Direct and inverse proportion are tested at both Foundation and Higher tier across AQA, Edexcel and OCR. At Foundation, you need to recognise and use simple proportional relationships. At Higher, you must set up and use algebraic formulas involving proportion — including cases where y is proportional to x², x³ or √x. These questions can carry up to five marks and often appear in context. This guide covers both types of proportion with clear definitions, methods and fully worked examples.
What Is Proportion?
Two quantities are proportional when they are connected by a consistent multiplying relationship.
Direct proportion
Two quantities are in direct proportion when one increases at the same rate as the other. If you double one, you double the other.
The symbol ∝ means "is proportional to". The constant k is called the constant of proportionality.
The graph of direct proportion is a straight line through the origin.
Inverse proportion
Two quantities are in inverse proportion when one increases as the other decreases. If you double one, you halve the other.
The graph of inverse proportion is a reciprocal curve (hyperbola) — it never touches the axes.
Higher-tier variations
At Higher level, y can be proportional to powers or roots of x:
- y ∝ x² → y = kx²
- y ∝ x³ → y = kx³
- y ∝ √x → y = k√x
- y ∝ 1/x² → y = k/x²
Step-by-Step Method
How to find the formula and solve proportion problems
- Write the proportionality statement using ∝ (e.g., y ∝ x²).
- Replace ∝ with = k to create a formula (y = kx²).
- Substitute the given pair of values to find k.
- Write the complete formula with the value of k.
- Use the formula to find the unknown value.
How to recognise proportion from a table
- Direct proportion: y/x is constant for all pairs.
- Inverse proportion: x × y is constant for all pairs.
Worked Example 1 — Foundation Level (Direct Proportion)
Question: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Step 1: y ∝ x → y = kx
Step 2: Substitute x = 4, y = 20: 20 = k × 4 → k = 5
Step 3: Formula: y = 5x
Step 4: When x = 7: y = 5 × 7 = 35
Check: The ratio y/x = 20/4 = 5 and 35/7 = 5 ✓ — consistent.
Worked Example 2 — Higher Level (Inverse Proportion with Powers)
Question: y is inversely proportional to x². When x = 2, y = 50. Find: (a) y when x = 5, (b) x when y = 8.
Step 1: y ∝ 1/x² → y = k/x²
Step 2: Substitute x = 2, y = 50: 50 = k/4 → k = 200
Step 3: Formula: y = 200/x²
(a) When x = 5: y = 200/25 = 8
(b) When y = 8: 8 = 200/x² → x² = 200/8 = 25 → x = 5 (taking the positive root)
Check: When x = 5, y = 8. xy² relationship: 2² × 50 = 200 and 5² × 8 = 200 ✓
Common Mistakes
- Forgetting to find k. You must always calculate the constant of proportionality using the given values before you can use the formula.
- Confusing direct and inverse. If one quantity increases while the other decreases, it is inverse proportion, not direct. Read the question carefully.
- Using the wrong power. "y is proportional to the square of x" means y = kx², not y = k × 2x. The word "square" refers to x², not 2x.
- Not taking the correct root. When solving for x in y = k/x², you get x² = k/y, so x = √(k/y). In context, x is usually positive.
- Assuming proportion means equal. y ∝ x does not mean y = x. There is always a constant k involved.
Exam Tips
- Write the proportionality statement and formula at the start of every answer. This earns a method mark immediately.
- On AQA and OCR papers, you may be asked "is this direct or inverse proportion?". Check whether x × y or y/x is constant.
- At Higher tier, questions often combine proportion with other topics. For example, you may need to find k, then use the formula to set up and solve a quadratic equation.
- Learn to recognise the graphs: direct proportion is a straight line through the origin; inverse proportion is a reciprocal curve. This helps in multiple-choice or graph-matching questions.
Practice Questions
Question 1 (Foundation): y is directly proportional to x. When x = 6, y = 18. Find y when x = 10.
Question 2 (Higher): y is proportional to x². When x = 3, y = 36. Find x when y = 100.
Question 3 (Higher): P is inversely proportional to √d. When d = 16, P = 10. Find P when d = 100.
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Related Topics
- Ratio Basics and Sharing
- Ratio Problem Solving in Context
- Linear Graphs and Equation of a Line
- Other Graphs: Cubic, Reciprocal, Exponential
Summary
- Direct proportion: y ∝ x means y = kx. Both quantities increase (or decrease) together.
- Inverse proportion: y ∝ 1/x means y = k/x. One increases as the other decreases.
- Always find the constant of proportionality k first using given values.
- At Higher tier, y can be proportional to x², x³, √x, 1/x² and other expressions.
- Direct proportion graphs are straight lines through the origin; inverse proportion graphs are reciprocal curves.
- Write the formula clearly at the start of your answer for method marks.
- Check your answer by verifying it satisfies the proportional relationship.