Sheet № 35 · Foundation + Higher · AQA · Edexcel · OCR
Direct & Inverse Proportion –
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 c
§Key definitions
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
§Formulas to memorise
y ∝ x means y = kx
y ∝ 1/x means y = k/x
y ∝ x² → y = kx²
y ∝ x³ → y = kx³
y ∝ √x → y = k√x
y ∝ 1/x² → y = k/x²
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.
Worked example
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Step 1: y ∝ x → y = kx
⚠ 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.
Bonus
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