1953 AHSME Problems/Problem 43

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If the price of an article is increased by percent $p$, then the decrease in percent of sales must not exceed $d$ in order to yield the same income. The value of $d$ is:

$\textbf{(A)}\ \frac{1}{1+p} \qquad \textbf{(B)}\ \frac{1}{1-p} \qquad \textbf{(C)}\ \frac{p}{1+p} \qquad \textbf{(D)}\ \frac{p}{p-1}\qquad \textbf{(E)}\ \frac{1-p}{1+p}$

Solution

It turns out that none of the listed solutions are the true solution for the problem as written, so we'll edit it slightly by replacing "percent" with "proportion". To solve the modified problem, note that the price of the article is $1+p$ times what it was originally, so that demand must be at least $\frac{1}{1+p}$ times what it was originally to yield at least the same income. Since $1-d = \frac{1}{1+p}$, this implies that $d = \frac{p}{1+p}$. Thusly, our answer is $\boxed{\text{(C)}}$, and we are done.

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