2011 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2011 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 12A solutions, or check the answer key.

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Concepts:weighted mean ratio and proportion

Difficulty rating: 1040

At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12,$ $15,$ and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?

$12$

$\dfrac{37}{3}$

$\dfrac{88}{7}$

$13$

$14$

Solution:

Take the grade sizes in the ratio $4 : 2 : 1$ for third, fourth, and fifth grades. The weighted average is $\dfrac{4 \cdot 12 + 2 \cdot 15 + 1 \cdot 10}{7} = \dfrac{48 + 30 + 10}{7} = \dfrac{88}{7}.$

Thus, the correct answer is C.

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