Direct Proportion
Two quantities are in direct proportion when they grow together
at a fixed rate: double one and the other doubles too, halve
one and the other halves, treble one and the other trebles. They always
keep step, linked by a single multiplier:
y = kx
The number k is the
constant of proportionality: the fixed amount of
y for every one unit of x.
Buy twice as many sweets, pay twice as much. Walk for twice as long at a steady
pace, go twice as far. That steady "for every one" rate is the heart of it.
A shop sells apples at a fixed price each. Three apples cost £6:
= £6
Because cost and number of apples are in direct proportion, one
apple must cost 6 \div 3 = 2 pounds. Now scaling is easy:
5 apples cost 2 \times 5 = 10 pounds, and 10 apples cost
2 \times 10 = 20 pounds. Find the price of one, and you can
price any number.
The unitary method: find ONE, then scale
The cleanest way to use a proportion is the unitary method:
find the value of one unit first, then multiply by however many you want.
It works every time because the rate per unit never changes.
Worked example. If 3 pens cost £1.20, then one pen costs
1.20 \div 3 = 0.40 (40p), so 5 pens cost
0.40 \times 5 = 2.00 (£2.00).
Worked example. A car travels 150 km on 10 litres of fuel. Per one litre that
is 150 \div 10 = 15 km. So on 4 litres it goes
15 \times 4 = 60 km, and on 25 litres it goes
15 \times 25 = 375 km.
Worked example. 4 identical bricks weigh 6 kg. One brick weighs
6 \div 4 = 1.5 kg, so 10 bricks weigh
1.5 \times 10 = 15 kg.
A jug of lemonade uses 2 lemons per cup of water. The lemons and the water are in
direct proportion — keep the rate the same and the drink tastes the same.
per cup → double it →
Want 3 cups? That is 2 \times 3 = 6 lemons. Want to make a
giant 10-cup batch for a party? 2 \times 10 = 20 lemons.
The rate "2 lemons per cup" is the constant k.
When y is directly proportional to
x:
- they are linked by y = kx;
- the graph is a straight line through the origin with gradient k;
- find k = y \div x from one matching pair, then use it for any other;
- the unitary method is the same idea — find the value of one unit first, then scale up.
See it: a straight line through the origin
Each dot is "buy n, pay n times the
price". Because every item costs the same, the dots line up perfectly straight
and the line aims right at the origin — buy nothing and you
pay nothing. A dearer item just tilts the line steeper. Press
Refresh for a new price.
Drive the line yourself
Pull the slider to change the constant k. The line
always passes through the origin — a bigger k just
tilts it steeper.
When is it NOT direct proportion?
Not everything that grows together is in direct proportion. A taxi that charges
a £3 flag-fall plus £1 per km is not directly proportional: at 0 km you
still pay £3, so the graph crosses the cost axis above zero instead of going through the
origin. The unitary method only works when "nothing costs nothing".
Two traps to avoid:
- Find the cost of ONE first, then multiply. Don't try to jump straight
from "5 cost £10" to "8 cost £?" — first get one (£2), then scale (2 \times 8 = 16).
- Check the origin. True direct proportion means 0 of something costs 0.
If there's a fixed charge, joining fee, or starting amount that you pay even for zero items,
it is not direct proportion — and the line will not pass through the origin.