1999 AHSME Problems/Problem 11

Problem

The student locker numbers at Olympic High are numbered consecutively beginning with locker number . The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number and four centers to label locker number . If it costs $137.94 to label all the lockers, how many lockers are there at the school?

Solution

Solution 1

The locker labeling requires \frac{137.94}{0.02}=6897 digits. Lockers 1 through 9 require 9 digits total, lockers 10 through 99 require 2 \times 90=180 digits, and lockers 100 through 999 require 3 \times 900=2700 digits. Thus, the remaining lockers require 6897-2700-180-9=4008 digits, so there must be \frac{4008}{4}=1002 more lockers, because they each use 4 digits. Thus, there are 1002+999=2001 student lockers, or answer choice \boxed{\textbf{(A)}}.

Solution 2

Since all answers are over , work backwards and find the cost of the first lockers. The first lockers cost dollars, while the next lockers cost . Lockers through cost , and lockers through inclusive cost .

This gives a total cost of . There are dollars left over, which is enough for digits, or more four digit lockers. These lockers are and , leading to answer .

See Also

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