Direct Proportion Graphs –

Direct proportion graphs are tested across both Foundation and Higher GCSE Maths papers. Recognising a proportional relationship from a graph and finding the constant of proportionality are essential skills that link ratio, algebra, and graphical work.

What Are Direct Proportion Graphs?

Two quantities are in direct proportion if when one doubles, the other doubles; when one trebles, the other trebles. The relationship can be written as y = kx, where k is the constant of proportionality. Graphically, this produces a straight line that passes through the origin (0, 0).

The gradient of the line equals k. For example, if a graph shows that 4 litres of paint costs £20, the gradient is 20 ÷ 4 = 5, so k = 5 and the relationship is cost = 5 × litres.

Not every straight-line graph represents direct proportion. If the line does not pass through the origin (for example, y = 2x + 3), the relationship is linear but not directly proportional. This distinction is a common exam question.

Key Formulas

Step-by-Step Method

1. Check whether the graph is a straight line passing through the origin — if so, it shows direct proportion.
2. Choose a clear point on the line (not the origin) and read its coordinates (x, y).
3. Calculate the constant of proportionality: k = y / x.
4. Write the equation as y = kx.
5. Use this equation to find unknown values by substituting.

Worked Example 1 — Foundation Level

Question: A graph shows the cost of apples. At 3 kg the cost is £4.50. Show that cost is directly proportional to weight and find the cost of 7 kg.

Working:

Step 1 — The graph is a straight line through the origin, so cost is directly proportional to weight.

Step 2 — k = 4.50 / 3 = 1.50.

Step 3 — The equation is C = 1.5w.

Step 4 — For 7 kg: C = 1.5 × 7 = £10.50.

Answer: £10.50

Worked Example 2 — Higher Level

Question: y is directly proportional to x. When x = 8, y = 20. Find the value of y when x = 14.

Working:

Step 1 — y = kx. Substitute: 20 = k × 8, so k = 20 / 8 = 2.5.

Step 2 — The equation is y = 2.5x.

Step 3 — When x = 14: y = 2.5 × 14 = 35.

Answer: y = 35

Worked Example 3 — Exam Style

Question: The graph below shows the relationship between distance (d km) and fuel used (f litres). The line passes through the origin and the point (50, 4). (a) Write a formula for f in terms of d. (b) How much fuel is needed for 120 km? (c) How far can the car travel on 10 litres? (5 marks)

Working:

(a) k = f / d = 4 / 50 = 0.08, so f = 0.08d.

(b) f = 0.08 × 120 = 9.6 litres.

(c) 10 = 0.08d, so d = 10 / 0.08 = 125 km.

Answer: (a) f = 0.08d (b) 9.6 litres (c) 125 km

Common Mistakes

• Assuming any straight line shows direct proportion. The line must pass through the origin. A line like y = 3x + 2 is linear but not directly proportional.
• Using the origin to calculate k. Since 0/0 is undefined, always use a point where both x and y are non-zero.
• Mixing up k and 1/k. If 5 litres costs £10, then k = 10/5 = 2 (cost per litre), not 5/10 = 0.5. Make sure you divide the y-value by the x-value.

Exam Tips

• If the question says "directly proportional," immediately write y = kx and find k from the given information.
• On graph questions, check that the line starts at (0, 0) before concluding direct proportion.
• The constant k has real-world meaning — state it in context (e.g. "£2 per litre" or "0.08 litres per km").

Practice Questions

Q1 (Foundation): y is directly proportional to x. When x = 5, y = 15. Find y when x = 9.

Q2 (Foundation): A graph passes through the origin and (4, 12). Write the equation.

Q3 (Higher): The cost of ribbon is directly proportional to its length. 2.5 metres costs £3.75. Find the cost of 8 metres.

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Related Topics

• Direct and Inverse Proportion
• Linear Graphs and Equation of a Line
• Real-Life Graphs

Summary

• Direct proportion means y = kx, and the graph is a straight line through the origin.
• The constant k equals y divided by x, and it represents the gradient of the line.
• If the line does not pass through the origin, the relationship is not directly proportional.
• Use k to find unknown values by substituting into y = kx.
• Always state what k represents in context when answering exam questions.