The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, SAT-II, Math Kangaroo 7-8.

Direct variation: two quantities Q and P vary directly with one another when their dependence is modeled by a line that passes through the origin.

figure

Let us choose two different values for the quantity x:

equation

Then, if y depends on x linearly (with y-intercept zero):

equation

because the y-intercept is zero this is also the slope of the line:

equation

When a problem states that two quantities vary directly it means that, as they both change, their ratio does not change.

Inverse variation: two quantities vary inversely with one another when their dependence is an equilateral hyperbola.

Points on an equilateral hyperbola have the property that the product of their coordinates is constant.

figure

When a problem states that two quantities vary inversely it means that, as they both change, their product does not change.

Let us choose two different values for the quantity x:

equation

Then, if y depends inversely on x:

equation

Example:

If y varies directly with the square of x and inversely with the square root of z, what happens to the value of y if we quadruple z and triple x?

In order to keep track of the process, let us focus on two possible combinations of the variables x, y and z. Let us label each combination:

equation

equation

Based on the definitions of direct and inverse variation, we can write an expression that does not change in value as the three variables change:

equation

We want to find ynew.

We know how x and z change:

equation

equation

equation

equation

equation

and the relation between the new value of y and the old value of y is:

equation