Inverse Variation at a Glance

If you thought direct variation was a cinch, welcome to inverse variation. When two variables vary inversely, that means what goes up goes down, and what does down goes up. This numerical seesawing goes on and on till you get dizzy or solve the problem—whatever comes first.

In algebra-land, here's the general form for inverse variation:

This states y varies inversely with x, where k is still a constant of variation. "Inversely" means that as x increases, y decreases, and as x decreases, y increases.

Another way to say this is y is inversely proportional to x. Or that y is in inverse proportion to x.

We can also rearrange that inverse variation formula as k = xy. Notice that when x increases, y decreases so that the product k remains the same.

Still having fun, inversions and all? Check out the sample problems below.

Sample Problem

If y varies inversely with x, and y = 8 when x = 2, find y when x = 4.

First, write the general form for inverse variation.

Now plug in our values for x and y to find k.

Solve for k.

k = 16

Plug k back into the general form for inverse variation.

To finish up, we find y when x = 4.

Which means y = 4.

Sample Problem

If y is inversely proportional to x², and y = 4 when x = 5, find y when x = 10.

Write the general form for inverse variation. This time, we're told y varies inversely with x², not just plain-old x.

Now plug in our values for x and y to find k.

Solve for k.

k = 100

Plug k back into the general form for inverse variation.

Now find y when x = 10.

Sample Problem

For gases at a constant temperature—sewer gas, for example—volume varies inversely with pressure. If a gas has a volume of 50.0 L at a pressure of 2.00 atm, find its volume at a pressure of 4.00 atm.

First, write the general form for inverse variation.