Direct and inverse proportion
OCR J560 Higher tests algebraic proportion at A03 (problem solving) level. You must set up the equation, find the constant of proportionality, and use it.
Notation
- y is directly proportional to x: y ∝ x. Equation: y = kx.
- y is inversely proportional to x: y ∝ 1/x. Equation: y = k/x.
- y is proportional to x²: y = kx². Inversely to x²: y = k/x².
- y proportional to √x: y = k√x.
The symbol ∝ is read "is proportional to". The constant k (the constant of proportionality) is found from one given pair (x, y).
Method (every proportion question)
- Write the equation with k.
- Substitute the given pair to find k.
- Rewrite the equation with k filled in.
- Use the equation to answer.
✦Worked example— Worked example — inverse proportion
The pressure P of a gas is inversely proportional to its volume V. When V = 10 cm³, P = 12 N/cm².
Step 1: P = k/V. Step 2: 12 = k/10 → k = 120. Step 3: P = 120/V. Step 4: When V = 8, P = 120/8 = 15 N/cm². When P = 30, V = 120/30 = 4 cm³.
✦Worked example— Worked example — square law
The kinetic energy E is proportional to v². When v = 4, E = 80.
E = kv² → 80 = 16k → k = 5. So E = 5v². When v = 6, E = 5 × 36 = 180.
Recognising from data
Two clues:
- Direct: doubling x doubles y; the ratio y/x is constant.
- Inverse: doubling x halves y; the product xy is constant.
OCR Higher often gives a small table and asks "is y proportional to x or x²?" — test the ratios.
OCR mark scheme conventions
- M1 for writing the proportional equation with k.
- A1 for finding k correctly.
- A1 for the final answer in context.
- "Show clearly" demands the k value to be stated explicitly.
⚠Common mistakes
- Using y = kx when the question says inverse — should be y = k/x.
- Forgetting to square (or square-root) when the relationship is x² or √x.
- Skipping the k step and trying to scale directly — fails for square laws.
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