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Question 1125943: A set of 7 positive integers has a unique mode of 1, a mean of 5, and a median of 6. What is the largest possible value for any of the integers in the set.
Found 2 solutions by Boreal, greenestamps:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! mean of 5 with 7 integers can have total of 35. But, a unique mode of 1 means at least 2 of those integers have to be 1. The median of 6 means that 6 is the fourth from the bottom, and there are 3 integers higher than that.
The bottom 3 integers are 1,1, and one other.For the largest possible value of any integer, the third integer below the median has to be smaller or 1.
That makes 3 integers of 1 each, and 1 of 6. The total is 35, so two of the remaining three are 7 and 8, because there is a unique mode of 1. That makes the total 24 (7,8,6,1,1,1). The largest possible value is 11.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
There is one error in the analysis of the problem by the other tutor, leading to the wrong answer to the question.
The mean of the 7 numbers is 5, so their sum is 35.
The median is 6, so the 4th number is 6; there are 3 numbers smaller than (or equal to) 6 and 3 larger than (or equal to) 6.
Since we want to find the largest possible value for the largest number, the smallest 3 numbers should all be 1.
So the first 4 numbers are 1, 1, 1, and 6. Their sum is 9, so the sum of the other 3 numbers must be 26.
Again since we want to find the largest possible value for the largest number, the first two of the three numbers larger than or equal to 6 should be as small as possible. Since the unique mode is 1 and there are 3 of them, we can only have one more 6.
So the first two numbers after the median are 6 and 7. The sum of the first 6 numbers is then 1+1+1+6+6+7=22, so the largest possible value for the largest number in the set is 35-22 = 13.
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