Short answer
Inverse proportion describes a relationship where one quantity increases at exactly the same rate as another decreases. Doubling one variable halves the other; trebling one reduces the other to one-third. The key feature is that the two quantities always multiply together to give the same constant value.
At a glance
- Key stage
- Key Stage 3
- Subject
- Ratio
- Type
- How-to guide
- For
- Students
- Read time
- 5 min
- Last updated
- 8 October 2026
Where this fits
- Key Stage 3Years 7–9This article
- GCSEYears 10–11
Method at a glance
- Identify the two quantities
- Multiply a given pair to find the constant k
- Divide k by the new value of one quantity to find the other
What is the difference between direct and inverse proportion?
In direct proportion, both quantities change in the same direction — double one and the other also doubles. In inverse proportion, they change in opposite directions — double one and the other halves.
| Situation | Direct or inverse? | Why |
|---|---|---|
| More paint at £5 per litre → higher total cost | Direct | Double the litres, double the cost |
| More workers on a job → job finished in less time | Inverse | Double the workers, half the time |
| More speed on a journey → less time to arrive | Inverse | Double the speed, half the time |
| More sandwiches bought → higher total price | Direct | Double the sandwiches, double the cost |
The quick test: multiply the two values from any pair in your table. If the product is always the same, the relationship is inverse proportion.
How do you recognise inverse proportion from a table?
Check that x × y = constant for every row.
Example: Is this table inverse proportion?
| Hours worked (x) | Items produced (y) | x × y |
|---|---|---|
| 2 | 60 | 120 |
| 3 | 40 | 120 |
| 4 | 30 | 120 |
| 6 | 20 | 120 |
Every product equals 120, so yes, this is inverse proportion.
How do you solve an inverse proportion problem?
Use the constant product (k = x × y) to find the missing value.
Method:
- Identify the two quantities.
- Multiply a given pair to find the constant k.
- Divide k by the new value of one quantity to find the other.
Worked example: 4 workers take 15 days to paint a building. How long would it take 6 workers?
- Quantities: workers (w) and days (d).
- Constant: k = 4 × 15 = 60.
- With 6 workers: d = 60 ÷ 6 = 10 days.
Check: 4 × 15 = 60 and 6 × 10 = 60 ✓
Another example: A car travelling at 60 mph takes 3 hours to complete a journey. How long does the same journey take at 90 mph?
- k = 60 × 3 = 180.
- Time at 90 mph = 180 ÷ 90 = 2 hours.
What does an inverse proportion graph look like?
In direct proportion, the graph is a straight line through the origin. In inverse proportion, the graph is a curved line called a reciprocal curve or hyperbola — it curves downwards steeply on the left and flattens out towards the right, never touching either axis.
The shape makes sense: as x gets very large, y gets very small (but never reaches zero), and vice versa.
Can inverse proportion involve non-integer answers?
Yes — the constant k is not always a whole number, and neither is the answer.
Example: 8 machines fill a warehouse in 5.5 hours. How long would 11 machines take?
- k = 8 × 5.5 = 44.
- Time = 44 ÷ 11 = 4 hours.
The division worked out neatly here, but it does not always do so. Leave your answer as a fraction or a decimal with appropriate accuracy.
What assumptions are important in inverse proportion problems?
Inverse proportion problems rely on the assumption that every worker/machine/vehicle works at the same constant rate. If three of six workers are less efficient, the actual time would be longer than the calculation predicts. In real life, these assumptions are rarely perfect, but for exam purposes they are always stated (or implied) to hold.
Another assumption: there is no "set-up time" that stays constant regardless of worker count. Exam problems are always designed so the constant-product method works directly.
Frequently asked questions
How do I know whether to multiply or divide when solving an inverse proportion problem?
Ask: "As one quantity goes up, does the other go up or go down?" If down, the proportion is inverse. To find the new value, calculate k = x × y from a known pair, then divide k by the given new value of the other variable. Never multiply; that would be direct proportion.
Does doubling always halve in inverse proportion?
Yes — that is the defining feature. If the product x × y is constant (= k), then when x is multiplied by any factor n, y must be divided by n to keep the product at k. This holds for any multiplier, not just 2: trebling x means y becomes one-third; multiplying x by 1.5 means y is multiplied by 2/3.
What if the problem says "the number of days taken varies inversely with the number of workers"?
This is just formal mathematical language for the same relationship. "Varies inversely" means inverse proportion. Set up the constant k = workers × days from the given information, then use k to find the unknown.
How is inverse proportion different at GCSE compared to KS3?
At GCSE, inverse proportion is expressed algebraically as y = k/x, and you may need to find k using coordinates or solve algebraically for unknowns. At KS3, the constant-product method (k = x × y) is sufficient. The idea is identical — GCSE simply adds the formal notation.
For more worked examples and a Professor Pi walkthrough of inverse proportion problems, visit aitutors.me.
Key terms
- direct proportion
- inverse proportion
- multiply
- Check
- Another example
- curved line
- reciprocal curve
- same constant rate