2013 AMC 8 Problem 16

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8^\text{th}-graders to 6^\text{th}-graders is 5:3, and the ratio of 8^\text{th}-graders to 7^\text{th}-graders is 8:5. What is the smallest number of students that could be participating in the project?

\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 79 \qquad \textbf{(E)}\ 89

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.