2015 AMC 8 Problem 16

Question

In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If

\dfrac{1}{3}

of all the ninth graders are paired with

\dfrac{2}{5}

of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?

\dfrac{2}{15}

\dfrac{4}{11}

\dfrac{11}{30}

\dfrac{3}{8}

\dfrac{11}{15}

Answer: B

Concepts:ratio and proportion fraction

Difficulty rating: 1300

Small hint:

Let

s

and

n

be the numbers of sixth and ninth graders.

Big hint:

The paired sixth graders and paired ninth graders are the same number of pairs.

Video solution:

Click to load, then click again to play

Written solution:

Let there be

s

sixth graders and

n

ninth graders. The number of paired ninth graders equals the number of paired sixth graders, so

\dfrac13 n=\dfrac25 s.

This gives

5n=6s,

so

n:s=6:5.

The total number of students is therefore proportional to

11

parts.

The paired ninth graders make up

\dfrac13\cdot\dfrac6{11}=\dfrac2{11}

of all students, and the paired sixth graders make up

\dfrac25\cdot\dfrac5{11}=\dfrac2{11}

of all students. Altogether,

\dfrac4{11}

of the students have a buddy.

Thus, B is the correct answer.