Quantitative Comparison: Secrets of Surface Area and Volume in a Comparison Problem

It’s time to once again practice a problem for the GRE exam. This time, the professional GRE preppers at Testmasters bring you a step-by-step solution to solving surface area, volume, and quantitative comparison problems.

First, a brief note about quantitative comparison problems. For these problems, the questions can come from any area of math, but you will have to determine which quantity is larger, if both are equal, or if a relationship cannot be determined with the given information. Sometimes, you will have to calculate values for each quantity to compare them, but other times, you will not be able to find exact values because there may be variables involved. In these instances, it helps to know what happens when certain math operations are performed.

Today’s problem is a quantitative comparison problem that uses surface area and volume of a cube. The GRE does not provide any formulas, so you will have to know them before taking the exam. If you write the formulas down every time you use them, memorizing them becomes much easier.

Below are two quantities. If quantity A is larger, the answer is A. If quantity B is larger, the answer is B. If they are both equal, the answer is C, and if a relationship cannot be determined, the answer is D.

The length of a side of cube A measures 4, and the length of each side of cube B measures 3.

Quantity A

The ratio of the surface area of cube A to the volume of cube A.

Quantity B

The ratio of the surface area of cube B to the volume of cube B.

First, find the surface area of each of the cubes. The surface area is equal to sa formula. Plug in the side length for each of the cubes into the formula.

cube a sa

cube b sa

Now, recall that the volume of a cube is volume formula. Plug in the side lengths for each of the cubes into this volume formula to find the volumes of the respective cubes.

cube a v

cube b v

Because the question asked for the ratio of each cube’s surface area to its volume, divide the surface area of cube A by its volume and repeat the process for cube B.

cube a sa over v

cube b sa over v

Because the ratio of the surface area of cube B to its volume is greater than the same ratio for the surface area and the volume of cube A, the answer is (B).

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