Proportion — equal ratios
Two quantities are in direct proportion when their ratio is constant. As one doubles, so does the other. OCR J560 tests this routinely on Foundation papers (recipe-style scaling) and on Higher (algebraic proportion notation).
The "unitary method"
Find the value of one unit, then multiply.
Example: "5 pens cost £1.75. What do 12 pens cost?"
- Cost of 1 pen = 1.75 ÷ 5 = £0.35.
- Cost of 12 pens = 12 × 0.35 = £4.20.
The "scale-factor method"
Spot the multiplier directly.
Example: "8 oranges cost £2.40. What do 24 oranges cost?"
- 24 ÷ 8 = 3 (multiplier).
- Cost = 3 × 2.40 = £7.20.
Recipe scaling
Recipe for 4 people uses 200 g flour. For 6 people: 200 × 6/4 = 300 g.
The scaling factor is 6/4 = 1.5.
Best-buy comparisons
Compare price per unit (or units per pound).
Example: "Pack A: 6 cans for £4.50. Pack B: 10 cans for £7.00. Which is better value?"
- A: 4.50/6 = 75 p per can.
- B: 7.00/10 = 70 p per can.
- B is cheaper per can → B is better value.
Direct proportion algebraically (Higher)
If y is directly proportional to x:
- y ∝ x
- y = kx for some constant k
Find k from one (x, y) pair, then use the formula.
Inverse proportion (Higher)
If y is inversely proportional to x:
- y ∝ 1/x
- y = k/x
As x doubles, y halves.
Example: "Time taken is inversely proportional to number of workers." 4 workers take 6 hours → k = 4 × 6 = 24. With 8 workers, time = 24/8 = 3 hours.
OCR mark scheme conventions
- M1 for setting up the proportion (e.g. 1.75/5 or scale factor).
- A1 for the answer with correct units.
- For algebraic proportion: B1 for ∝ statement, M1 for k, A1 for final value.
⚠Common mistakes
- Adding when you should multiply (proportion is multiplicative).
- Forgetting units (£, kg) on the final answer.
- Confusing direct with inverse — read the question carefully.
- Best-buy errors: comparing different "per" quantities (per pack vs per item).
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