R10: Direct and Inverse Proportion
Solve problems involving direct and inverse proportion; graphical and algebraic representations
📋 Key Concepts
Direct Proportion: y is directly proportional to x means y = kx. As x increases, y increases at a constant rate.
Inverse Proportion: y is inversely proportional to x means y = k/x. As x increases, y decreases.
| Proportion Type | Equation | Graph |
|---|---|---|
| Direct | y = kx | Straight line through origin |
| Inverse | y = k/x | Curve (hyperbola) |
📝 Direct Proportion
y ∝ x means y = kx
Find k using one pair of values, then use the equation.
Find k using one pair of values, then use the equation.
Example 1
y is directly proportional to x. When x = 5, y = 30.
a) Find the equation connecting y and x.
b) Find y when x = 8.
Solution:
a) y = kx
30 = k × 5
k = 6
Equation: y = 6x
b) y = 6 × 8 = 48
Example 2
The cost of apples is directly proportional to the weight. 3 kg costs £4.50. Find the cost of 7 kg.
Solution:
Cost = k × weight
£4.50 = k × 3 kg
k = £1.50 per kg
Cost of 7 kg = £1.50 × 7 = £10.50
📝 Inverse Proportion
y ∝ 1/x means y = k/x
As x doubles, y halves. The product xy = k is constant.
As x doubles, y halves. The product xy = k is constant.
Example 3
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.
Solution:
y = k/x
15 = k/4
k = 15 × 4 = 60
y = 60/x
When x = 10: y = 60/10 = 6
Example 4
The time to complete a job is inversely proportional to the number of workers. 5 workers take 12 hours. How long for 8 workers?
Solution:
Time × Workers = k (constant)
k = 12 × 5 = 60
Time = 60/8 = 7.5 hours
📝 Recognising Proportion from Tables
Direct Proportion: y/x is constant
Inverse Proportion: x × y is constant
Inverse Proportion: x × y is constant
Example 5
State whether each table shows direct or inverse proportion:
Table A: x: 2, 4, 6 | y: 10, 20, 30
Table B: x: 2, 4, 8 | y: 24, 12, 6
Solution:
Table A: y/x = 10/2 = 5, 20/4 = 5, 30/6 = 5 (constant)
Direct proportion
Table B: x × y = 2×24 = 48, 4×12 = 48, 8×6 = 48 (constant)
Inverse proportion
📝 Graphs of Proportion
Direct: Straight line through origin, gradient = k
Inverse: Curve approaching axes but never touching them
Inverse: Curve approaching axes but never touching them
Example 6
Sketch the graph of y = 2x (direct proportion).
Solution:
Straight line through (0,0) with gradient 2.
Points: (0,0), (1,2), (2,4), (3,6)
🧮 Worked Examples — Exam Style
Proportion questions: (1) write the model, (2) find k from one pair, (3) rewrite the equation, (4) answer. Forwards or backwards — same recipe.
y ∝ x. When x = 12, y = 30. Find x when y = 20.
y = kx: 30 = 12k → k = 2.5 → y = 2.5x
20 = 2.5x → x = 8
M1 for k, A1 for 8. The reverse direction (find x) is just equation solving on the same model.
A recipe for 6 people uses 300 g of rice. How much for 9 people?
Direct proportion: 1 person = 50 g → 9 × 50 = 450 g
M1 for the unitary step, A1 for 450. Recipes, currency and fuel are the standard contexts.
Exam wording to expect: "y is proportional to x", "Find an equation for y in terms of x", tables to classify.
❓ Practice Questions
Q1: y ∝ x. When x = 7, y = 42. Find y when x = 10.
Q2: y ∝ 1/x. When x = 6, y = 5. Find y when x = 15.
Q3: 4 workers build a wall in 9 days. How long for 6 workers?
Q4: A car travels 240 miles in 4 hours at constant speed. How far in 7 hours?
Q5: Does the table show direct or inverse proportion? x: 3, 6, 9 | y: 8, 4, 2.67
Q6: y ∝ x². When x = 3, y = 18. Find y when x = 6.
Q7: P is inversely proportional to V. When V = 4, P = 30. Find P when V = 8.
