# 1954 AHSME Problems/Problem 11

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## Problem 11

A merchant placed on display some dresses, each with a marked price. He then posted a sign “$\frac{1}{3}$ off on these dresses.” The cost of the dresses was $\frac{3}{4}$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was:

$\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4}$

## Solution

Let $x$ be the marked price. Then $\frac{2}{3}x$ would be the selling price of the dresses. The cost of the dresses was $\frac{3}{4}$ of the selling price, or $\frac{3}{4}\frac{2}{3}x \implies \frac{1}{2}x$. So the ratio of the cost to the marked price is $\frac{\frac{1}{2}x}{x}$, or $\frac{1}{2} \implies \fbox\textbf{(A)} \frac{1}{2}$.

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