# 2016 AMC 10A Problems/Problem 5

## Problem

A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box? $\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 144$

## Solution 1

Let the smallest side length be $x$. Then the volume is $x \cdot 3x \cdot 4x =12x^3$. If $x=2$, then $12x^3 = 96 \implies \boxed{\textbf{(D) } 96.}$

## Solution 2

As seen in the first solution, we end up with $12x^3$. Taking the answer choices and dividing by $12$, we get $(A) 4$, $(B) 4 \frac{2}{3}$, $(C) 5 \frac{1}{3}$, $(D) 8$, $(E) 12$ and the final answer has to equal $x^3$. The only answer choice that works is $(D)$.

## Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/bc-somFWrbg


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## Video Solution

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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 