R8: Ratio & Linear Functions
Relate ratios to fractions and linear functions; represent ratios graphically
📋 Key Concepts
Definition: A ratio relationship can be represented as a linear function y = kx, where k is the constant of proportionality (gradient).
Key Relationships
- Ratio a:b means y/x = b/a
- The graph of a direct proportion is a straight line through the origin
- The gradient equals the constant of proportionality
y = kx (direct proportion)
k = y/x (gradient of the line)
k = y/x (gradient of the line)
📝 Graphs of Direct Proportion
Features: The graph is a straight line that passes through the origin (0,0). The gradient is the constant of proportionality.
Example 1
y is directly proportional to x. When x = 4, y = 12.
a) Find the equation connecting y and x.
b) Sketch the graph.
Solution:
a) y = kx
12 = k × 4, so k = 3
Equation: y = 3x
b) The graph is a straight line through (0,0) with gradient 3.
Points: (1,3), (2,6), (4,12)
📝 Finding the Gradient
Method: The gradient (k) can be found from any point on the line using k = y/x.
Example 2
A graph shows the relationship between cost (£) and number of items. The line passes through (5, 15). Find the cost per item.
Solution:
Gradient = y/x = 15/5 = 3
Cost per item = £3
Equation: Cost = 3 × Number of items
📝 Ratio and Linear Functions
Connection: A ratio a:b corresponds to a gradient of b/a when plotting y against x.
Example 3
The ratio of blue paint to red paint is 2:5. Express this as a linear relationship.
Solution:
If we plot Blue (y) against Red (x):
Blue/Red = 2/5
Blue = (2/5) × Red
y = 0.4x (gradient = 0.4)
For every 5 units of red, you need 2 units of blue.
📝 Interpreting Graphs
Example 4
A graph shows distance (km) against time (hours). The line passes through (2, 80). What is the speed?
Solution:
Speed = gradient = distance/time = 80/2 = 40
Speed = 40 km/h
Equation: Distance = 40 × Time
Example 5
A conversion graph shows pounds (£) against dollars ($). £50 = $65. Write the equation.
Solution:
Gradient = $65/£50 = 1.3
Equation: $ = 1.3 × £
📝 From Graph to Ratio
Method: Read coordinates from the graph and express as a ratio in simplest form.
Example 6
A graph of y against x passes through (6, 8). Express y:x as a ratio.
Solution:
Ratio y:x = 8:6
Simplify: 8:6 = 4:3 (divide by 2)
Answer: y:x = 4:3
🧮 Worked Examples — Exam Style
Direct proportion IS a linear function through the origin: y = kx. The ratio y : x equals k : 1 — the gradient IS the ratio.
Petrol cost is proportional to litres: 20 litres cost £28. Write the function and find the cost of 32 litres.
k = 28/20 = 1.4 → C = 1.4L → 32 × 1.4 = £44.80
M1 for k = 1.4, A1 for the equation, A1 for £44.80. The gradient £1.40/litre is the unit price.
y : x = 3 : 5. Write y in terms of x and find y when x = 45.
y = (3/5)x → y = 27 when x = 45
M1 for the 3/5 multiplier, A1 for 27. Ratio-to-function: the gradient is first : second.
Exam wording to expect: "y is proportional to x", "write y in terms of x", or a ratio embedded in a graph question.
❓ Practice Questions
Q1: y is proportional to x. When x = 3, y = 18. Find the equation.
Q2: A graph passes through (4, 10). What is the gradient?
Q3: The ratio of flour to butter is 3:2. Write this as a linear function (flour in terms of butter).
Q4: A distance-time graph passes through (3, 120). Find the speed in km/h.
Q5: The ratio x:y is 5:7. What is the gradient of the line y plotted against x?
Q6: Two quantities in ratio 4 : 7. If x = 12, find y, and write y as a function of x.
Q7: A graph of cost against items passes through (5, 12). Write the cost function.