✅ Answers
- y = 60 (y = 6x)
- y = 2 (y = 30/x)
- 6 days
- 420 miles
- Inverse proportion (x × y ≈ 24)
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. k = 42 ÷ 7 = 6 → y = 6 × 10 = 60
Q2. k = 6 × 5 = 30 → y = 30/15 = 2
Q3. Work = 4 × 9 = 36 worker-days → 36 ÷ 6 = 6 days
Q4. 60 mph × 7 = 420 miles
Q5. x × y = 24, 24, 24.02 — constant product → inverse
Q6. k = y/x² = 18/9 = 2 → y = 2 × 36 = 72
Q7. k = PV = 120 → P = 120/8 = 15
📌 Key Facts to Memorise
- Direct: y = kx (product not constant; graph = straight line through origin)
- Inverse: y = k/x (product x × y constant; graph = hyperbola)
- Direct square: y = kx²; inverse square: y = k/x²
- Classify from a table: check y/x, y/x², x × y — the one that stays constant names the model
- Always find k BEFORE answering — k is the method mark
🗺️ In the Exam
Proportion: 3–5 marks, Papers 2/3. Tables-for-classification test AO2; worded problems (workers, recipes, pressure-volume) test modelling.
Links to: multiplicative reasoning (R6), graphs (R8), equations (R13).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Direct proportion: y = kx, divide to find k
- Inverse proportion: y = k/x, multiply to find k
- Check: Direct - y/x constant; Inverse - xy constant
- Graphs: Direct = straight line; Inverse = curve
- Real problems: speed/distance (direct), workers/time (inverse)
🧠 Problem-Solving Strategies
Problem-Solving
For direct proportion: y = kx (constant ratio y/x). For inverse proportion: y = k/x (constant product xy). Always identify which type by checking: both increase = direct; one up one down = inverse. Use a table to organise values and find k first.
Multi-Step Problem
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 12, and find x when y = 60.
Solution: y = k/x. 15 = k/4, so k = 60. When x = 12: y = 60/12 = 5. When y = 60: 60 = 60/x, so x = 1.
⚠️ Common Errors
Watch Out!
1. Wrong: For inverse proportion, writing y = kx and solving as direct proportion Correct: Inverse proportion means y = k/x, not y = kx. If x doubles, y halves.
2. Wrong: When y = k/x and x = 4, y = 15, writing k = 4 × 15 = 60 and then y = x/60 Correct: k = x × y = 60, but the formula is y = 60/x (k goes in the numerator).
3. Wrong: Assuming all relationships where one value decreases are inverse proportion Correct: Verify by checking if xy is constant (inverse) or y/x is constant (direct).
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The time (t hours) to complete a journey is inversely proportional to the average speed (s mph). When s = 40, t = 3. (a) Find t when s = 60. (b) The driver needs to arrive 30 minutes earlier than the original journey. What speed is needed? (c) Explain why this model breaks down for very high speeds.
(a) t = k/s. 3 = k/40, so k = 120. When s = 60: t = 120/60 = 2 hours.
(b) New time = 3 − 0.5 = 2.5 hours. 2.5 = 120/s, so s = 120/2.5 = 48 mph.
(c) The model assumes speed is the only factor. At very high speeds, factors like traffic, speed limits, and road conditions make the model unrealistic.
Mark scheme: M1 for finding k, A1 k = 120, M1 for t when s = 60, A1 t = 2, M1 for reverse calculation, A1 48 mph, A1 explanation
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The number of workers (w) on a project and the days (d) to complete it are in inverse proportion. 6 workers take 10 days.
(a) How long do 15 workers take?
(b) The manager says "Doubling the workers always halves the time." Is this realistic?
(c) In practice, 9 workers took 8 days instead of the predicted value. Calculate the predicted value and suggest why the actual time differs.
Answers: (a) k = 6 × 10 = 60. d = 60/15 = 4 days. (b) Not realistic — some tasks cannot be split and coordination takes time. (c) Predicted: 60/9 ≈ 6.67 days. Actual 8 days is longer because tasks may not be parallelizable and communication overhead increases.
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