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O Level · Mathematics D 4024

Direct and inverse proportion

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Cambridge O Level Mathematics (Syllabus D) 4024

Direct and inverse proportion

The big idea: If y is proportional to something, then y equals a constant times that something. Every proportion question has the same three steps: write the equation with k, find k from the given pair, then use it.

Summary — What This Chapter Covers

Use the symbol ∝ (“is proportional to”).
Express direct proportion algebraically: y ∝ x, x², x³, √x, ∛x gives y = kx, kx², kx³, k√x, k∛x.
Express inverse proportion algebraically: y = k/x, k/x², k/√x, …
Find the constant k and use the equation to find unknown quantities.

1. Direct Proportion

Syllabus 2.8 · Paper 1 & 2

y = k × (something)

“y is directly proportional to x” is written y ∝ x and means y = kx. When x doubles, y doubles. The same idea works with powers and roots of x.

In words	Symbols	Equation
y is proportional to x	y ∝ x	y = kx
y is proportional to the square of x	y ∝ x²	y = kx²
y is proportional to the cube of x	y ∝ x³	y = kx³
y is proportional to the square root of x	y ∝ √x	y = k√x
y is proportional to the cube root of x	y ∝ ∛x	y = k∛x

Worked Example

y is directly proportional to the square of x. When x = 2, y = 12. (a) Find y when x = 5. (b) Find x when y = 147 and x > 0.

Practice Question 1

(a) y ∝ √x and y = 6 when x = 4. Find x when y = 15.
(b) y ∝ x³ and y = 40 when x = 2. Find y when x = 3.
(c) y ∝ ∛x and y = 6 when x = 8. Find y when x = 125.

2. Inverse Proportion

Syllabus 2.8 · Paper 1 & 2

y = k ÷ (something)

“y is inversely proportional to x” is written y ∝ 1/x and means y = k/x. When x doubles, y halves.

Inverse proportion y ∝ 1/x → y = k/x · y ∝ 1/x² → y = k/x² · y ∝ 1/√x → y = k/√x

Worked Example

F is inversely proportional to the square of d. F = 20 when d = 3. (a) Find F when d = 6. (b) Find d when F = 45 and d > 0.

Practice Question 2

(a) y is inversely proportional to x, and y = 4 when x = 6. Find y when x = 3.
(b) The time taken to build a wall is inversely proportional to the number of workers. 6 workers take 15 days. How long would 9 workers take?

3. The Effect of a Change

Syllabus 2.8 · Paper 1 skill

Apply the same operation to the multiplier

² ² ²

Practice Question 3

³ ²

Memorise

Three steps

Direct

² ³

Inverse

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Direct and inverse proportion

Direct and inverse proportion

y = k × (something)

y = k ÷ (something)

Apply the same operation to the multiplier

Three steps

Direct

Inverse

Past papers that test Direct and inverse proportion

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