GMAT Question of the Day – Problem Solving – Geometry

If s is between 0 and 9, how many different equilateral triangles with side s can be formed that have an area which is an integer value?

A. 3

B. 4

C. 5

D. 6

E. 7

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Correct Answer: E.

GMAT Question of the Day Solution

This GMAT question of the day is tough but not impossible. What makes it tough? You have to consider a few different things while keeping in mind the limitations imposed by the question. First off – what’s the area of an equilateral triangle? Now consider what numbers will create an integer value for the area? We have to get rid of the 4 in the denominator and the \( \sqrt{3} \) in the numerator. Let’s look at one issue at a time:

1. To cancel the 4 you need to have a at least a 4 in the numerator. It doesn’t need to be a 4 but could be an 8 or a 12 or a 16. Anything with at least one 4 as a factor. Keep in mind the limitation that S must be less than 9. So that leaves 2, 4, 6, and 8 each of which when squared will have at least one 4.

2. We also need to either cancel or transform the \( \sqrt{3} \). We can do this by either dividing or multiplying by \( \sqrt{3} \). This means that S must have \( \sqrt[4]{3} \) in the numerator or the denominator.

Putting both together you have these options for S:

\( 2\sqrt[4]{3} \), \( 4\sqrt[4]{3} \), \( 6\sqrt[4]{3} \)

\( \frac{2}{\sqrt[4]{3}} \), \( \frac{4}{\sqrt[4]{3}} \), \( \frac{6}{\sqrt[4]{3}} \), \( \frac{8}{\sqrt[4]{3}} \)

Note that \( 8\sqrt[4]{3} \) will not work because it is greater than 9. That leaves seven different triangles.