OML Search

In these lessons, we will learn about inverse variation and how to solve applications that involve inverse variation.

Related Topics:

Direct Variation Word Problems, More Algebra Word Problems

There are many situations in our daily lives that involve

For example, the number of days required to build a bridge is varies inversely to the number of workers. As the number of workers increases, the number of days required to build would decrease.

In general, when two variables

In notation, inverse variation is written as

*Example:*

Suppose that *y* varies inversely as *x* and that *y* = 8 when *x* = 3.

a) Form an equation connecting *x* and *y*.

b) Calculate the value of *y* when *x* = 10.

*Solution:*

i.e. *xy* = *k* where *k* is a non-zero constant

a) Substitute *x* = 3 and *y* = 8 into the equation to obtain *k*

3 × 8 = *k* ⇒ *k* = 24

The equation is *xy* = 24

b) When *x* = 10, 10 × *y* = 24 ⇒ *y* =

*Example:*

Suppose that *y* varies inversely as *x* ^{2} and that *y* = 10 when *x* = .

a) Find the equation connecting *x* and *y* .

b) Find the value of *y* when *x* = 3.

*Solution:*

i.e. *yx*^{2} = *k*

a) Substitute *x* = and *y* = 10 into the equation to obtain *k*

The equation is *yx*^{2} =

b) When *x* = 3,

Examples:

1. y varies inversely as x. y = 4 when x = 2. Determine the inverse variation equation. Then determine y when x = 16.

2. The time, t, required to empty a tank varies inversely as the rate, r, of pumping. If a pump can empty a tank in 2.5 hours at a rate of 400 gallons per minute, how long will it take to empty a tank at 500 gallons per minute?

3. The force, F, needed to break a board varies inversely with the length, L, of the board. If it takes 24 pounds of pressure to break a board 2 feet long, how many pounds of pressure would it take to break a board that is 5 feet long?

4. y varies inversely as the square root of x. y = 6 when x = 16. Determine the inverse variation equation. Then determine y when x = 4.

Example: y varies inversely as x. y = 3 when x = 10. Determine the inverse variation equation. Then determine y when x = 6.

Example: y varies inversely as x. y = 1/2 when x = 2/3. Find the variation constant and the inverse variation equation. Then determine y when x = 2/15.

The following video gives some practical examples of direct variation and indirect/inverse variation.

Example: On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string has a frequency of 400 cycles per second. Find the constant of proportionality and the frequency of a 10-inch string.

Recognizing Direct and Inverse Variation

Example: Let x

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.