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Difference between revisions of "1994 AJHSME Problems/Problem 23"

(Created page with "==Problem== If <math>X</math>, <math>Y</math> and <math>Z</math> are different digits, then the largest possible <math>3-</math>digit sum for <math>\begin{tabular}{ccc} X & X &...")
 
 
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<math>\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ \qquad \text{(E)}\ ZZY</math>
 
<math>\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ \qquad \text{(E)}\ ZZY</math>
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==Solution==
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The sum can be rewritten as <math>113X + 10Y</math>. To get the largest possible sum, we maximize the hundreds digit, <math>X</math>. If <math>X=9</math>, the sum is a <math>4</math>-digit number, so we let <math>X=8</math> and <math>113(8)+10Y = 904+10Y</math>. To continue maxmimizing this sum, we can let <math>Y=9</math>, a different digit from <math>X</math>, and <math>904+10(9)=994</math>, which has the form <math>\boxed{\text{(D)}\ YYZ}</math>.
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==See Also==
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{{AJHSME box|year=1994|num-b=22|num-a=24}}
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{{MAA Notice}}

Latest revision as of 23:14, 4 July 2013

Problem

If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for

$\begin{tabular}{ccc} X & X & X \\ & Y & X \\ + & & X \\ \hline \end{tabular}$

has the form

$\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ \qquad \text{(E)}\ ZZY$

Solution

The sum can be rewritten as $113X + 10Y$. To get the largest possible sum, we maximize the hundreds digit, $X$. If $X=9$, the sum is a $4$-digit number, so we let $X=8$ and $113(8)+10Y = 904+10Y$. To continue maxmimizing this sum, we can let $Y=9$, a different digit from $X$, and $904+10(9)=994$, which has the form $\boxed{\text{(D)}\ YYZ}$.

See Also

1994 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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
All AJHSME/AMC 8 Problems and Solutions

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