2008 AMC 12B Problem 5

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

If there are

r

roses and

c

carnations, then

3r + 2c = 50

Since

2c

and

50

are even,

3r

is even, so

r

must be even; find how many even

r

work

Let

r

be the number of roses and

c

the number of carnations, so

3r + 2c = 50

with

r, c \ge 0.

Because

2c

and

50

are even,

3r

must be even, forcing

r

to be even. The largest possible

r

is

16

(since

3 \cdot 17 \gt 50

), so

r \in \{0, 2, 4, \ldots, 16\}.

That gives

9

values of

r,

each determining a bouquet.

Thus, the correct answer is C.

Problem 5 in other years