GCSE MATHS • HIGHER TIER

Direct Proportion Made Simple

Recognise proportional relationships, find the constant of proportionality and solve scaling problems.

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Direct Proportion in Four Moves

1

Recognise

Check that the two quantities increase or decrease at the same rate.

2

Find k

Use a known pair of values to find the constant of proportionality.

3

Substitute

Write the proportional equation and substitute the required value.

4

Check

Check that the relationship and final answer are sensible.

What You’ll Learn

Understand what direct proportion means.

Recognise directly proportional quantities.

Use the equation y = kx.

Find the constant of proportionality.

Solve direct proportion problems using unit rates.

Interpret direct proportion graphs.

Apply direct proportion to real-life problems.

Key Rules

Direct proportion means two quantities change at the same rate.

When one quantity doubles or triples, the other does the same.

Write direct proportion as:

y∝x

Use the equation :

y=kx

Worked Proof: Direct Proportion

Step-by-Step Method

  • 1. Recognise that the two quantities are directly proportional.
  • 2. Write the direct proportion equation:
    y=kx
  • 3. Substitute the known values of (x) and (y).
  • 4. Calculate the constant of proportionality:
    k=y÷x
  • 5. Rewrite the equation using the value of \(k\).

More Worked Examples

1

Question: y is directly proportional to x. When x = 4, y = 12. Find y when x = 10.

Step 1: Find k.

12 = 4k
12 = 4k
k = 3
Step 2: Write the equation.
y = 3x
Step 3: Substitute x = 10
y = 3 × 10 = 30
Answer: 30

2

Question: y is directly proportional to x. When x = 5, y = 20. Find y when x = 12

20 = 5k

k = 4
y = 4x
y = 4 × 12 = 48
Answer: 48

3

Question: A recipe uses 300g of flour for 12 cakes. How much flour is needed for 20 cakes?

300 ÷ 12 = 25g per cake

25 × 20 = 500g
Answer: 500g

4

Question: A car travels 150 km using 15 litres of fuel. How much fuel is needed for 250 km?

15 ÷ 150 = 0.1 litres per km

250 × 0.1 = 25 litres
Answer: 25 litres.
  • Common GCSE Mistakes
  • Forgetting to find the constant of proportionality first.
  • Assuming a relationship is proportional when it is not.
  • Making arithmetic errors when finding or using k.
  • Substituting values incorrectly into the proportional equation.
  • Exam Tips
  • Check first that the quantities are directly proportional.
  • For algebra questions, write y = kx before substituting.
  • Use the known x and y values to find k accurately.
  • For practical problems, finding the value for one unit can be efficient.
  • A direct proportion graph must be a straight line through the origin.
  • Substitute your answer back into the relationship to check it.

Skills Used in Direct Proportion

Direct Proportion

Ratio

Algebra

Constant of Proportionality

Constant of Proportionality

Graphs

Scaling

Problem Solving

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Frequently Asked Questions

Two quantities are directly proportional when they increase or decrease at the same rate.

The source uses y ∝ x and the equation y = kx.

k is the constant of proportionality.

Use a known pair of x and y values in y = kx and solve for k.

It is a straight line passing through the origin (0,0).

The source gives examples including shopping, fuel consumption, wages, recipes, construction, engineering and science.