2001 AMC 10 Problem 16

1. The mean appears in both clues, so make it the unknown. Call the mean $m$.
2. "The mean is $10$ more than the least" means the least number is $m - 10$.
3. "The mean is $15$ less than the greatest" means the greatest number is $m + 15$.
4. For three numbers, the median is the middle one, so the middle number is $5$.
5. The mean of three numbers is their sum divided by $3$, so their sum is $3m$. Write the sum both ways:
$$(m - 10) + 5 + (m + 15) = 3m.$$
6. The left side simplifies to $2m + 10$. So $2m + 10 = 3m$, which gives $m = 10$.
7. The sum is $3m = 30$.
8. Check: the numbers are $0$, $5$ and $25$. Their mean is $30 \div 3 = 10$, which is $10$ more than $0$ and $15$ less than $25$.

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

When several numbers are all described in relation to one quantity, let that quantity be the unknown. Then there is only one letter and one equation. The fact "sum $=$ $3 \times$ mean" is used twice: to set up the equation and to get the final answer.