2008 AMC 10B Problem 8

Question

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?

1

7

9

16

17

Let

r

be the number of roses; then

3r+2c=50

Since

2c

is even and

50

is even,

r

must be even, and

3r\le 50

If

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,

so

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.

Problem 8 in other years