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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.

Let us choose two different values for the quantity x:

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

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

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.

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:

Then, if y depends inversely on x:

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:

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:

We want to find ynew.

We know how x and z change:

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