Number and ratio · GCSE Maths
Inverse proportion
GCSE Maths inverse proportion: y = k/x, product k = xy constant, and worker–days problems when one quantity rises as the other falls.
Inverse proportion: as one doubles, the other halves. xy = k — the product stays constant, not the ratio.
The important bits
What you need to know
- 1
Two quantities are in inverse proportion if when one increases the other decreases so that their product stays constant: xy = k.
- 2
y ∝ 1/x means y = k/x. The graph is a reciprocal curve, not a straight line through the origin.
- 3
If 6 workers take 10 days, the job is 60 worker-days. 4 workers take 60 ÷ 4 = 15 days. Name the conserved product.
- 4
To find k, multiply the given pair: y = 12 when x = 5 gives k = 60. Then when x = 8, y = 60/8 = 7.5.
- 5
Speed and time for a fixed distance are inversely proportional: double the speed, halve the time. Distance = speed × time links the three.
- 6
y is proportional to 1/x² means y = k/x². Halving x quadruples y. Read the power on x in the denominator.
- 7
Do not use direct proportion on worker problems where more people means fewer days — that is inverse unless the question states otherwise.
- 8
Check inverse proportion: multiply x and y for every pair in a table. If the product is constant, it is inverse proportion.
Quotations worth analysing
Short evidence. Real method.
“xy = k”
The product is fixed. Students who divide y by x and expect a constant gradient have applied direct proportion to an inverse problem.
“As one doubles, the other halves”
Only true for y ∝ 1/x. For y ∝ 1/x², doubling x quarters y. Match the verbal description to the equation.
“Worker-days stay constant”
6 × 10 = 4 × 15 = 60. The job size is the product. Adding workers without dividing days is the direct-proportion mistake.
Go deeper
Worker and day problems
It takes 8 people 12 days to paint a hall. How long for 6 people? Total work = 8 × 12 = 96 person-days. Six people need 96 ÷ 6 = 16 days. More people, fewer days — inverse. If the question asks how many people for 10 days: 96 ÷ 10 = 9.6 people, which in context means 10 people with a fractional overrun or the question expects 9.6 as the mathematical answer. Pumps filling a tank: 3 pumps in 8 hours is 24 pump-hours. 4 pumps need 6 hours. Always write the product sentence: “total work = people × days = constant.” Speed–time for fixed distance: 60 mph for 2 hours is 120 miles. At 40 mph, time = 120/40 = 3 hours. Distance is the product speed × time.
Go deeper
y = k/x and the reciprocal graph
If y is inversely proportional to x and y = 10 when x = 4, k = 40 and y = 40/x. When x = 8, y = 5; when x = 2, y = 20. Plotting y against x gives a curve falling from left to right, never touching the axes. Plotting y against 1/x gives a straight line through the origin with gradient k — useful on Higher papers. y ∝ 1/x²: when x = 2, y = 18, k = 18 × 4 = 72, y = 72/x². When x = 4, y = 72/16 = 4.5. Doubling x from 2 to 4 quarters y from 18 to 4.5.
Go deeper
Inverse versus direct in word problems
Direct: more hours, more pay at fixed rate. Inverse: more workers, fewer days for the same job. Direct: more kilometres, more fuel at constant mpg if consumption per km is fixed — actually fuel ∝ distance is direct. Inverse: fixed fuel tank, higher speed might mean fewer hours if distance fixed — time ∝ 1/speed. Read the story. If the total amount of work, distance, or material is fixed and you split it among more units, think inverse. If each unit adds more total, think direct. Mixed questions: “partially proportional” is rare at GCSE; stick to the keyword “inversely proportional” or “product is constant.”
See the idea in action
y is inversely proportional to x. When x = 6, y = 15. Find y when x = 10 and x when y = 6. Step 1: xy = k. 6 × 15 = 90, so k = 90 and y = 90/x. Step 2: When x = 10, y = 90/10 = 9. Step 3: When y = 6, 6 = 90/x, so x = 90/6 = 15. Check products: 6 × 15 = 90, 10 × 9 = 90, 15 × 6 = 90.
Exam technique
Turn knowledge into marks
Write xy = k before dividing. Inverse proportion divides after the product — do not find a “per one” rate and multiply like direct proportion.
Common mistakes
Do not give these marks away
- 01
Using y = kx instead of y = k/x when the question says inversely proportional.
- 02
Adding or subtracting workers and days instead of keeping the product constant.
- 03
Applying inverse proportion when the total is not fixed (e.g. more hours worked means more pay — direct).
5 machines take 12 hours. How long for 8 machines (inverse proportion)?
A7.5 hours
B19.2 hours
C4 hours
D60 hours
Show the answer
7.5 hours. Product 5 × 12 = 60 machine-hours. 60 ÷ 8 = 7.5 hours. 19.2 is 8 × 12 ÷ 5 — multiplied instead of using constant product.
Quick questions
If this is the bit you searched
What is the difference between direct and inverse proportion?
Direct: ratio y/x is constant, y = kx. Inverse: product xy is constant, y = k/x. One doubles–doubles, the other doubles–halves.
How do I find k for inverse proportion?
Multiply the given x and y: k = xy. Then use y = k/x or x = k/y for other values.
Is the graph of inverse proportion a straight line?
No. y = k/x is a reciprocal curve. y against 1/x is a straight line through the origin if y ∝ 1/x.
Can y be inversely proportional to x²?
Yes: y = k/x². Then xy² = k or y ∝ 1/x². Halving x quadruples y. Read the exact wording in the question.