Direct Proportion

Two quantities are in direct proportion when they increase and decrease at the same rate. If one doubles, the other doubles too. The relationship is written y = kx, where k is the constant of proportionality.

What Does Direct Proportion Mean?

Two quantities are directly proportional if their ratio stays constant.

When $x$ doubles, $y$ doubles. When $x$ triples, $y$ triples.

Real-life examples:

The more apples you buy, the more you pay (price ∝ quantity)
The further you drive at constant speed, the more fuel you use
The number of workers and the total wages paid (same hourly rate)

The Formula

Direct Proportion

y = kx

y and x are the two quantities; k is the constant of proportionality (always the same for a given relationship)

To find $k$ : divide a known value of $y$ by the matching value of $x$ .

$k = \frac{y}{x}$

Finding the constant k

5 kg of apples cost £3.50.

$k = \frac{3.50}{5} = 0.70 \quad \text{(£0.70 per kg)}$

Cost for 8 kg: $y = 0.70 \times 8 = £5.60$

Recognising Direct Proportion Graphs

A directly proportional graph:

Is a straight line
Passes through the origin (0, 0)

If the graph doesn't pass through (0, 0), the relationship is linear but not directly proportional.

Table Check

A quick check: if $\frac{y}{x}$ gives the same value for every pair in a table, the quantities are in direct proportion.

$x$	$y$	$y/x$
2	6	3
5	15	3
10	30	3 ✓

Real-World Examples

Real world: scaling a recipe

A pasta recipe for 4 people uses 300 g of pasta.

$k = 300 \div 4 = 75 \text{ g per person}$

For 10 people: $75 \times 10 = 750 \text{ g}$

Caterers and batch cooks use this direct proportion to eliminate guesswork when scaling any recipe.

Real world: fuel costs on a road trip

A car consumes 7 litres per 100 km.

For a 450 km drive: $\dfrac{7}{100} \times 450 = 31.5 \text{ litres}$

Fuel consumption and distance are directly proportional — doubling the journey doubles the fuel needed.

Real world: currency exchange

The exchange rate is €1 = $1.10.

To convert €85: $85 \times 1.10 = \$ 93.50$

Banks, travel apps, and online shops all rely on direct proportion to convert currencies instantly.

Practice Problems

A car travels 120 km in 2 hours at constant speed. How far will it travel in 5 hours?

Show answer

k = 120/2 = 60 km/h. Distance = 60 × 5 = 300 km.

A recipe uses 200 g of flour to make 8 biscuits. How much flour is needed for 20 biscuits?

Show answer

k = 200/8 = 25 g per biscuit. Flour = 25 × 20 = 500 g.

Is this table showing direct proportion? x: 3, 6, 9 — y: 7, 14, 21

Show answer

y/x = 7/3 ≈ 2.33 for all pairs. The ratio is constant, so yes — directly proportional.