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In this lesson, we will learn about indirect variation and how to solve applications that involve indirect variation.

Related Topics:

Direct Variation Word Problems

Joint and Combined Variation

More Algebra Word Problems

There are many situations in our daily lives that involve **inverse variation** (indirect variation).

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 *x* and *y* are such that

*xy* = *k* where *k* is a non-zero constant, we say that *y ***varies inversely **with *x*.

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,

The following video defines inverse variation and shows how to solve some inverse variation problems.

1. y varies inversely as x. Given 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.

This video provides an example of how to solve a basic inverse variation problem.

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

Inverse variation problem with a change in variable.

Example: m varies inversely as t. Given m = 9 when t = 6. Find the variation constant and the inverse variation equation. Then determine m when t = 27.

Inverse variation problem with fractional variation constant.

This video provides an example of how to solve a inverse variation problem when k is a fraction

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.

This video provides an inverse variation example relating the number of workers to the number of hours it takes to complete a job.

The time, t, required to do a job varies inversely with the number of people, p, working on the job.
If it takes 6 hours for 8 workers to complete a job. How long will it take if there are 9 workers?

This video provides an inverse variation example relating the loudness in decibels and the distance from a sound source.

The loudness, L, of sound measured in decibels, dB, is inversely proportional to the square of the distance, d, in feet from the sound source.
If a person is 15 feet from a leaf blower, the sound is 100 dB. How loud is the leaf blower if someone is 45 feet away.
Round to the nearest tenth.

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.

How to tell if two variables vary inversely.

Recognizing Direct and Inverse Variation

This video looks a inverse variation: identifying inverse variations from ordered pairs, writing inverse variation equations, graphing inverse variations, and finding missing values. It includes several examples.

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

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