Proportion is one of those GCSE topics that sits across both Foundation and Higher tiers but means slightly different things at each level. At Foundation, proportion is mostly about ratio problems and the practical formula triangle. At Higher, proportion goes algebraic — you write equations like y ∝ x² and solve for unknowns. This guide covers everything you need for both tiers.
What Proportion Actually Means
Two quantities are in proportion when their ratio stays the same as both quantities change.
Direct Proportion
In direct proportion, as one quantity increases, the other increases at the same rate. The ratio between them stays constant.
If y is directly proportional to x, we write:
y ∝ x
This is the same as saying y = kx where k is a constant (called the constant of proportionality).
Examples in Everyday Life
- Cost is directly proportional to weight when you buy apples per kilogram. 2 kg costs twice as much as 1 kg. 5 kg costs five times as much.
- Distance is directly proportional to time when you travel at a constant speed.
- Total wage is directly proportional to hours worked at a fixed hourly rate.
Inverse Proportion
In inverse proportion, as one quantity increases, the other decreases — and they decrease in a specific way: their product stays constant.
If y is inversely proportional to x, we write:
y ∝ 1/x
This is the same as saying y = k/x where k is the constant of proportionality.
Examples in Everyday Life
- Time taken is inversely proportional to speed for a fixed journey. Travel twice as fast → take half the time.
- Number of days to complete a project is inversely proportional to the number of workers (assuming all workers contribute equally).
- Pressure is inversely proportional to volume for a fixed amount of gas at fixed temperature (this is Boyle's Law from physics, but it appears in GCSE maths problems too).
Foundation Tier Proportion Problems
At Foundation tier, proportion problems are usually concrete and involve real numbers, not algebra. Standard methods include:
The Unitary Method
Find the value of one unit, then scale up to the value you need.
Example: 6 pens cost £4.50. How much do 10 pens cost?
- Cost of 1 pen: £4.50 ÷ 6 = £0.75
- Cost of 10 pens: £0.75 × 10 = £7.50
Direct Proportion Tables
For straightforward direct proportion, set up a ratio table and scale.
Example: A recipe for 4 people uses 200 g of flour. How much flour for 7 people?
- For 4 people: 200 g
- For 1 person: 50 g
- For 7 people: 50 × 7 = 350 g
Inverse Proportion at Foundation
These usually involve workers or speed.
Example: 8 workers take 6 days to build a wall. How long would 4 workers take?
This is inverse proportion: fewer workers means more days.
- Total work = 8 × 6 = 48 worker-days
- For 4 workers: 48 ÷ 4 = 12 days
Higher Tier Proportion — The Algebraic Approach
At Higher tier, proportion becomes algebraic. You see equations like y ∝ x² or y ∝ √x and you need to:
- Rewrite the proportion statement as an equation with a constant k
- Use given values to find k
- Use the equation with k to answer the question
The Four Standard Forms
| Statement | Equation |
|---|---|
| y ∝ x (direct proportion to x) | y = kx |
| y ∝ x² (direct proportion to x²) | y = kx² |
| y ∝ 1/x (inverse proportion to x) | y = k/x |
| y ∝ 1/x² (inverse proportion to x²) | y = k/x² |
Other forms you will see: y ∝ √x becomes y = k√x. y ∝ x³ becomes y = kx³.
Worked Example 1 — Direct Proportion
The variable y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.
Step 1: Write the equation. y = kx²
Step 2: Substitute the given values to find k.
45 = k × 3² = 9k → k = 5
Step 3: Use the equation.
y = 5x². When x = 5: y = 5 × 25 = 125.
Worked Example 2 — Inverse Proportion
The variable p is inversely proportional to q². When q = 4, p = 3. Find p when q = 6.
Step 1: Write the equation. p = k/q²
Step 2: Substitute to find k.
3 = k/16, so k = 48.
Step 3: Use the equation.
p = 48/q². When q = 6: p = 48/36 = 4/3.
Worked Example 3 — Square Root Proportion
The time T for a pendulum to swing is directly proportional to the square root of its length L. A pendulum of length 64 cm takes 1.6 seconds. How long is a pendulum that takes 2.4 seconds?
Step 1: Write the equation. T = k√L
Step 2: Substitute. 1.6 = k√64 = 8k. So k = 0.2.
Step 3: Use the equation. 2.4 = 0.2√L. So √L = 12, giving L = 144 cm.
Proportion Graphs
Each form of proportion gives a distinctive shape on a graph.
- Direct proportion (y = kx): a straight line through the origin, gradient k.
- Direct proportion to x² (y = kx²): a parabola through the origin.
- Inverse proportion (y = k/x): a hyperbola — two branches in the top-right and bottom-left quadrants. Never touches either axis.
- Inverse proportion to x² (y = k/x²): similar shape to y = k/x but falls off faster.
Recognising the graph shape is a common 2-mark question. Memorise the four shapes.
How to Spot Direct vs Inverse Proportion in a Question
Reading the question carefully is half the battle. Watch for the trigger words:
Direct proportion phrases:
- "y is directly proportional to x"
- "y varies directly with x"
- "as x doubles, y doubles"
- "y is proportional to x"
Inverse proportion phrases:
- "y is inversely proportional to x"
- "y is proportional to 1/x"
- "as x doubles, y halves"
- "y varies inversely with x"
A subtler clue: if doubling one quantity should make a journey faster or a job quicker, it is usually inverse proportion. If doubling one quantity should make the total bigger, it is usually direct.
Finding the Constant of Proportionality
Every proportion question on Higher tier asks you to find k at some point. The recipe is always:
- Write the equation in the form y = k × (something)
- Substitute the given pair of values
- Solve for k
Sometimes the question asks you for k explicitly. Sometimes it just asks you to find the equation. Either way, you have to find k first.
Where Students Lose Marks
Confusing direct and inverse. Re-read the question. The word "inversely" is the critical signal.
Wrong power of the variable. "Proportional to x²" is not the same as "proportional to x". Substitute the actual power.
Forgetting to substitute back into the new equation. Students sometimes find k correctly but then forget to use the new equation. Always write the equation with k in place before plugging in the second set of values.
Missing the negative root. When solving for a quantity that has been squared, both positive and negative roots are valid algebraically. The question usually only wants the positive value (a length, a time, a positive quantity) — but be ready to justify why you have discarded the negative root.
Rounding too early. Like every multi-step problem, keep precision until the final step.
Quick Self-Check
- If y is directly proportional to x and y = 12 when x = 4, what is y when x = 9?
- If p is inversely proportional to q and p = 6 when q = 5, what is p when q = 15?
- If A is directly proportional to r² and A = 50 when r = 5, what is A when r = 8?
Answers: (1) 27 (since k = 3, so y = 3 × 9); (2) 2 (since k = 30, so p = 30/15); (3) 128 (since k = 2, so A = 2 × 64).
If those flowed easily, you have got the technique. The exam questions are this method applied to messier numbers and dressed in a context (engines, springs, gas pressure, etc.).
Related Topics
Proportion connects directly to:
- Simultaneous equations — sometimes the constant of proportionality appears in a system you have to solve
- How to Solve Quadratic Equations — proportion to x² frequently rearranges into a quadratic
- GCSE Maths Formulas You Must Know — the four standard proportion equations are core
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