If y is directly proportional to x, then y = kx: doubling x doubles y. If y is inversely proportional to x, then y = k/x: doubling x halves y, and the product xy stays constant. Find the constant k from one pair of values, write the equation, then use it to find any other value. At Higher tier, proportion can involve powers and roots, such as y = kx².
Proportion equations
- Direct: y ∝ x means y = kx (y ÷ x is constant)
- Inverse: y ∝ 1/x means y = k/x (xy is constant)
- Direct with a square: y ∝ x² means y = kx²
- Find k by substituting a known pair of values
Direct proportion
When y is directly proportional to x, written y ∝ x, the equation is y = kx, where k is the constant of proportionality. Find k by substituting a known pair: if y = 30 when x = 5, then 30 = 5k, so k = 6 and y = 6x. Then use the equation to find any other value.
Inverse proportion
When y is inversely proportional to x, written y ∝ 1/x, the equation is y = k/x. As x increases, y decreases, and the product xy is always k. If y = 15 when x = 4, then k = 4 × 15 = 60, so y = 60/x.
Recognising the type
From a table, check whether y ÷ x is constant (direct) or x × y is constant (inverse). In a context, ask whether doubling one quantity doubles or halves the other. More workers on a job means fewer days, which is inverse proportion; more cakes need more eggs, which is direct.
Proportion with powers and roots (Higher)
Some quantities vary with a square, a cube or a square root. If y is proportional to the square of x, then y = kx². Substitute to find k as before: if y = 36 when x = 3, then 36 = 9k, so k = 4 and y = 4x².
Worded problems
Many questions do not mention k. If 6 workers take 12 days, the total work is 72 worker-days, so 8 workers take 72 ÷ 8 = 9 days. Thinking about what stays constant makes these problems straightforward.
Worked examples
Example 1. y is directly proportional to x. When x = 5, y = 30. Find y when x = 8.
- y = kx: 30 = 5k, so k = 6 and y = 6x.
- When x = 8, y = 48.
Answer: y = 48.
Example 2. y is inversely proportional to x. When x = 4, y = 15. Find y when x = 6.
- y = k/x: k = 4 × 15 = 60, so y = 60/x.
- When x = 6, y = 60 ÷ 6.
Answer: y = 10.
Example 3. A table shows x = 2, 4, 6 with y = 30, 15, 10. Is this direct or inverse proportion? Give the equation.
- x × y = 60 each time, so the product is constant.
Answer: Inverse proportion: y = 60/x.
Example 4. 6 workers can build a wall in 12 days. How long would 8 workers take, working at the same rate?
- The total work is 6 × 12 = 72 worker-days.
- 72 ÷ 8.
Answer: 9 days.
Example 5. y is directly proportional to x². When x = 3, y = 36. Find y when x = 5.
- y = kx²: 36 = 9k, so k = 4.
- When x = 5, y = 4 × 25.
Answer: y = 100.
Common mistakes
- Using y = kx for inverse proportion. Inverse proportion is y = k/x.
- Forgetting to find k first. Substitute a known pair to find k before using the equation.
- Squaring the wrong thing. For y ∝ x², square x when finding k and when using the equation.
- Treating any decreasing relationship as inverse proportion. Check that the product xy is constant.
- Not stating the equation when asked. Give the formula, such as y = 6x, not just the value of k.
How proportion is examined
Proportion questions turn up at Foundation and Higher on Edexcel GCSE and iGCSE papers, with powers and roots added at Higher. Questions ask you to:
- find a formula for y in terms of x
- use the formula to find a value of y or x
- decide whether a table or context shows direct or inverse proportion
- solve worded problems, such as workers and days
The direct and inverse proportion worksheet has answers for every question, and the GCSE proportion engine provides more.
See proportion graphs for the graphical side, and ratio for sharing in a given ratio.
Frequently asked questions
What does y ∝ x mean?
y is directly proportional to x, so y = kx for some constant k.
What is the formula for inverse proportion?
y = k/x, which means xy = k.
How do you find the constant of proportionality?
Substitute a known pair of values into the formula and solve for k.