The Constant of Proportionality

One number hidden in the table

Here are three pairs of numbers: when x = 2, y = 6; when x = 4, y = 12; and when x = 6, y = 18. Divide each y by its x: 6 ÷ 2 = 3, 12 ÷ 4 = 3 and 18 ÷ 6 = 3. The answer is 3 every time. Every y is its x multiplied by the same number, 3, so y is in direct proportion to x.

The pairs (2, 6), (4, 12) and (6, 18) lie on a straight line through 0, and each y is 3 times its x.

Give the number a name: k

Whenever y is in direct proportion to x, there is one fixed number that every x is multiplied by to give its y. We call that number k and write y = kx, which means y = k × x. The number k is called the constant of proportionality. For the pairs above, k = 3, so y = 3x.

To find k, divide any y by its x: k = y / x. Once you know k, you can find the y for any x. When x = 10, y = 3 × 10 = 30.

The symbol $\propto$

There is a symbol for "is proportional to": $\propto$. The statement $y \propto x$ is read "y is proportional to x", and it means y = kx for some fixed number k. It says that such a number exists without saying what it is.

To find k, you need one pair of values. If $y \propto x$ and y = 15 when x = 5, then k = 15 / 5 = 3 and y = 3x.

Inverse proportion has a constant too

In inverse proportion, the quotient is not fixed, but the product is. 2 workers take 12 days to build a wall, 4 workers take 6 days and 6 workers take 4 days: 2 × 12 = 4 × 6 = 6 × 4 = 24 every time. Here the constant is the product, x × y = 24, so k = 24 and y = 24 / x.

In the drawing below, the left graph shows direct proportion, y = 2x, and the right graph shows inverse proportion, x × y = 8, drawn as a rectangle x wide and y tall. Start at x = 2 and double it.

Drag x from 2 to 4. On the left, y / x stays 2 while y doubles. On the right, x × y stays 8 while y halves, and the rectangle keeps its area.

The usual mistakes

Multiplying instead of dividing to find k. If y = 12 when x = 4, then k = 12 / 4 = 3, not 12 × 4 = 48. The constant is what x is multiplied by to give y, so undo the multiplication.

Subtracting instead of dividing. 12 − 4 = 8 is how much bigger y is than x, but y = kx is about multiplying: k = 12 / 4 = 3.

Reading $y \propto x$ as y = x. y = x is only the case k = 1. The symbol $\propto$ allows any fixed number k.

Checking only one pair. Any one pair of numbers gives a quotient. Two quantities are in direct proportion only if every pair gives the same quotient, so check them all.