Difference between revisions of "2001 AMC 12 Problems/Problem 4"

(New page: the mean of 3 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?)
 
Line 1: Line 1:
the mean of 3 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?
+
== Problem ==
 +
The mean of three numbers is <math>10</math> more than the least of the numbers and <math>15</math>
 +
less than the greatest. The median of the three numbers is <math>5</math>. What is their
 +
sum?
 +
 
 +
<math>\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36</math>
 +
 
 +
== Solution ==
 +
Let <math>m</math> be the mean of the three numbers. Then the least of the numbers is <math>m − 10</math>
 +
and the greatest is <math>m + 15</math>. The middle of the three numbers is the median, 5. So
 +
<math>\dfrac{1}{3}[(m-10) + 5 + (m + 15)] = m</math>, which implies that <math>m=10</math>.
 +
Hence, the sum of the three numbers is <math>3(10) = 30</math>, and the answer is <math>\text{(D)}</math>.
 +
 
 +
== See Also ==
 +
{{AMC12 box|year=2001|num-b=3|num-a=5}}

Revision as of 14:13, 16 February 2008

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 $m − 10$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212)) 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 $\text{(D)}$.

See Also

2001 AMC 12 (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
Invalid username
Login to AoPS