Direct & Inverse Proportion

Buy more notebooks, pay more money; hire more workers, finish in fewer days. Both changes are suspiciously tidy — tidy enough to predict with one formula. Ratios and proportion laid the groundwork; now we install the engine.

Direct proportion: the same multiplier, together

A notebook costs 3. Buy 1 and pay 3, buy 2 and pay 6, buy 5 and pay 15. Double the quantity and the total doubles — that is direct proportion, written as

$y = kx$

There is exactly one test: $\frac{y}{x}$ always equals the same number. Here $6 \div 2 = 15 \div 5 = 3$ , and that constant 3 is the constant of proportionality $k$ . On a graph, direct proportion is a straight line through the origin.

Inverse proportion: the product is conserved

A 60 km trip: at 60 km/h it takes 1 hour, at 30 km/h it takes 2 hours, at 20 km/h it takes 3 hours. Slower speed, longer time — but every product is $60 \times 1 = 30 \times 2 = 20 \times 3 = 60$ . That is inverse proportion, written as

$xy = k$

The test: $xy$ always equals the same number. The worksite version: 4 workers finish in 6 days, 8 workers finish in 3 days — the product is 24 worker-days either way.

Recognizing it from a table

Given a data table, divide first, then multiply:

$x$	1	2	4
$y$	6	12	24

$\frac{y}{x}$ is 6 in every column — direct proportion, $k = 6$ , so $y = 6x$ . If dividing gives no constant, try multiplying: for instance $x = 1, 2, 3$ with $y = 60, 30, 20$ keeps the product at 60 — inverse proportion confirmed.

The two faces of the constant k

Think of $k$ as the exchange rate of the relationship: for direct proportion $k = \frac{y}{x}$ (how much $y$ each unit of $x$ buys), for inverse proportion $k = xy$ (the conserved total). Once you find $k$ , every empty cell in the table can be computed.

InteractiveProportion Lines

Slope m1Intercept b2

y = x + 2

y-intercept

(0, 2)

x-intercept

(-2, 0)

Set $b$ to 0 and drag $m$ : every line through the origin is a declaration of direct proportion, and the bigger $m$ is, the more $y$ each step of $x$ buys. Drag $b$ away from 0 and the line stays straight but stops being proportional — see the reminder below.

Only lines through the origin count

A taxi charging 10 up front plus 2 per kilometer has a straight-line relationship with distance — but it is not proportional: riding 0 km still costs 10. Direct proportion demands $y = 0$ when $x = 0$ ; only starting together from zero deserves the name.

Check yourself

Quick quiz

1. 3 pens cost 12. How much do 7 of the same pens cost?
2. 6 workers need 10 days to build a road. How long do 12 workers need?
3. Which pair is in direct proportion?