R7: Proportion
Understand proportion as equality of ratios; relate ratios to fractions
📋 Key Concepts
Definition: A proportion states that two ratios are equal. If a:b = c:d, then the two ratios are in proportion.
Key Terms
- Direct proportion: As one quantity increases, the other increases at the same rate
- Inverse proportion: As one quantity increases, the other decreases
- Proportional: Quantities that change in a constant ratio
If a:b = c:d, then a×d = b×c
Cross-multiplication to check proportion
Cross-multiplication to check proportion
📝 Direct Proportion
Direct Proportion: y is directly proportional to x means y = kx, where k is a constant.
Example 1
y is directly proportional to x. When x = 4, y = 12. Find y when x = 9.
Solution:
Step 1: Find k
y = kx, so 12 = k × 4
k = 12 ÷ 4 = 3
Step 2: Use k to find y
y = 3 × 9 = 27
Answer: y = 27
Example 2
5 notebooks cost £8. How much do 12 notebooks cost?
Solution:
Cost is directly proportional to number of notebooks.
One notebook = £8 ÷ 5 = £1.60
12 notebooks = 12 × £1.60 = £19.20
Answer: £19.20
📝 Checking Proportion
Method: Cross-multiply to check if two ratios are in proportion.
Example 3
Are 6:10 and 9:15 in proportion?
Solution:
Cross-multiply: 6 × 15 = 90 and 10 × 9 = 90
Since both products equal 90, the ratios are in proportion.
Answer: Yes
Alternative: Simplify both - 6:10 = 3:5 and 9:15 = 3:5
📝 Finding Missing Values
Method: Use cross-multiplication when one value is missing in a proportion.
Example 4
Find x if 3:5 = 12:x
Solution:
Cross-multiply: 3 × x = 5 × 12
3x = 60
x = 60 ÷ 3 = 20
Answer: x = 20
Example 5
The ratio of flour to sugar in a recipe is 5:3. If you use 250g of flour, how much sugar is needed?
Solution:
Set up proportion: 5:3 = 250:x
One part = 250 ÷ 5 = 50g
Sugar = 3 × 50 = 150g
Answer: 150g sugar
📝 Ratios and Fractions
Converting: A ratio a:b can be expressed as fractions a/(a+b) and b/(a+b) for parts of a whole.
Example 6
A ratio of red:blue marbles is 3:7. What fraction of the marbles are red?
Solution:
Total parts = 3 + 7 = 10
Fraction red = 3/10
Fraction blue = 7/10
🧮 Worked Examples — Exam Style
Direct proportion: both quantities rise together (y = kx). Inverse: one rises as the other falls (y = k/x). Exam questions expect you to find k first, then answer.
y is directly proportional to x. When x = 4, y = 10. Find y when x = 14.
y = kx → 10 = 4k → k = 2.5
So y = 2.5x → y = 2.5 × 14 = 35
M1 for finding k, A1 for 35. Writing the proportional statement first (y = kx) banks the method mark even if k goes wrong.
4 workers build a wall in 6 days. How long would 8 workers take (same rate)?
More workers → fewer days, so this is INVERSE proportion.
Total work = 4 × 6 = 24 worker-days. 24 ÷ 8 = 3 days.
M1 for 4 × 6, A1 for 3. Doubling workers halves the time — sanity-check the direction before answering.
Exam wording to expect: "y is (inversely) proportional to x", symbol ∝, or "best value" money comparisons.
❓ Practice Questions
Q1: y ∝ x. When x = 6, y = 18. Find y when x = 15.
Q2: Are 8:12 and 12:18 in proportion?
Q3: Find x if 4:7 = 20:x
Q4: A mixture uses sand and cement in ratio 5:1. What fraction of the mixture is cement?
Q5: 3 kg of apples cost £4.50. How much do 7 kg cost?
Q6: y ∝ x. When x = 2, y = 8. Find y when x = 9.
Q7: 5 tins of soup cost £6.25. How much do 8 tins cost?
✅ Answers
- y = 45 (k = 3)
- Yes, both simplify to 2:3
- x = 35
- 1/6
- £10.50
🧮 Worked Examples — Harder Exam Style
8 pens cost £3.60. Work out the cost of 13 pens.