✅ Answers
- y = 6x
- 2.5 (or 5/2)
- Flour = (3/2) × Butter or F = 1.5B
- 40 km/h
- 7/5 = 1.4
📋 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 = 18 ÷ 3 = 6 → y = 6x
Q2. Through origin → gradient = 10/4 = 2.5
Q3. F = (3/2)B → F = 1.5B
Q4. 120 ÷ 3 = 40 km/h (assuming the graph starts at the origin)
Q5. Gradient = y/x = 7/5 = 1.4
Q6. k = 7/4 = 1.75 → y = 1.75 × 12 = 21; function y = 1.75x
Q7. k = 12/5 = 2.4 → C = 2.4n
📌 Key Facts to Memorise
- Direct proportion ↔ y = kx ↔ straight line through the origin
- Gradient = unit rate = the ratio (k : 1)
- From a graph: k = gradient = any point’s y ÷ x
- y : x = a : b gives y = (a/b)x
- Non-zero intercept means NOT direct proportion (fixed charges)
🗺️ In the Exam
Proportion-as-graphs: 2–4 marks on Papers 2/3. The fixed-charge version (y = mx + c) versus pure proportion (y = kx) is the AO2 discriminator.
Links to: proportion (R7/R10), straight lines (A9/A10), ratio (R4).
🧭 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
- Direct proportion graphs always pass through the origin
- The gradient k = y/x for any point on the line
- Ratio a:b gives gradient b/a when plotting y against x
- Read coordinates carefully from graphs
- Equation of direct proportion: y = kx (no constant term)
🧠 Problem-Solving Strategies
Problem-Solving
A ratio expressed as a linear function takes the form y = kx where k is the constant of proportionality. The ratio y:x corresponds to the gradient of the line. Use a table to find the multiplier between corresponding values. Check your answer by substituting back into the relationship.
Multi-Step Problem
The ratio of y to x is 3:5. Express y as a function of x. When x = 35, find y. When y = 27, find x.
Solution: y/x = 3/5, so y = 3x/5. When x = 35: y = 3(35)/5 = 21. When y = 27: 27 = 3x/5, so x = 27 × 5/3 = 45.
⚠️ Common Errors
Watch Out!
1. Wrong: If y:x = 2:3, writing y = 2x instead of y = ⅔x Correct: y/x = 2/3, so y = (2/3)x. The ratio 2:3 means y is ⅔ of x, not 2 times x.
2. Wrong: Confusing the ratio y:x with x:y when writing the function Correct: y:x = 3:5 means y = 3k and x = 5k for some k, giving y = (3/5)x. Read the ratio order carefully.
3. Wrong: Assuming the linear function must pass through the origin Correct: Only direct proportion relationships pass through the origin. If the relationship has an offset (e.g. y = 2x + 3), it is linear but NOT directly proportional.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The cost (£C) of hiring a van is directly proportional to the number of days (d). Hiring for 5 days costs £225. Write C as a function of d. A customer has a budget of £315. How many full days can they hire the van for? The company adds a £40 insurance fee. Write the new cost function and find how many full days the customer can now afford.
Step 1: C = kd. 225 = k × 5, so k = 45. Therefore C = 45d.
Step 2: For £315: 315 = 45d, so d = 7 days.
Step 3: New cost: C = 45d + 40.
Step 4: 315 = 45d + 40, so 45d = 275, d = 6.11... So 6 full days.
Mark scheme: M1 for setting up proportion, A1 for k = 45, A1 for C = 45d, M1 for solving, A1 for 7 days, M1 for new function, A1 for 6 full days
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Two taxi companies charge differently. Company A: C = 1.80d + 3 (d = distance in miles). Company B: C = 2.40d.
(a) For what distance do both companies charge the same amount?
(b) Which company is cheaper for a 5-mile journey?
(c) A customer says "Company B is always better value because there's no booking fee." Is this correct? Explain.
Answers: (a) 1.80d + 3 = 2.40d → 3 = 0.60d → d = 5 miles (b) At 5 miles both charge £12 — they're equal (c) No — for journeys under 5 miles, Company A is cheaper (e.g. 2 miles: A = £6.60, B = £4.80... wait: A = 1.80×2+3 = £6.60, B = 2.40×2 = £4.80, so B is actually cheaper for short trips). For journeys over 5 miles, A is cheaper. The customer is wrong.
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