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Suppose ten distinct, positive integers have a median of \(12\). ("Distinct integers" means that no two integers are the same.)

What is the smallest the average of those ten integers could be?

 

\(\phantom{median of 10}\)

 Jun 5, 2022
 #1
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The smallest possible average is 25/3.

 Jun 5, 2022
 #2
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Since there are ten distinct integers, the median must be the average of the highest number (in the set) below 12 and the smallest number above 12 (in the set).

 

You will want the four smallest numbers below 12:      1, 2, 3, and 4.

You also want the five smallest numbers above 12:     13, 14, 15, 16, and 17.

 

To get the mean of the highest number below 12 and the smallest number above 12, these two middle numbers must be 11 and 13.

 

So, the numbers are:  1, 2, 3, 4, 11, 13, 14, 15, 16, and 17.

 

You will need to find the mean of these numbers to get your answer.

 Jun 6, 2022

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