Inverse Proportion

Two quantities are in inverse proportion when one increases as the other decreases by the same factor. The product of the two quantities is always constant, and the formula is y = k/x.

What is Inverse Proportion?

Two quantities are inversely proportional when one increases as the other decreases at the same rate.

If you double one quantity, the other halves. Their product stays constant.

Real-life examples:

Workers and time: more workers → less time to finish a job
Speed and journey time: driving faster → journey takes less time
Gear teeth: a bigger gear turns more slowly

The Formula

Inverse Proportion

y = \frac{k}{x}

y and x are the two quantities; k is the constant of proportionality (= x × y for any pair)

To find $k$ : multiply any matching pair of values. $k = xy$ .

Workers and time

4 workers can paint a fence in 6 hours.

Find k: $k = 4 \times 6 = 24$

How long for 3 workers? $y = 24 / 3 = 8$ hours

How long for 8 workers? $y = 24 / 8 = 3$ hours

Graph Shape

An inversely proportional graph is a hyperbola — a curve that gets closer to the axes but never touches them.

Unlike direct proportion, it does not pass through the origin.

Comparing Direct and Inverse

Feature	Direct ( $y = kx$ )	Inverse ( $y = k/x$ )
One doubles…	the other doubles	the other halves
Graph	straight line through origin	hyperbola
Constant	$y/x = k$	$xy = k$

Real-World Examples

Real world: speed and travel time

A journey is 240 km long. At 60 km/h it takes 4 hours.

$k = 60 \times 4 = 240$

At 80 km/h: $t = 240 \div 80 = 3 \text{ hours}$

Speed and travel time are inversely proportional — driving faster saves time, and the product (speed × time) stays constant at 240.

Real world: workers and a project deadline

4 bricklayers can build a wall in 6 days.

$k = 4 \times 6 = 24$

To finish in 3 days: $n = 24 \div 3 = 8 \text{ bricklayers needed}$

Construction managers use this to decide how many workers to hire when a deadline moves.

Real world: gear ratios on a bicycle

A cyclist uses a front gear with 52 teeth and swaps rear gears between 13 and 26 teeth.

With 13 teeth: $52 \div 13 = 4$ wheel turns per pedal turn.

With 26 teeth: $52 \div 26 = 2$ wheel turns per pedal turn.

Gear size and the number of wheel turns per pedal are inversely proportional — a smaller rear gear gives a higher gear ratio and more speed on flat roads.

Practice Problems

6 volunteers can clean a park in 4 hours. How long would it take 8 volunteers?

Show answer

k = 6 × 4 = 24. Time = 24 ÷ 8 = 3 hours.

At 60 km/h a journey takes 3 hours. How long at 90 km/h?

Show answer

k = 60 × 3 = 180. Time = 180 ÷ 90 = 2 hours.

A table shows x: 2, 4, 6 and y: 12, 6, 4. Is this inverse proportion?

Show answer

xy = 24 for all pairs (2×12, 4×6, 6×4). Yes — the product is constant, so inverse proportion.