1999 AMC 12 Problem 11

Question

The student lockers at Olympic High are numbered consecutively beginning with locker number

1.

The plastic digits used to number the lockers cost

2

cents apiece. Thus, it costs

2

cents to label locker number

9

and

4

cents to label locker number

10.

If it costs

\$137.94

to label all the lockers, how many lockers are there at the school?

2001

2010

2100

2726

6897

Divide the total cost by the cost per digit to get the number of digits

Subtract the digits used by lockers

1

-

9,

10

-

99,

and

100

-

999

Labeling costs

\frac{\$137.94}{\$0.02} = 6897

digits. Lockers

1

-

9

use

9

digits, lockers

10

-

99

use

2 \cdot 90 = 180

digits, and lockers

100

-

999

use

3 \cdot 900 = 2700

digits.

The remaining digits number

6897 - 2700 - 180 - 9 = 4008,

which label

\frac{4008}{4} = 1002

four-digit lockers. In all there are

999 + 1002 = 2001

lockers.

Thus, the correct answer is A.

Problem 11 in other years