Direct and Inverse Proportion

Proportion is a key topic in AQA GCSE Mathematics and describes how two quantities change in relation to each other. Understanding direct and inverse proportion allows you to solve problems ranging from simple recipes to complex scientific relationships. Higher-tier students must also be comfortable setting up and using algebraic equations for proportion.

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Direct Proportion

Two quantities are in direct proportion if, when one increases, the other increases at the same rate. If one doubles, the other doubles. If one trebles, the other trebles.

Key Features of Direct Proportion

Property	Description
Relationship	As one quantity increases, the other increases at the same rate
Graph	A straight line through the origin
Equation	y = kx, where k is the constant of proportionality
Ratio	The ratio y/x is always constant
Notation	y is directly proportional to x is written as y is proportional to x

Using the Unitary Method

The unitary method is the simplest approach at Foundation tier: find the value of one unit, then multiply.

Worked Example 1: 5 pens cost 3.50 pounds. How much do 8 pens cost?

• Cost of 1 pen = 3.50 / 5 = 0.70 pounds
• Cost of 8 pens = 0.70 x 8 = 5.60 pounds

Worked Example 2: A car travels 150 miles on 20 litres of fuel. How far can it travel on 35 litres?

• Distance per litre = 150 / 20 = 7.5 miles
• Distance on 35 litres = 7.5 x 35 = 262.5 miles

Exam Tip: The unitary method works for almost all proportion problems at Foundation tier. Find the value of ONE, then scale up or down.

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Setting Up Direct Proportion Equations [H]

For Higher tier, you need to set up equations using the constant of proportionality, k.

Method

1. Write the proportional relationship (e.g. y = kx).
2. Substitute known values to find k.
3. Write the full equation.
4. Use it to find unknown values.

Worked Example 3: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.

1. y = kx
2. 20 = k x 4, so k = 5
3. y = 5x
4. When x = 7: y = 5 x 7 = 35

Worked Example 4: y is directly proportional to the square of x. When x = 3, y = 36. Find y when x = 5.

1. y = kx squared
2. 36 = k x 9, so k = 4
3. y = 4x squared
4. When x = 5: y = 4 x 25 = 100

graph TD
    A[Identify the relationship] --> B[Write the equation y = kx or y = kx squared etc.]
    B --> C[Substitute known values to find k]
    C --> D[Write the complete equation]
    D --> E[Substitute new value to find the unknown]

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Inverse Proportion

Two quantities are in inverse proportion if, when one increases, the other decreases at the same rate. If one doubles, the other halves.

Key Features of Inverse Proportion

Property	Description
Relationship	As one quantity increases, the other decreases
Graph	A curved line (reciprocal/hyperbola) that never touches the axes
Equation	y = k/x, where k is the constant of proportionality
Product	The product y times x is always constant
Notation	y is inversely proportional to x

Using the Unitary Method for Inverse Proportion

Worked Example 5: 6 workers can build a wall in 12 days. How long would it take 9 workers?

• Total worker-days = 6 x 12 = 72
• Time for 9 workers = 72 / 9 = 8 days

Worked Example 6: A journey takes 3 hours at 60 mph. How long does it take at 90 mph?

• Total distance = 60 x 3 = 180 miles
• Time at 90 mph = 180 / 90 = 2 hours

Exam Tip: The key test for inverse proportion: if you multiply the two quantities together and always get the same answer, they are inversely proportional.

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Setting Up Inverse Proportion Equations [H]

Worked Example 7: y is inversely proportional to x. When x = 2, y = 10. Find y when x = 5.

1. y = k / x
2. 10 = k / 2, so k = 20
3. y = 20 / x
4. When x = 5: y = 20 / 5 = 4

Worked Example 8: y is inversely proportional to x squared. When x = 4, y = 5. Find y when x = 10.

1. y = k / (x squared)
2. 5 = k / 16, so k = 80
3. y = 80 / (x squared)
4. When x = 10: y = 80 / 100 = 0.8

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Comparing Direct and Inverse Proportion

Feature	Direct Proportion	Inverse Proportion
As x increases...	y increases	y decreases
Equation	y = kx	y = k / x
Graph through origin?	Yes	No
Constant	y / x = k	y times x = k
Real-world example	Cost and quantity	Speed and time

graph LR
    A[Proportion] --> B[Direct]
    A --> C[Inverse]
    B --> D[y = kx]
    B --> E[Straight line through origin]
    C --> F[y = k/x]
    C --> G[Reciprocal curve]

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Proportion in Context

Worked Example 9: The number of hours of sunshine, h, is directly proportional to the number of ice creams sold, n. On a day with 6 hours of sunshine, 240 ice creams were sold. How many ice creams would be sold on a day with 10 hours of sunshine?

1. n = kh
2. 240 = k x 6, so k = 40
3. n = 40h
4. When h = 10: n = 40 x 10 = 400 ice creams

Exam Tip: On the exam, look for key phrases: "directly proportional" means y = kx; "inversely proportional" means y = k/x; "proportional to the square" means y = kx squared. Underline these phrases in the question.

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