1 pen: £3.60 ÷ 8 = £0.45
13 pens: 13 × £0.45 = £5.85
M1 for £3.60 ÷ 8, A1 for £5.85. If the numbers divide awkwardly (e.g. cost of 13 from a 12-pack price), find the price of 1 item via the HCF-friendly route instead.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. y = kx, k = 18 ÷ 6 = 3 → y = 3 × 15 = 45
Q2. 8 : 12 = 2 : 3 and 12 : 18 = 2 : 3 → yes, in proportion. Cross-check: 8 × 18 = 144 = 12 × 12 ✓
Q3. 4 → 20 is × 5, so x = 7 × 5 = 35
Q4. Parts: 5 + 1 = 6. Cement = 1 part of 6 = 1/6
Q5. £4.50 ÷ 3 = £1.50 per kg → 7 × £1.50 = £10.50
Q6. k = 8 ÷ 2 = 4 → y = 4 × 9 = 36
Q7. £6.25 ÷ 5 = £1.25 per tin → 8 × £1.25 = £10
📌 Key Facts to Memorise
- Direct proportion (∝): both rise together — y = kx
- Inverse proportion: one rises as the other falls — y = k/x
- Unitary method: find the value of ONE item first, then scale
- Best buys: compare price per unit (or amount per £) — same unit both sides
- Direction check before answering: doubling workers HALVES the time
🗺️ In the Exam
Proportion questions: 3–5 marks, usually Paper 2/3 (calculator): best buys, recipes, exchange rates, workers-and-days. The "show it is inversely proportional" version appears at Higher.
Links to: ratio (R4), compound measures (R11), percentages (R9).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Cross-multiply to solve proportional equations
- Check proportion by simplifying both ratios
- For direct proportion, y = kx (find k first)
- To find a missing value: multiply diagonally, divide by the remaining value
- Remember: "proportional to" means "a constant multiple of"
🧠 Problem-Solving Strategies
Problem-Solving
For direct proportion, use the unitary method: find the value for 1 unit, then scale up. Set up equivalent fractions to check proportionality: a/b = c/d. For proportional reasoning, use the "double number line" or "multiplier" approach. If two quantities are in proportion, their ratio stays constant.
Multi-Step Problem
5 painters can paint a fence in 8 days. How long would 12 painters take, assuming they all work at the same rate?
Solution: Total work = 5 × 8 = 40 painter-days. For 12 painters: time = 40 ÷ 12 = 3⅓ days (3 days 8 hours).
⚠️ Common Errors
Watch Out!
1. Wrong: Assuming more workers means more time (e.g. 12 painters take longer than 5) Correct: More workers means LESS time — this is inverse proportion, not direct proportion. Total work is constant.
2. Wrong: Setting up proportion as 5/8 = 12/x (treating it as direct proportion) Correct: For inverse proportion: 5 × 8 = 12 × x, so x = 40/12 = 3.33. The product is constant, not the ratio.
3. Wrong: Confusing the order in a proportion calculation: e.g. 5/12 = x/8 Correct: Match corresponding values: 5 workers → 8 days, 12 workers → x days. For inverse: 5 × 8 = 12x.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A water tank leaks at a constant rate. After 3 hours, 15 litres have leaked. The tank holds 240 litres when full. It is currently 75% full. How long until the tank is completely empty? Give your answer in hours and minutes.
Step 1: Rate of leak = 15 ÷ 3 = 5 litres per hour.
Step 2: Current volume = 75% of 240 = 0.75 × 240 = 180 litres.
Step 3: Time to empty = 180 ÷ 5 = 36 hours = 36 hours 0 minutes.
Mark scheme: M1 for finding rate, A1 for 5 l/hr, M1 for 75% calculation, A1 for 180 litres, M1 for ÷5, A1 for 36 hours
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A café sells coffee. When the price is £2.40, they sell 180 cups per day. When the price rises to £2.80, they sell 150 cups per day.
(a) Is the relationship between price and cups sold direct proportion? Explain.
(b) Calculate the daily revenue at each price.
(c) Which price gives higher revenue? Is this always the best strategy for the café?
Answers: (a) No — if direct proportion, doubling price would halve sales. £2.40/£2.80 = 0.857 but 180/150 = 1.2, so not proportional (b) Revenue at £2.40: 180 × 2.40 = £432. Revenue at £2.80: 150 × 2.80 = £420 (c) £2.40 gives higher revenue, but the café must also consider profit (cost per cup), not just revenue
📝 Exam Questions by Topic
🎬 Video Resources
Ready to ace your GCSE Mathematics exams?
Get the best revision books and guides to boost your grades.