2001 AMC 10 Problem 16

Question

The mean of three numbers is

10

more than the least of the numbers and

15

less than the greatest. The median of the three numbers is

5.

What is their sum?

5

20

25

30

36

Let

m

be the mean; then the least is

m-10

and the greatest is

m+15

The median is

5,

so

\dfrac13\big((m-10)+5+(m+15)\big)

=m

Let

m

be the mean. The least number is

m-10

and the greatest is

m+15,

with median

5.

Since the mean of the three is

m,

\begin{aligned} &\tfrac13\big((m-10)+5+(m+15)\big) \\ &\quad =m, \end{aligned}

which gives

m=10.

The sum is

3m=30.

Thus, the correct answer is D.

Problem 16 in other years