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There is a group of seven positive integers whose average is 5, whose median is 4, and whose only modes are 2 and 8. What are all the integers?

 Apr 25, 2020
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Seven positive integers, mean = 5, median = 4, and two modes 2 and 8.

 

Since there are an odd number of numbers and 4 is a median, at least one of the numbers is 4 and three of the numbers must be larger than 4 and three of the numbers must be smaller than 4.

 

Since both 2 and 8 are modes, either both are used twice or both are used 3 times.

 

When I tried to use them three times, I couldn't get anything to work.

 

So, using them twice, I have 2, 2, 4, 8, 8, giving a sum of 24.

For seven numbers to have a mean of 5, their sum must be 7 x 5  =  35.

I need another 11, split between a number smaller than 4 and a number larger than 4.

If I try 3 for the smaller number, this means that I have to use 8 for the larger number, but I can't because that would mess up the modes.

If the number is 1, than the other number must be 10.

 

1, 2, 2, 4, 8, 8, 10

 Apr 25, 2020

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