The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?
The arithmetic mean of the numbers $3, 5, 7, a,$ and $b$ is equal to $\frac{3+5+7+a+b}{5}=\frac{15+a+b}{5}=15$. Solving for $a+b$, we get $a+b=60$. Dividing by $2$ to find the average of the two numbers $a$ and $b$ gives $\frac{60}{2}=\fbox{30}$.