1994 AJHSME Problems/Problem 23

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 (ProblemsAnswer KeyResources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png