# GMAT Math Example Problem with Variable Proportions

Today, a GMAT Math expert from Testmasters will explain a GMAT Quantitative Reasoning example problem dealing with variable proportions. Consider the following question:

A chemical container is being filled with benzene at a constant rate. It took 2 hours to fill 3/8 of its total volume. How much more time will it take to finish filling the container?

We can recognize this as a proportion problem since we have two different variables: hours and fraction of container filled, each with two different values. Since we can logically deduce that the more time that passes, the more of our container will be filled, this must be a direct proportion (when one variable increases, so too does the other, and at a constant ratio). A direct proportion can be solved by setting up the following equation:

$\frac{x_{1}}{y_{1}}=\frac{x_{2}}{y_{2}}$

Here, our first time is 2 hours and our second time is unknown. Our first fraction of the container filled is 3/8 and our second is 5/8:

$x_{1}=2$
$y_{1}=\frac{3}{8}$
$x_{2}=unknown$
$y_{2}=1-\frac{3}{8}=\frac{5}{8}$

For clarification, we get that fraction 5/8 since the question asks how much time it would take to finish filling the container. Since we have filled 3/8 already, 5/8 remains.

Then we set up the following:

$\frac{2}{\frac{3}{8}}=\frac{x_{2}}{\frac{5}{8}}$

To divide by a fraction, we multiply each of our numerators by the reciprocal of the denominator:

$\frac{16}{3}=\frac{8y_{1}}{5}$

By cross multiplying, we get:

$24y_{1}=80$
$y_{1}=3\frac{1}{3}$

1/3 of an hour is 20 minutes, giving us a final answer of 3 hours and 20 minutes.

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