2011 AMC 12A Problem 4

Question

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

Answer: C

Concepts:weighted mean ratio and proportion

Difficulty rating: 1040

Small hint:

Let the number of fifth graders be

1,

so fourth graders is

2

and third graders is

4

Big hint:

The average is

\dfrac{4 \cdot 12 + 2 \cdot 15 + 1 \cdot 10}{4 + 2 + 1}

Solution:

Take the grade sizes in the ratio

4 : 2 : 1

for third, fourth, and fifth grades. The weighted average is

\begin{aligned} \dfrac{4\cdot12+2\cdot15+10}{7} \\ = \dfrac{48+30+10}{7} \\ = \dfrac{88}{7}. \end{aligned}

Thus, the correct answer is C.