Short answer
When y is proportional to a power of x, write y = kxⁿ, substitute one known pair of values to find the constant k, then use y = kxⁿ to answer the rest. The most common cases at GCSE Higher are y ∝ x², y ∝ x³, y ∝ √x and y ∝ 1/x².
At a glance
- Key stage
- GCSE
- Subject
- Ratio and proportion
- Type
- How-to guide
- For
- Students
- Read time
- 4 min
- Last updated
- 10 September 2026
Where this fits
- Key Stage 3Years 7–9
- GCSEYears 10–11This article
Method at a glance
- Write down the equation form (e.g
- Substitute the known pair of values (x₀, y₀) and solve for k
- Rewrite the equation with k inserted, then use it to find any unknown
What does y ∝ xⁿ mean?
The symbol ∝ means "is proportional to." If y ∝ xⁿ, then doubling x multiplies y by 2ⁿ — not by 2. This is what distinguishes power proportion from simple direct proportion.
| Relationship | Equation form | Doubling x multiplies y by |
|---|---|---|
| y ∝ x (direct) | y = kx | 2 |
| y ∝ x² | y = kx² | 4 |
| y ∝ x³ | y = kx³ | 8 |
| y ∝ √x | y = k√x | √2 ≈ 1.41 |
| y ∝ 1/x (inverse) | y = k/x | ½ |
| y ∝ 1/x² (inverse square) | y = k/x² | ¼ |
How do you find the constant of proportionality k?
Method — three steps:
- Write down the equation form (e.g. y = kx²).
- Substitute the known pair of values (x₀, y₀) and solve for k.
- Rewrite the equation with k inserted, then use it to find any unknown.
Worked example 1 — y ∝ x²:
y is proportional to x². When x = 3, y = 45. Find y when x = 5.
- y = kx²
- 45 = k × 3² = 9k → k = 5
- Equation: y = 5x². When x = 5: y = 5 × 25 = 125
Check: 3→45, 5→125. Ratio of x values = 5/3. Ratio of y values = 125/45 = 25/9 = (5/3)². ✓
Worked example 2 — y ∝ √x:
y is proportional to the square root of x. When x = 16, y = 20. Find x when y = 35.
- y = k√x
- 20 = k√16 = 4k → k = 5
- Equation: y = 5√x. When y = 35: 35 = 5√x → √x = 7 → x = 49
Check: √49 = 7; y = 5 × 7 = 35 ✓
How do you handle inverse square proportion?
Inverse square proportion (y ∝ 1/x²) appears in physics-style GCSE maths problems — the intensity of light or sound decreasing with distance.
Worked example — y ∝ 1/x²:
y is inversely proportional to the square of x. When x = 2, y = 18.
- y = k/x²
- 18 = k/4 → k = 72
- Equation: y = 72/x².
Find y when x = 6: y = 72/36 = 2
What happens to y when x is tripled? y = 72/(3x)² = 72/9x² = (1/9) × y. So y is divided by 9. ✓ (inverting 3² = 9)
What if the problem gives you two unknowns?
Sometimes a question gives neither k explicitly nor a complete pair (x, y) — instead it gives a proportional relationship between two situations. You do not need to find k.
Worked example — ratio method:
y ∝ x³. When x doubles, what happens to y?
y₁ = kx₁³ and y₂ = k(2x₁)³ = 8kx₁³.
So y₂/y₁ = 8. y multiplies by 8.
This works for any power proportion without needing a specific value of k.
What are common mistakes in power proportion questions?
| Mistake | Example of error | Correct approach |
|---|---|---|
| Using direct proportion rule | y ∝ x² but writing y₂/y₁ = x₂/x₁ | y₂/y₁ = (x₂/x₁)² |
| Forgetting to square k's calculation | 45 = k × 3 instead of k × 9 | Always square (or cube, or root) x first |
| Sign error in inverse square | k/x² but writing kx² | Write the equation clearly before substituting |
| Not checking the answer | No verification step | Substitute your answer back into y = kxⁿ |
Frequently asked questions
How do I know which power to use?
The question tells you: "y is proportional to the square of x" means y ∝ x². "y is inversely proportional to the cube of x" means y ∝ 1/x³. Read the wording carefully and write the equation form before doing any calculation.
Can I use a table to find k?
Yes. If you have a table of x and y values, compute y/x² (for y ∝ x²) for each row. If the relationship holds, all entries in the y/x² column will be equal — that constant value is k. This is a useful check and can reveal the relationship type even when it is not stated.
Does power proportion appear on GCSE Foundation or Higher only?
Simple direct and inverse proportion (y ∝ x, y ∝ 1/x) appear on both tiers. Power proportion (y ∝ x², y ∝ x³, y ∝ √x, y ∝ 1/x²) is a Higher-tier topic. You will not see y ∝ x² on a Foundation-only paper, but it is a standard question type on Higher papers.
What happens to y when x is halved (for y ∝ x²)?
Halving x multiplies x by ½, so x² is multiplied by (½)² = ¼. Therefore y is multiplied by ¼ — it becomes one-quarter of its original value. The general rule: if x is multiplied by a factor f, then y is multiplied by fⁿ for y ∝ xⁿ.
Need step-by-step guidance on proportion problems? Professor Pi is ready to help at aitutors.me.
Key terms
- y multiplies by 8
- square
- cube
- Higher-tier