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A list of ten distinct, positive integers has a median of 5.5.  What is the smallest possible average of the ten positive integers?  Explain your ansswer in complete sentences.

 Mar 25, 2023
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To find the smallest possible average of ten positive integers with a median of 5.5, we can start by considering the definition of the median. The median of a list of numbers is the middle value when the numbers are listed in order. Therefore, we can assume that the fifth and sixth numbers in the list are both equal to 5.5, since there are 5 numbers less than 5.5 and 4 numbers greater than 5.5.

Since the list contains ten distinct, positive integers, we can choose the remaining 8 numbers to be as small as possible to minimize the average. The smallest positive integer is 1, and we can choose the next 7 integers to be consecutive, starting with 2. This gives us a list of 1, 2, 3, 4, 5.5, 5.5, 6, 7, 8, 9.

The sum of these numbers is 51, so the average is 51/10 = 5.1. Therefore, the smallest possible average of ten positive integers with a median of 5.5 is 5.1.

 Mar 25, 2023

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