Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, 
one at the rate of $3$ feet per second, the other at $2$ feet per second. 
They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. 
$\textbf{(A)}\ 24\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 18$
First, note that it will take $30$ seconds for the first swimmer to reach the other side and $45$ seconds for the second swimmer to reach the other side.  Also, note that after $180$ seconds (or $3$ minutes), both swimmers will complete an even number of laps, essentially returning to their starting point.

At this point, find the number of meeting points in the first $3$ minutes, then multiply by four to get the answer.  From the graph (where the x-axis is the time in seconds and the y-axis is distance from one side of the pool), there are five meeting points, so the two swimmers will pass each other $20$ times, which is answer choice $\boxed{\textbf{(C)}}$.