Variation problems involve formulas which show the relationship between two or more variables. In some formulas, when one value increases, another value decreases. An example of such a formula is h = 20/b. In such cases, we say that h varies inversely with b or h is inversely proportional to b. In general, if y varies inversely with x then y = k/x for some value of k, k ≠ 0.

**Example:** Suppose y varies inversely with x. If y = 5 when x = 14, what is x when y = 8 ?

Set up the equation for inverse variation and solve for k.

y = k/x

5 = k/14

70 = k

Substitute 70 for k and 8 for y into the original equation and solve for x.

8 = 70/x

8x = 70

x = 70/8

x = 35/4

**Example:** If y varies inversely with x and y = 6 when x = 5, what is y when x = 12 ?

Set up the equation for inverse variation and solve for k.

y = k/x

6 = k/5

30 = k

Substitute 30 for k and 12 for x in the original equation and solve for y.

y = 30/12

y = 5/2

**Example:** On a dance floor, the amount of floor space per person is inversely proportional to the number of people. If n represents the number of people and s represents the amount of square feet of floor space per person, then s = k/n represents the relationship between s and n.

If there are 12 people on the dance floor and each of them have 11 square feet of floor space, how much floor space will each person have if there are 20 people on the dance floor?

Substitute 12 for n and 11 for s and solve for k.

11 = k/12

132 = k

Now substitute 132 for k and 20 for n and solve for s.

s = 132/20

s = 6.4 feet2

**Example:** If y varies inversely with the square of x and y is 6 when x is 4, what is y when x is 10 ?

The equation for this is y = k/x2 (notice the difference from y = k/x since the variation is with the square of x)

Substitute the value for y and x and solve for k.

6 = k/(42)

6 = k/16

96 = k

Now substitute 16 for k and 10 for x in the original equation and solve for y.

y = 96/(102)

y = 96/100

y= 24/25 or 0.96

In closing, the process for solving inverse variation problems is to set up the equation for inverse variation and solve for k. Then substitute the value for k and the other known variable and solve for the unknown variable.

This guide should help assist those with confusion on the topic of inverse variation.