1958 AMC 12 Problem 45

Question

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:

x

cannot exceed

70

y

can equal

2x

the amount of the check cannot be a multiple of

5

the incorrect amount can equal twice the correct amount

the sum of the digits of the correct amount is divisible by

9

Write the correct and incorrect amounts in cents

Their difference simplifies to

99(y-x)

In cents, the incorrect amount minus the correct amount is

\begin{aligned} &(100y+x)-(100x+y)\\ &\qquad=99(y-x). \end{aligned}

Since

\$17.82

is

1782

cents,

99(y-x)=1782,

so

y-x=18.

The two-digit values

x=18,\ y=36

satisfy this relation and have

y=2x.

Thus that equality can occur.

Therefore, the correct answer is B.

Problem 45 in other years