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Suppose ten distinct, positive integers have a median of 10. ("Distinct integers" means that no two integers are the same.)

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

 

P.S. When you are answering, please include equations. Not just the answer.

 Jan 24, 2021
 #1
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_ _ _ 10 _ _ _          three integers below 10    three above 10    choose the smallest possible to make the average as small as possib;e

 

123 10 11 12 13    av =7.42

 Jan 24, 2021
 #4
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The problem specifies that there are 10 integers in the list.  This is nice reasoning, though. 

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Guest Jan 24, 2021
 #5
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...you are right!

_  _ _ _ 9 11 _ _ _ _

 

 

1 2 3 4  9 11  12 13 14 15     would fit the bill

Guest Jan 24, 2021
 #2
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If the median of an even number of distinct integers is ten, then the two middle numbers have to sum to 20.

Making them 9 and 11 would be ideal so we can use the rest of the small one-digit numbers for the below numbers and the small two-digit numbers directly above 11(what I'm saying is using these numbers will minimize the other numbers)

To minimize the others, we make them 1, 2, 3, 4, 9, 11, 12, 13, 14, and 15.

The average of those numbers is 84/10 or 8.4.

 Jan 24, 2021
 #6
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Hey, MooMoo, look at my answer posted just below yours a few seconds later. 

 

Great minds think alike .... and so do ours!   

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Guest Jan 24, 2021
 #7
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wow i saw it,

same answer, kind of the same explantion,

Great mind do think alike!

MooMooooMooM  Jan 24, 2021
 #3
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Suppose ten distinct, positive integers have a median of 10. ("Distinct integers" means that no two integers are the same.)

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

 

P.S. When you are answering, please include equations. Not just the answer.  

 

 

I don't have an equation, but I'll include my reasoning.  Maybe you can make the equations. 

 

The median is the integer that the number of integers smaller than it is equal to the number of intergers larger than it.  To get the smallest average, we'll need the smallest integers.  Since there's an even number of integers in the list, the median has to fall between two integers.  9 and 11 are the smallest that will give us 10. 

 

Now you just use the smallest numbers that include those two.   I get 1, 2, 3, 4, 9, 11, 12, 13, 14, 15. 

 

The integers in that list total 84 so the average of them is 8.4.  

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 Jan 24, 2021
 #8
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Attn:  skc123,

 

Do not ask a question more than once in a short space of time.

If you do repost after a day or two then you get the address of the old post and ask for it to be answered there.

 

This is the 3rd time you have asked this question in the space of about 1 hour.

 

DO NOT WASTE PEOPLE'S TIME LIKE THIS AGAIN!

 Jan 24, 2021
edited by Melody  Jan 24, 2021
 #9
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Thank you to all the people who have answered.   laugh

 Jan 24, 2021

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