?Foundation
Q25 machines can complete a job in 12 hours. How long would 6 machines take?Foundation
Q3y is inversely proportional to x. When x = 3, y = 20. Find y when x = 5.Foundation
Q4A car makes a fixed journey. At 40mph the journey takes 3 hours. How long would the same journey take at 60mph?Foundation
Q58 workers can complete a task in 10 days. How many days would 4 workers take to complete the same task?Foundation
Higher (Grade 5–7)
Q6y is inversely proportional to x. When x = 6, y = 15. Find x when y = 9.Higher
Q7It takes 6 taps 5 hours to fill a tank. How long would 15 taps take to fill the same tank?Higher
Q8p is inversely proportional to q. When q = 4, p = 18. Find the constant of proportionality, and write a formula connecting p and q.Higher
Q9y is inversely proportional to x². When x = 2, y = 9. Find y when x = 3.Higher
Q10The time T for a journey is inversely proportional to speed s. At 25mph the journey takes 6 hours. Find the time taken at 50mph.Higher
Higher — Hard (Grade 8–9)
Q11y is inversely proportional to x. When x = 15, y = 8. Find x when y = 20.Grade 8–9
Q12F is inversely proportional to d². When d = 3, F = 20. Find F when d = 6.Grade 8–9
Q13y is inversely proportional to x³. When x = 2, y = 10. Find the value of x when y = 1.25.Grade 8–9
Q14Two quantities p and q are in inverse proportion. If p increases by 25%, by what percentage does q decrease?Grade 8–9
Q15The number of days needed to complete a job is inversely proportional to the number of workers. 5 workers take 18 days. How many workers are needed to finish the job in 6 days?Grade 8–9
Answers
Foundation (Q1–Q5)
Q16 hours(3 × 8 = 24 worker-hours, 24 ÷ 4)
Q210 hours(5 × 12 = 60, 60 ÷ 6)
Q312(k = 60, y = 60⁄x, 60 ÷ 5)
Q42 hours(40 × 3 = 120 miles, 120 ÷ 60)
Q520 days(8 × 10 = 80, 80 ÷ 4)
Higher (Q6–Q10)
Q6x = 10(k = 90, x = 90⁄y, 90 ÷ 9)
Q72 hours(6 × 5 = 30, 30 ÷ 15)
Q8k = 72, p = 72⁄q
Q94(k = 36, y = 36⁄x², 36 ÷ 9)
Q103 hours(k = 150, T = 150⁄s, 150 ÷ 50)
Higher — Hard (Q11–Q15)
Q11x = 6(k = 120, x = 120⁄y, 120 ÷ 20)
Q12F = 5(k = 180, F = 180⁄d², 180 ÷ 36)
Q13x = 4(k = 80, 1.25 = 80⁄x³, x³ = 64)
Q1420% decrease(q_new = k ÷ 1.25p = 0.8 × k⁄p = 0.8q)
Q1515 workers(5 × 18 = 90 worker-days, 90 ÷ 6)
Common mistakes
Common Mistake 1
Confusing inverse and direct proportion
If one quantity increasing causes the other to decrease (like speed and journey time), that's inverse proportion — treating it as direct proportion gives an answer that's the wrong way round.
Common Mistake 2
Adding instead of multiplying to find the constant
The constant of proportionality for inverse proportion comes from multiplying the two known values (k = xy), not adding or subtracting them.
Common Mistake 3
Forgetting to square or cube x
For y ∝ 1⁄x², forgetting to square x before dividing k by it is a common slip — always work out x² (or x³) first.
Common Mistake 4
Expecting a straight line graph
Inverse proportion always produces a curve, never a straight line — if a graph is straight, the relationship is direct proportion (or not proportional at all).
Exam tips
💡 Exam Tip 1
Always write the proportion equation first
Writing "y = k⁄x" (or the correct power) as your first line of working secures method marks, even if a later calculation slip costs the final answer.
💡 Exam Tip 2
For work/rate problems, find the total first
"Workers × time" or "taps × time" style problems are usually quickest by finding the constant total (e.g. total worker-hours) before dividing by the new value.
💡 Exam Tip 3
Sense-check the direction of change
If x increases, y should decrease (for positive k) — if your final answer increases alongside x, you've likely mixed up direct and inverse proportion.
💡 Exam Tip 4
Take the correct root for power proportions
When solving for x in y = k⁄x² or y = k⁄x³, rearrange to isolate the power first, then take the square root or cube root as the final step.
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