# 1961 AHSME Problems/Problem 15

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## Problem

If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is: $\textbf{(A)}\ \frac{x^3}{y^2}\qquad \textbf{(B)}\ \frac{y^3}{x^2}\qquad \textbf{(C)}\ \frac{x^2}{y^3}\qquad \textbf{(D)}\ \frac{y^2}{x^3}\qquad \textbf{(E)}\ y$

## Solution

Let $k$ be the number of articles produced per hour per person. By using dimensional analysis, $$\frac{x \text{ hours}}{\text{day}} \cdot x \text{ days} \cdot \frac{k \text{ articles}}{\text{hours} \cdot \text{person}} \cdot x \text{ people} = x \text{ articles}$$ Solving this yields $k = \frac{1}{x^2}$. Using dimensional analysis again, the number of articles produced by $y$ men working $y$ hours a day for $y$ days is $$\frac{y \text{ hours}}{\text{day}} \cdot y \text{ days} \cdot \frac{\frac{1}{x^2} \text{ articles}}{\text{hours} \cdot \text{person}} \cdot y \text{ people} = \frac{y^3}{x^2} \text{ articles}$$ The answer is $\boxed{\textbf{(B)}}$.

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