The idea

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.

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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

One fixed multiplier

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:

y = kx

The unitary method is 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.

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