2008 AMC 10B Problem 8

Problem 8 of 25EasierNumber Theory

A class collects

\$50

to buy flowers for a classmate who is in the hospital. Roses cost

\$3

each, and carnations cost

\$2

each. No other flowers are to be used. How many different bouquets could be purchased for exactly

\$50?

Show solution

Solution

r

roses and

c

carnations are bought, then

3r+2c=50.

Because

2c

and

50

are even,

3r

must be even, so

r

is even. Also

3r\le 50,

r\le 16.

The even values

r=0,2,4,\ldots,16

each give a valid

c,

which is

9

bouquets. Thus, the correct answer is C.