2016 AMC 10A Problem 5

A rectangular box has integer side lengths in the ratio

1:3:4.

Which of the following could be the volume of the box?

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Solution

Let

s

be the side length of the smallest side. Then the other two sides are

3s

and

4s.

The volume is therefore

s \cdot 3s \cdot 4s = 12s^3.

Testing out values of

s,

we see that if

s = 2,

then

12s^3 = 96,

which is an answer choice. Thus, the correct answer is D .

Concepts: volume · ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.