2008 AMC 10B Problem 8

Below is the professionally curated solution for Problem 8 of the 2008 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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Concepts:Diophantine Equation parity

Difficulty rating: 1080

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 $\$50?$

$1$

$7$

$9$

$16$

$17$

Solution:

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.

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