GCSE Maths Direct & Inverse Proportion Notes
In direct proportion two quantities increase together; in inverse proportion one increases as the other decreases. A constant k links them.
Key concepts
- Direct proportion
- y is proportional to x means y = kx — double x and y doubles.
- Inverse proportion
- y is inversely proportional to x means y = k ÷ x — double x and y halves.
- Constant of proportionality (k)
- The fixed number linking the quantities; find it using one known pair of values.
- Recognising the type
- Grow together = direct; one up while the other down = inverse.
Key formulas
Direct proportion
y = kx
Inverse proportion
y = k ÷ x
Worked example
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
- 1Write the relationship: y = kx.
- 2Substitute known values: 20 = k × 4, so k = 5.
- 3The equation is y = 5x.
- 4When x = 7: y = 5 × 7 = 35.
Answer: y = 35
Common mistakes
✗ Confusing direct and inverse proportion.
✓ Direct uses y = kx; inverse uses y = k ÷ x.
✗ Skipping the step of finding k.
✓ Always work out the constant k first from the given pair of values.
✗ Only scaling by the ratio without checking the type.
✓ Simple scaling only works for direct proportion, not inverse.
Quick check
Try these, then reveal the answers.
If y = kx and y = 12 when x = 3, find k.Reveal answer
4
If y ∝ 1/x and y = 6 when x = 2, find k.Reveal answer
12
Is 'more workers, less time' direct or inverse?Reveal answer
Inverse.
Exam tips
- Write the proportion equation with k before substituting numbers.
- State the full equation (e.g. y = 5x) as part of your working — it earns marks.
- Read the context to decide whether quantities grow together or apart.
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Frequently asked questions
What is the constant of proportionality?+
The fixed multiplier k that links the two quantities; you find it from a known pair of values.
How do I know if a problem is inverse proportion?+
One quantity increases as the other decreases — for example, more people finishing a job in less time.
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