# 1961 AHSME Problems/Problem 15

## 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 1

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 $\boxed{\textbf{(B)}}$.

## Solution 2 (Simple logic)

The question is based on the assumption that each person, each hour, each day, will be produce a constant number of items (maybe fractional).

So it takes $x$ men $x$ hours to produce $\frac{x}{x}=1$ item in a day.

In a similar manner, 1 man, 1 hour, for a day, can produce $\frac{1}{x^2}$ items. So $y$ men, $y$ hours a day, for $y$ days produce $\frac{y^3}{x^2}$ items. Therefore, the answer is $\boxed{B}$.

Dimensional analysis is definitely the most rigid, but if you know the ending units (e.g. you know that density is measured in $g/cm^2$ or something like that, you can just treat is as simple proportions and equations.

~hastapasta

## What's happening here? Why isn't the answer "B" ?

Notice that if we change the problem to $x$ men produce $x$ items a day, $y$ men produces how many items a day, then the answer would be $y$. In this case, it would be a direct variation. However, notice that direct variations only have two factors --- an independent and dependent variable each (cause-effect, $x$- $y$). However, there are 3 factors, not 1, that are contributing to how many items are produced in the original problem. This is a joint variation problem, not a direct variation problem. This is the reason why the answer is $\boxed{B}$ (also see that the base unit (1 man/1 hour/1 day) is $\frac{1}{x^2}$).

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