Direct proportion

Why "directly proportional" means multiplied by a fixed number, how the unitary method and the formula y = kx are the same idea, and how to spot when a relationship is (and isn't) direct proportion.

Ratio, proportion & rates of change · Direct & inverse proportion

Direct proportion

Why "directly proportional" means multiplied by a fixed number, how the unitary method and the formula y = kx are the same idea, and how to spot when a relationship is (and isn't) direct proportion.

Why it works

Two quantities are in direct proportion when one is always the other multiplied by the same fixed number. Double one and the other doubles; triple one and the other triples; halve one and the other halves. In symbols,

y \propto x

means

y = kx

for some constant

k

— the constant of proportionality. That single number carries the whole relationship: it is "how much

y

per one

x

".

The unitary method is just finding $k$ . "12 bricks weigh 30 kg — what do 20 weigh?" First find what one brick weighs:

30 \div 12 = 2.5

kg. That

2.5

is exactly

k

\text{weight} = 2.5 \times \text{bricks}

. Then any number of bricks is one multiplication away:

20 \times 2.5 = 50

kg. Whether you call it "cost per pen", "grams per cookie" or "

k

", it is the same move: divide to get the value of one, multiply up to the amount you want.

Scaling is multiplicative, never additive. If 3 kg costs £7.50, then 5 kg does not cost £9.50 ("2 more, so add 2"). Quantities in proportion scale by multiplying: 1 kg costs £2.50, so 5 kg costs

5 \times 2.50 = £12.50

. When a value goes up by equal jumps, check the ratio, not the difference — proportion means

\frac{y}{x}

is constant.

How to recognise direct proportion. In a table,

y \div x

gives the same answer in every column (that answer is

k

). On a graph, direct proportion is a straight line through the origin — through the origin matters, because no

x

means no

y

. A line like

y = 2x + 3

is straight but not direct proportion: at

x = 0

you'd still have

y = 3

, and doubling

x

does not double

y

.

Best value is direct proportion in disguise. To compare 300 g for £1.86 with 500 g for £3.20, you can't compare the totals — the boxes are different sizes. Compare the rate: price per gram (or per 100 g). The lower rate is the better value, whatever the box size.