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Difference between revisions of "1958 AHSME Problems/Problem 45"

(Created page with "== Problem == A check is written for <math> x</math> dollars and <math> y</math> cents, <math> x</math> and <math> y</math> both two-digit numbers. In error it is cashed for <mat...")
 
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== Solution ==
 
== Solution ==
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<math>\fbox{}</math>
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The correct total number of cents is <math>100x+y</math>. Due to the error, it is cashed <math>100y+x</math> cents. We have <math>100y+x-100x-y=1782</math>. Simplifying, we have <math>y-x=18</math>. Looking at the answer choices:
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A:  <math>y=89, x=71</math> is a counterexample
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B:  Then <math>y=36, x=18</math>, correct.
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C:  As long as <math>y</math> is a multiple of 5, the original amount is a multiple of 5. A counterexample is <math>x=15, y=33</math>.
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D:  Then the correct amount is 17.82, and the incorrect amount is 35.64. Obviously incorrect.
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E:  This is not fixed. A counterexample is <math>x=11,y=29</math>.
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Therefore, the correct answer is <math>\boxed{\textbf{(B)}\ {y}\text{ can equal }{2x}\qquad\\}</math>
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--Ethanol 2012
  
 
== See Also ==
 
== See Also ==

Revision as of 03:19, 27 September 2025

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 $&#036;17.82$. 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

See Also

1958 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 44 		Followed by
Problem 46
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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