2001 AMC 12 Problems/Problem 4

The following problem is from both the 2001 AMC 12 #4 and 2002 AMC 10A #16, so both problems redirect to this page.

Problem

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?

$\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36$

Solution

Let $m$ be the mean of the three numbers. Then the least of the numbers is $m-10$ and the greatest is $m + 15$ . The middle of the three numbers is the median, 5. So $\dfrac{1}{3}[(m-10) + 5 + (m + 15)] = m$ , which implies that $m=10$ . Hence, the sum of the three numbers is $3(10) = 30$ , and the answer is $\text{(D)}$ .

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2001 AMC 12 Problems/Problem 4

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