Direct Variation vs. Inverse Variation

SAT Math Tutorial

Direct Variation vs. Inverse Variation

Intro

The SAT often asks questions about direct variation and inverse variation. These questions use wording like “…varies directly with…”, “…varies inversely with…”, “…is directly proportional to…” and “…is inversely proportional to…”

The formulas for direct and inverse variation are as follows.

Direct Variation

As one variable increases, the other variable increases proportionately.

    \[ y=kx \]

Inverse Variation

As one variable increases, the other variable decreases proportionately.

    \[ y=\frac{k}{x} \]

Here is a typical SAT variation problem.

Sample Problem

If x varies directly with y, and x=15 when y=72, then what is the value of x when y=48?

24


22.5


10


4.8


5


Solution

The first step for this problem is to determine whether to use the direct variation formula or the inverse variation formula.

In this case, the problem states that “x varies directly with y.” Thus, the relationship between x and y is a direct one, requiring the direct variation formula, y=kx.

Next, find the value for k by substituting the appropriate values for x and y into the direct variation formula.

    \begin{align*} y&=kx \notag \\ 72&=k\cdot15 \notag \\ \frac{72}{15}&=k \notag \\ 4.8&=k \notag \end{align*}

Once the value of k is determined, substitute it into y=kx to see the true relationship between x and y.

    \[ y=4.8x \]

Finally, the question asks for “the value of x when y=48.” So substitute 48 for y in the equation above and solve for x.

    \begin{align*} y&=4.8x \notag \\ 48&=4.8x \notag \\ \frac{48}{4.8}&=x \notag \\ 10&=x \notag \end{align*}

Thus, the answer is x=10.

Challenge Questions
How would the answer change if the question stated that “x varies inversely with y?” What if it stated that “x varies inversely as the square of y?”



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