Question:
What is the average of all positive integers that have three digits when written in base $5$, but two digits when written in base $8$? Write your answer in base $10$.

Answer:
If an integer $n$ has three digits in base $5$, then $5^2\le n<5^3$. If an integer $n$ has two digits in base $8$, then $8^1\le n<8^2$. The overlap of these intervals is $$\{25,26,27,28,\ldots,61,62,63\}.$$The average of the integers in this set is $\frac{25+63}{2} = \boxed{44}$.