1958 AHSME Problems/Problem 45

Problem

A check is written for $x$ dollars and $y$ cents, $x$ and $y$ both two-digit numbers. In error it is cashed for $y$ dollars and $x$ cents, the incorrect amount exceeding the correct amount by $$17.82$ . Then:

$\textbf{(A)}\ {x}\text{ cannot exceed }{70}\qquad \\ \textbf{(B)}\ {y}\text{ can equal }{2x}\qquad\\ \textbf{(C)}\ \text{the amount of the check cannot be a multiple of }{5}\qquad \\ \textbf{(D)}\ \text{the incorrect amount can equal twice the correct amount}\qquad \\ \textbf{(E)}\ \text{the sum of the digits of the correct amount is divisible by }{9}$

Solution

The correct total number of cents is $100x+y$ . Due to the error, it is cashed $100y+x$ cents. We have $100y+x-100x-y=1782$ . Simplifying, we have $y-x=18$ . Looking at the answer choices:

A: $y=89, x=71$ is a counterexample

B: Then $y=36, x=18$ , correct.

C: As long as $y$ is a multiple of 5, the original amount is a multiple of 5. A counterexample is $x=15, y=33$ .

D: Then the correct amount is 17.82, and the incorrect amount is 35.64. Obviously incorrect.

E: This is not fixed. A counterexample is $x=11,y=29$ .

Therefore, the correct answer is $\boxed{\textbf{(B)}\ {y}\text{ can equal }{2x}\qquad\\}$

--Ethanol 2012

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 44	Followed by Problem 46
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1958 AHSME Problems/Problem 45

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