Variation

To solve a variation problem, it is necessary to write a variation equation.  First understand that all variation problems must include a value for k which is the constant of variation (sometimes called the constant of proportionality.)  Notice in the following three types of variation that k is in present in each of the equations.

If there is a direct variation, such as d varies directly as t, then the equation would be .
If there is an inverse variation, such as V varies inversely as T, then the equation would be .

If there is a joint variation (two or more variances), such as y varies jointly as x and z, then the equation would be .

Example:

If x varies directly as y, and x = 9 when y = 3, find x when y = 12.

First write the appropriate variation equation.  Since it is a direct variation, write

Next substitute the first situation, x = 9 when y = 3, into the equation and find k the constant of variation.  Since , then .

Now rewrite the direct variation equation using the value of, to get.

Finally substitute the value of y = 12 into the new direct variation equation to get  and find that.

Variation applications involve reading a problem and setting up a variation equation to solve the problem.

Example:

Current, C, in an electrical current is inversely proportional to the resistance, r.  The current is 20 amperes when the resistance is 5 ohms.  Find the current when the resistance is 10 ohms.

First write the appropriate variation equation, .

Next find the constant of variation, k, by substituting the values of the first situation given into the variation equation.
Since C = 20 when r = 5, then  and .

Now rewrite the variation equation using the value of .  The new variation equation would be .

Finally find the current when r = 10 using the new variation equation with the value of  k that was found,  and find that amperes.

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 21-Sep-2010