If more workers are used to do a job, what happens to the time taken?
More workers = faster work.
In inverse proportion, as one increases, the other decreases.
Inverse Proportion
Inverse proportion describes a relationship where one quantity increases while the other decreases. It is the opposite of direct proportion and is based on ideas from ratio, with relationships often written in the form y = k/x.
Overview
Two quantities are in inverse proportion when one increases and the other decreases so that their product remains constant.
\( y \propto \frac{1}{x} \quad \text{and} \quad y = \frac{k}{x} \)
\( xy = k \)
\( xy = k \)
Inverse proportion often appears in problems involving time, speed, workers, pipes, area and pressure.
What you should understand after this topic
- Understand what inverse proportion means
- Understand how inverse proportion differs from direct proportion
- Use y = k/x to solve problems
- Use xy = k to solve problems
- Solve missing-value problems involving inverse proportion
Key Definitions
Inverse Proportion
When one quantity increases while the other decreases so that the product stays constant.
Inverse Proportional To
\( y \propto \frac{1}{x} \)
Constant of Proportionality
The fixed value \( k \) in \( y = \frac{k}{x} \) or \( xy = k \).
Product
The result of multiplying two values together.
Reciprocal Relationship
One quantity is proportional to the reciprocal of the other.
Inverse Graph Shape
The graph is a curve, not a straight line.
Key Rules
Multiply to find the constant
\( k = xy \)
Use inverse formula
\( y = \frac{k}{x} \)
One goes up, the other goes down
That is the key pattern.
Graph is not a straight line
Inverse proportion makes a curve.
Quick Recognition
Words
“is inversely proportional to”
Equation
\( y = \frac{12}{x} \)
Product check
If \( xy \) stays constant, it is inverse proportion.
Graph pattern
A decreasing curve, not a straight line.
How to Solve
Step 1: Recognise inverse proportion
In inverse proportion, as one value increases, the other decreases so that their product stays constant. It is useful to compare this with direct proportion.
\( y \propto \frac{1}{x} \)
Exam thinking: If one doubles and the other halves, it is inverse proportion.
Step 2: Write the equation
Convert the proportional relationship into an equation.
\( y = \frac{k}{x} \)
\( xy = k \)
Both forms are equivalent and can be used in calculations.
Step 3: Find the constant \(k\)
Use given values to calculate \(k\).
If \( x = 4, y = 6 \):
\( k = xy = 4 \times 6 = 24 \)
Exam tip: Multiply the pair of values to find \(k\).
Step 4: Use the equation
Substitute new values into the equation.
\( y = \frac{24}{x} \)
If \( x = 8 \): \( y = 3 \)
Step 5: Graph of inverse proportion
The graph is a curve that gets closer to the axes but never touches them.
Exam tip: Inverse proportion graphs never cross the axes.
Key idea
The graph is not a straight line and has two branches.
Step 6: How to recognise inverse proportion
See direct proportion for comparison.
From words
Look for 'inversely proportional'.
From a table
Check if \( xy \) is constant.
From an equation
Looks like \( y = \frac{k}{x} \).
From context
More workers → less time, higher speed → less time.
Step 7: Exam method summary
- Recognise inverse proportion.
- Write \( y = \frac{k}{x} \).
- Find \(k\) using known values.
- Substitute to find the answer.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on recognising and solving problems involving inverse proportion.
Edexcel
State whether \( y \) is inversely proportional to \( x \) in the equation \( y = \frac{6}{x} \).
Edexcel
Given that \( y \propto \frac{1}{x} \) and \( y = 4 \) when \( x = 3 \), find \( y \) when \( x = 6 \).
Edexcel
Given that \( y \) is inversely proportional to \( x \), and \( y = 10 \) when \( x = 2 \), find the constant of proportionality.
Edexcel
The time taken to complete a journey is inversely proportional to the speed. If a journey takes 5 hours at 60 km/h, how long will it take at 100 km/h?
Edexcel
Write the relationship \( y \propto \frac{1}{x} \) as an equation using a constant of proportionality.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on forming and solving equations involving inverse proportion.
AQA
Given that \( y \propto \frac{1}{x} \) and \( y = 8 \) when \( x = 5 \), find an equation connecting \( y \) and \( x \).
AQA
If \( y \) is inversely proportional to \( x \), and \( y = 12 \) when \( x = 4 \), find \( x \) when \( y = 3 \).
AQA
The number of workers needed to complete a job is inversely proportional to the time taken. If 6 workers take 10 days, how many workers are required to complete the job in 4 days?
AQA
The pressure of a gas varies inversely with its volume. If the pressure is 200 kPa when the volume is 5 m^3, find the pressure when the volume is 8 m^3.
AQA
Explain how you recognise an inverse proportion from a table or graph.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, algebraic modelling, and graphical interpretation.
OCR
Given that \( y \propto \frac{1}{x} \), and \( y = 9 \) when \( x = 2 \), find the value of \( y \) when \( x = 6 \).
OCR
Write \( y \propto \frac{1}{x} \) as an equation involving a constant \( k \).
