Problem: The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\,$ fish for various values of $n\,$. $\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array}$ In the newspaper story covering the event, it was reported that (a) the winner caught $15$ fish; (b) those who caught $3$ or more fish averaged $6$ fish each; (c) those who caught $12$ or fewer fish averaged $5$ fish each. What was the total number of fish caught during the festival? Solution: Let $f$ be the total number of fish caught during the festival. We can see that $f = f'+196$ where $f'$ is the number of fish caught by those who caught between $3$ and $13$ fish exclusive. If we solve for $f'$, we are done. To do this. let $t$ be the total number of people at the event. By (b) notice that $\frac{f'+3\cdot 23 + 13 \cdot 5 + 14 \cdot 2 + 15 \cdot 1}{t-21}= 6$. Simplifying, we get $f'= 6t-303$. On the other hand, by (c) we know that $\frac{f' + 3 \cdot 23 + 2 \cdot 7 + 1 \cdot 5}{t-8} = 5$. Simplifying, we get $f' = 5t-128$. Together, we find that $t = 175$ and so $f' = 747$ and so $f = 943$.