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 $\$50?$

$1$

$7$

$9$

$16$

$17$

Answer: C

Concepts:Diophantine Equation parity

Difficulty rating: 1270

Solution:

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.

2008 AMC 12B Problem 5

Answer: C

Difficulty rating: 1270

Solution:

Problem 5 in Other Years