1997 AMC 12 Problem 8

Question

Mientka Publishing Company prices its best seller Where’s Walter? as follows:

C(n)= \begin{cases} 12n, & 1\le n\le24,\\ 11n, & 25\le n\le48,\\ 10n, & 49\le n, \end{cases}

where

n

is the number of books ordered, and

C(n)

is the cost in dollars of

n

books. Notice that

25

books cost less than

24

books. For how many values of

n

is it cheaper to buy more than

n

books than to buy exactly

n

books?

3

4

5

6

8

Only the two price-break points can make a larger order cheaper

Compare

C(n)

with

C(25)

near the first break and with

C(49)

near the second

At the first break,

C(25)=275

is below

C(23)=276

and

C(24)=288,

giving

n=23

and

n=24.

At the second,

C(49)=490

is below

C(n)=11n

for

n=45,

n=46,

n=47,

n=48.

No other

n

works because each piece increases. There are

2+4=6,

so D is correct.

Problem 8 in other years