Direct proportion

Two quantities are in direct proportion when multiplying one of them by a number multiplies the other by the same number. Rice sold at a fixed price per kilogram behaves this way: twice the mass costs twice as much, and half the mass costs half as much.

Rice at a fixed price

Why it works: every kilogram costs the same \(\$2.40\), so the cost per kilogram is the same for every pair in the table. For any two pairs \((x_1,y_1)\) and \((x_2,y_2)\) of directly proportional quantities, \(\dfrac{y_1}{x_1}=\dfrac{y_2}{x_2}\), or equivalently \(\dfrac{y_2}{y_1}=\dfrac{x_2}{x_1}\).

Method

1. Check that the quantities change in the same ratio: the same rate and the same conditions throughout.
2. Unitary method: find the amount for one unit, then multiply by the number of units required.
3. Proportion method: write \(\dfrac{\text{new amount}}{\text{new number of units}}=\dfrac{\text{old amount}}{\text{old number of units}}\) and solve.

Example

A machine fills \(450\) bottles in \(6\) minutes at a steady rate. How many bottles does it fill in \(10\) minutes?

Unitary method: in \(1\) minute it fills \(\dfrac{450}{6}=75\) bottles, so in \(10\) minutes it fills \(75\times10=750\) bottles.

Proportion method: \(\dfrac{N}{10}=\dfrac{450}{6}\), so \(N=\dfrac{450}{6}\times10=750\).

Check: \(10\) minutes is longer than \(6\) minutes, so the answer must be more than \(450\). Two quantities that simply increase together, such as a child’s height and mass, are not automatically in direct proportion; the ratio must stay the same.