2020 AMC 10A Problem 2

Averages are hard to split apart, but totals are easy. So turn the average into a total first.

1. Five numbers with average $15$ add up to $5 \times 15 = 75$.
2. The three numbers we know add up to $3 + 5 + 7 = 15$.
3. So the other two make up the rest: $a + b = 75 - 15 = 60$.
4. The average of $a$ and $b$ is their total divided by $2$: $\frac{60}{2} = 30$.

The answer is $\boxed{\textbf{(C)}\ 30}$.

"Average $\times$ how many $=$ total" lets you switch between averages and totals. Totals can be added and subtracted piece by piece, so do the work with totals and switch back to an average only at the end. Notice that we never learn $a$ or $b$ by itself, and we do not need to.

Five numbers with mean $15$ have total $5 \cdot 15 = 75$. The three known numbers contribute $3 + 5 + 7 = 15$, so
$$a + b = 75 - 15 = 60.$$
The average of $a$ and $b$ is therefore $\frac{60}{2} = 30$.

The answer is $\boxed{\textbf{(C)}\ 30}$.