2012 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2012 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 12A solutions, or check the answer key.

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Concepts:ratio and proportion system of equations

Difficulty rating: 1040

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

$8$

$16$

$25$

$64$

$96$

Solution:

Let $b$ be the number of blueberries. Then there are $2b$ raspberries, $4 \cdot 2b = 8b$ cherries, and $3 \cdot 8b = 24b$ grapes.

The total is $b + 2b + 8b + 24b = 35b = 280,$ so $b = 8$ and there are $8b = 64$ cherries.

Thus, the correct answer is D.

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