SOLUTION: Anna writes a list of positive integers that are not necessarily all different. She, of course, includes her favorite number 68 on the list. the mean of all of the numbers is 56.

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Question 905637: Anna writes a list of positive integers that are not necessarily
all different. She, of course, includes her favorite number 68 on the list. the mean of all of the numbers is 56. Then, Alex erases a 68. The
mean of the remaining numbers is 55.
What is the largest number that could be on the list?
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
Anna writes a list of positive integers that are not necessarily
all different. She, of course, includes her favorite number 68 on
the list. the mean of all of the numbers is 56. Then, Alex erases
a 68. The mean of the remaining numbers is 55.
What is the largest number that could be on the list?
Let s = the sum of all the positive integers besides the 68.
Let n = the number of positive integers

Then 

%28s%2B68%29%2Fn+=+56
or
s%2B68=56n
or
s-56n=-68


After Alex erases the 68, there are only n-1 positive integers,
with sum s and average 55, so

s%2F%28n-1%29=55
or
s=55%28n-1%29
or
s=55n-55
or
s-55n=-55

Solve the system

system%28s-56n=-68%2Cs-55n=-55%29

The solution is s = 660, n=13

So there are 13 positive integers, counting the 68.
So the other 12 positive integers have sum 660.

The largest possible positive integer in the list would
be when 11 of the 12 positive integers other than 68 are
as small as a positive integer can be, which is 1.  So 
that would be when 11 of the others beside the 68 are 
all 1's and one of them is 660-11 = 649.

Answer: 649

Edwin