# Difference between revisions of "2015 AMC 8 Problems/Problem 16"

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 $\tfrac{1}{3}$ of all the ninth graders are paired with $\tfrac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?

$\textbf{(A) } \frac{2}{15} \qquad \textbf{(B) } \frac{4}{11} \qquad \textbf{(C) } \frac{11}{30} \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{11}{15}$

## Solution 1

Let the number of sixth graders be $s$, and the number of ninth graders be $n$. Thus, $n/3=2s/5$, which simplifies to $n=6s/5$. Since we are trying to find the value of $\frac{n/3+2s/5}{n+s}$, we can just substitute $n$ for $6s/5$ into the equation. We then get a value of $\boxed{\textbf{(B)}~\frac{4}{11}}$