1980 AHSME Problems/Problem 15

Problem

A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is

$\text{(A)} \ 1 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 25 \qquad \text{(D)} \ 26 \qquad \text{(E)} \ 100$

Solution

Say that the price of the item in cents is $x$ (so $x$ is a positive integer as well). The sales tax would then be $\frac{x}{25}$ , so $n=\frac{1}{100}\left x+\frac{x}{25}\right=\frac{26x}{2500}=\frac{13x}{1250}$ (Error compiling LaTeX. Unknown error_msg). Since $x$ is positive integer, the smallest possible integer value for $n=\frac{13x}{1250}$ occurs when $x=1250$ , which gives us the answer $\fbox{\text{(C)13}}$ .

1980 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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1980 AHSME Problems/Problem 15

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