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?)
 
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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?
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{{duplicate|[[2001 AMC 12 Problems|2001 AMC 12 #4]] and [[2001 AMC 10 Problems|2001 AMC 10 #16]]}}
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== Problem ==
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The mean of three numbers is <math>10</math> more than the least of the numbers and <math>15</math>
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less than the greatest. The median of the three numbers is <math>5</math>. What is their
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sum?
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<math>\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36</math>
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== Solution ==
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Let <math>m</math> be the mean of the three numbers. Then the least of the numbers is <math>m-10</math>
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and the greatest is <math>m + 15</math>. The middle of the three numbers is the median, 5. So
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<math>\dfrac{1}{3}[(m-10) + 5 + (m + 15)] = m</math>, which can be solved to get <math>m=10</math>.
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Hence, the sum of the three numbers is <math>3(10) = \boxed{(\text{D})30}</math>.
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== See Also ==
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{{AMC12 box|year=2001|num-b=3|num-a=5}}
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{{AMC10 box|year=2001|num-b=15|num-a=17}}
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{{MAA Notice}}

Latest revision as of 21:12, 25 January 2021

The following problem is from both the 2001 AMC 12 #4 and 2001 AMC 10 #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 can be solved to get $m=10$. Hence, the sum of the three numbers is $3(10) = \boxed{(\text{D})30}$.

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
2001 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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