OCR
A fixed distance is travelled at different speeds. If it takes 3 hours at 80 km/h, how long will it take at 120 km/h?
OCR
Two variables are inversely proportional. When \( x = 4 \), \( y = 15 \). Find the value of \( y \) when \( x = 10 \).
OCR
State the key features of a graph that represents inverse proportion.
Exam Checklist
Step 1
If you see “inversely proportional”, write \( y = \frac{k}{x} \).
Step 2
Use known values to find \( k \) by multiplying.
Step 3
Substitute the new value into the completed equation.
Step 4
Check whether the answer makes sense: bigger \( x \), smaller \( y \).
Most common exam mistakes
Wrong formula
Using \( y = kx \) instead of \( y = \frac{k}{x} \).
Wrong constant
Dividing to find \( k \) when you should multiply.
Pattern confusion
Not checking whether the product stays constant.
Reasonableness mistake
Getting a larger answer when the question should produce a smaller one.
Common Mistakes
These are common mistakes students make when working with inverse proportion in GCSE Maths.
Using the wrong formula
Incorrect
A student uses \(y = kx\) instead of \(y = \frac{k}{x}\).
Correct
Inverse proportion follows \(y = \frac{k}{x}\) or \(xy = k\). This is different from direct proportion, which uses \(y = kx\).
Finding k incorrectly
Incorrect
A student divides instead of multiplying to find k.
Correct
For inverse proportion, use \(k = x \times y\), not division. Multiply the values to find the constant.
Relying only on “one goes up, one goes down”
Incorrect
A student assumes inverse proportion just because one value increases and the other decreases.
Correct
Check that the product \(xy\) is constant. This confirms inverse proportion.
Confusing direct and inverse proportion
Incorrect
A student uses the method for direct proportion.
Correct
Direct proportion uses \(y = kx\), while inverse proportion uses \(y = \frac{k}{x}\). Always identify the relationship first.
Expecting a straight-line graph
Incorrect
A student draws a straight line graph.
Correct
Inverse proportion graphs are curves (hyperbolas), not straight lines. They approach the axes but do not touch them.
Try It Yourself
Practise solving problems involving inverse proportional relationships.
Foundation
Foundation Practice
Understand inverse proportion and solve simple problems.
4 workers take 12 hours to complete a job. How long would 2 workers take?
Fewer workers means more time.
Halving workers doubles time → 12 × 2 = 24 hours.
6 machines take 10 hours. How long do 3 machines take?
Machines halve → time doubles.
3 is half of 6, so time doubles → 20 hours.
If speed doubles, what happens to time taken for a fixed journey? (Give answer as 'halves' or 'doubles')
Think: faster means less time.
Inverse relationship → double speed → half time.
Which situation shows inverse proportion?
Think: more leads to less.
More workers → less time, so inverse proportion.
8 workers take 6 hours. How long do 4 workers take?
Workers halve → time doubles.
4 is half of 8 → time doubles → 12 hours.
A student says: 'If x doubles, y doubles in inverse proportion.' What is wrong?
Inverse means opposite effect.
In inverse proportion, if x doubles, y halves.
10 workers take 4 hours. How long do 5 workers take?
Workers halve.
5 is half of 10 → time doubles → 8 hours.
If x increases, what happens to y in inverse proportion?
Opposite behaviour.
Inverse proportion means one goes up, the other goes down.
3 taps fill a tank in 9 hours. How long do 9 taps take?
Taps triple.
If taps triple, time divides by 3 → 3 hours.
Higher
Higher Practice
Solve inverse proportion problems using equations and constant k.
y is inversely proportional to x. When x = 2, y = 12. Find y when x = 6.
Use y = k/x.
k = xy = 24 → y = 24 ÷ 6 = 4.
y ∝ 1/x. When x = 4, y = 10. Find y when x = 5.
Find k using xy.
k = 40 → y = 40 ÷ 5 = 8.
y ∝ 1/x. If y = 6 when x = 3, what is k?
k = xy.
k = 6 × 3 = 18.
y ∝ 1/x. When x = 8, y = 3. Find y when x = 2.
Find k first.
k = 24 → y = 24 ÷ 2 = 12.
Which equation represents inverse proportion?
Look for division.
Inverse proportion is y = k/x.
y ∝ 1/x. When x = 10, y = 2. Find x when y = 5.
Find k first.
k = 20 → 5 = 20 ÷ x → x = 4.
A student says y = 1/x + 3 is inverse proportion. Why is this wrong?
Extra +3 breaks the rule.
Inverse proportion must be y = k/x exactly.
y ∝ 1/x. When x = 5, y = 6. Find y when x = 15.
Find k first.
k = 30 → y = 30 ÷ 15 = 2.
If x is multiplied by 4, what happens to y in inverse proportion?
Opposite effect.
Inverse proportion → multiply x → divide y.
y ∝ 1/x. When x = 12, y = 5. Find x when y = 20.
Find k first.
k = 60 → 20 = 60 ÷ x → x = 3.
Games
Practise this topic with interactive games.
Games coming soon.