2011 AMC 10A Problem 5

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:mean ratio and proportion

Difficulty rating: 1070

Small hint:

Use a

4:2:1

ratio for the numbers of third, fourth, and fifth graders

Big hint:

Compute a weighted average with weights

4,2,1

Solution:

WLOG, let there be one fifth grader. This then tells us that there are two fourth graders and four third graders.

We can do this, since we are only interested in the average, which is not impacted by the exact number of students.

The total number of minutes the students spend running is

12 \cdot 4 + 15 \cdot 2 + 10 = 88

minutes. The total number of students is

1 + 2 + 4 = 7.

The average is then

\dfrac{88}{7}.

Thus, C is the correct answer.