1995 AHSME Problems/Problem 13

Problem

The addition below is incorrect. The display can be made correct by changing one digit $d$ , wherever it occurs, to another digit $e$ . Find the sum of $d$ and $e$ .

$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$

$\mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} }$

Solution

If we change $0$ , the units column would be incorrect.

If we change $1$ , then the leading $1$ in the sum would be incorrect.

However, looking at the $2$ in the hundred-thousands column, it would be possible to change the $2$ to either a $5$ (no carry) or a $6$ (carry) to create a correct statement.

Changing the $2$ to a $5$ would give $745586 + 859430$ on top, which equals $1605016$ . This does not match up to the bottom.

Changing the $2$ to a $6$ gives $746586 + 869430$ on top, which has a sum of $1616016$ . This is the number on the bottom if the $2$ s were changed to $6$ s.

Thus $d=2$ and $e=6$ . so $d+e= 8 \boxed{\mathrm{ (C)}}$

1995 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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
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1995 AHSME Problems/Problem 13

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