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Problem 92

AMC 10/12, early questions

Number theory Difficulty 3.4 Multiple choice

The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$ . The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four centers to label locker number $10$ . If it costs $137.94 to label all the lockers, how many lockers are there at the school?
$

Pick one

Official solution

its A

Solution 2
Since all answers are over $2000$ , work backwards and find the cost of the first $1999$ lockers. The first $9$ lockers cost $0.18$ dollars, while the next $90$ lockers cost $0.04\cdot 90 = 3.60$ . Lockers $100$ through $999$ cost $0.06\cdot 900 = 54.00$ , and lockers $1000$ through $1999$ inclusive cost $0.08\cdot 1000 = 80.00$ .
This gives a total cost of $0.18 + 3.60 + 54.00 + 80.00 = 137.78$ . There are $137.94 - 137.78 = 0.16$ dollars left over, which is enough for $8$ digits, or $2$ more four digit lockers. These lockers are $2000$ and $2001$ , leading to answer $\boxed{\textbf{(A)}}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.