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 .


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.


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

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This page was edited on 21-Sep-2010