SOLUTION: A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?

Question 1109303: A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?
Answer by greenestamps(13206) (Show Source): You can put this solution on YOUR website!

In a set of 11 positive integers, the 6th one is the median, which is 9:
x, x, x, x, x, 9, x, x, x, x, x

The unique mode is 8; let's use only two 8's and see what the largest integer in the set can be:
x, x, x, 8, 8, 9, x, x, x, x, x

The mean is 10, so the sum of all 11 numbers is 110. We are trying to find the largest possible value for a number in the set; that means we want all the other numbers to be as small as possible. Since 8 is the unique mode and we are trying to use only two of them, the other numbers must all be different; for the sum of all 11 numbers to be 110, we get this:
1, 2, 3, 8, 8, 9, 10, 11, 12, 13, 33

Now let's use three 8's; that will allow us to use some smaller numbers twice, making it possible that we might get a larger number than 33 in the set. Using the smallest possible numbers, keeping our median of 9 and unique mode of 8, now using three 8s, we get this:
1, 1, 8, 8, 8, 9, 9, 10, 10, 11, 35

Yes, using three 8's for the unique mode, we were able to get 35 for the largest number in the set.

What about four 8's for the unique mode? Using the same logic, we get this:
1, 8, 8, 8, 8, 9, 9, 9, 10, 10, 30.

With the fourth 8, the largest number we can get in the set is 30.

So the final answer is...

The largest possible number in the set is 35.
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