Proportion questions appear in virtually every GCSE Maths exam series. They test algebra, interpretation, and the ability to build an equation from a written description — skills that span the whole paper. AQA 8300, Edexcel 1MA1, and OCR J560 all include proportion on the higher tier, usually as a 4–6 mark question appearing in Paper 2 or Paper 3. Foundation students encounter simpler ratio and proportion problems, but the algebraic form described here is a higher-tier staple.
Once you understand the method, proportion questions follow a reliable pattern. You will always: set up an equation, find the constant of proportionality, then use it to answer the question.
What You Need to Know First
Before working through proportion, you should be able to:
- Substitute values into algebraic expressions
- Rearrange equations to isolate a variable
- Work with squares, square roots, cubes, and cube roots
- Recognise the proportionality symbol ∝ (read as "is proportional to")
- Interpret graphs — proportion questions sometimes ask you to match a relationship to a graph
Direct Proportion: Setting Up y = kx
When y is directly proportional to x, we write:
y ∝ x, which becomes the equation y = kx
Here, k is the constant of proportionality. Your job in a proportion question is always to find k first.
The Method (Direct Proportion)
- Write the proportionality statement using ∝
- Replace ∝ with = k to form an equation
- Substitute the given values to find k
- Rewrite the equation with the value of k
- Use the equation to find the unknown
Worked Example 1: y is directly proportional to x. When x = 5, y = 30. Find y when x = 8.
Step 1: Write the proportion: y ∝ x, so y = kx
Step 2: Substitute x = 5, y = 30: 30 = k × 5 k = 30 ÷ 5 = 6
Step 3: The equation is y = 6x
Step 4: Find y when x = 8: y = 6 × 8 = 48
Proportion with Powers
Higher-tier questions very commonly use squared or cubed relationships. The proportionality symbol works the same way.
y ∝ x² → y = kx²
y ∝ x³ → y = kx³
y ∝ √x → y = k√x
Worked Example 2: y is directly proportional to the square of x. When x = 3, y = 45. Find y when x = 5, and find x when y = 80.
Step 1: y ∝ x², so y = kx²
Step 2: Substitute x = 3, y = 45: 45 = k × 3² 45 = 9k k = 45 ÷ 9 = 5
Step 3: The equation is y = 5x²
Step 4: Find y when x = 5: y = 5 × 5² = 5 × 25 = 125
Step 5: Find x when y = 80: 80 = 5x² x² = 80 ÷ 5 = 16 x = √16 = 4
Proportion questions with powers are extremely common on AQA and Edexcel higher papers. Recognising that y ∝ x² leads to y = kx² — not y = kx with x squared as a separate step — is the key habit to build.
Inverse Proportion: Setting Up y = k/x
When y is inversely proportional to x, as x increases, y decreases. We write:
y ∝ 1/x, which becomes the equation y = k/x
This is sometimes written as y = k × (1/x), which is exactly the same thing.
The Method (Inverse Proportion)
The steps are identical — the only change is the equation you set up.
Worked Example 3: y is inversely proportional to x. When x = 4, y = 9. Find y when x = 12, and find x when y = 6.
Step 1: y ∝ 1/x, so y = k/x
Step 2: Substitute x = 4, y = 9: 9 = k/4 k = 9 × 4 = 36
Step 3: The equation is y = 36/x
Step 4: Find y when x = 12: y = 36/12 = 3
Step 5: Find x when y = 6: 6 = 36/x x = 36/6 = 6
Inverse Proportion with Powers
Just as with direct proportion, inverse proportion can involve powers:
y ∝ 1/x² → y = k/x²
Worked Example 4: P is inversely proportional to the square of d. When d = 2, P = 50. Find P when d = 5.
Step 1: P ∝ 1/d², so P = k/d²
Step 2: Substitute d = 2, P = 50: 50 = k/4 k = 50 × 4 = 200
Step 3: The equation is P = 200/d²
Step 4: Find P when d = 5: P = 200/25 = 8
Proportion and Graphs
Exam questions sometimes show a graph and ask you to identify the type of proportion. Here is what the common shapes look like:
- y = kx (direct, y ∝ x) — a straight line through the origin
- y = kx² (direct, y ∝ x²) — a curve starting at the origin, increasing steeply
- y = k/x (inverse) — a curved hyperbola; both branches in the first and third quadrants if k is positive
- y = k/x² (inverse, y ∝ 1/x²) — a steeper version of the hyperbola
Being able to recognise these graphs gives you a quick check: if your equation gives y = k/x and the graph looks like a straight line, something has gone wrong.
Common Mistakes Students Make
- Writing y = kx when the relationship involves x². If the question says "y is directly proportional to the square of x", the equation must be y = kx², not y = kx. Using the wrong equation means every subsequent answer is wrong.
- Forgetting to find k before answering the question. Some students try to work out ratios from the given values without forming an equation. This only works for the simplest linear cases — with powers, you must find k.
- Confusing direct and inverse proportion. With inverse proportion, y = k/x. Students sometimes write y = k × x by mistake, missing the division. Check: as x gets bigger, does y get bigger (direct) or smaller (inverse)?
- Not showing the equation with the value of k substituted. Mark schemes award a method mark for writing the equation (e.g. y = 5x²). If you skip straight to the answer without showing this, you risk losing marks.
- Taking the square root with only the positive value when x could be negative. If x² = 16, technically x = ±4. In a proportion context, x is usually positive — but read the question carefully.
Exam Tips for Direct and Inverse Proportion
- Write the proportion statement first, then the equation. Writing "y ∝ x², so y = kx²" takes a moment but keeps your working clear and earns the setup mark on AQA and Edexcel mark schemes.
- Find k before doing anything else. Every proportion question gives you one pair of values specifically so you can find k. Use them before reaching for the unknown.
- Edexcel and AQA often ask two parts: find the equation, then use it to find y or x. Do not rush to the second part — it depends entirely on getting k right in the first part.
- If asked to find x from y, rearrange the full equation. If y = 5x², and you need x, rearrange: x² = y/5, then x = √(y/5). Do not try to guess or work backwards informally.
- Check your answer makes sense. For direct proportion, bigger x should give bigger y. For inverse proportion, bigger x should give smaller y. A quick sanity check can save you from a sign or division error.
Practise Now
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