SOLUTION: A set of five 2-digit whole numbers has a mean of 16, a mode of 13, and a median greater than the mean. What is the largest possible value of the set?

Question 1020075: A set of five 2-digit whole numbers has a mean of 16, a mode of 13, and a median greater than the mean. What is the largest possible value of the set?
Answer by LinnW(1048) (Show Source): You can put this solution on YOUR website!
Since the mode is 13, we have at least two numbers in the set with value 13.
We are given that the median is greater than 16. Since there are five
numbers in the set, the third highest number is the median.
So the first two numbers are 13 13 .
Since the median is greater than 16, it must be at least 17.
Suppose the third number is 17.
Since the mean is 16, the sum of the 5 numbers must be 5*16 = 80
Let Z represent the sum of the two remaining numbers.
13 + 13 + 17 + Z = 80
43 + Z = 80
Z = 37
Since each of the two numbers must be 2 digits and >= 17, the two
numbers must be 18 and 19.
This makes our set 13 13 17 18 19
Now we need to determine if the median can be greater than 17.
Try 18 for the median.
If Z once again represents the sum of the remaining two numbers,
13 + 13 + 18 + Z = 80
44 + Z = 80
Z = 36
This would make the two numbers 18 and 18, but this is impossible
since this would make the mode to be 18.
Try 19 for the median.
If Z once again represents the sum of the remaining two numbers,
13 + 13 + 19 + Z = 80
45 + Z = 80
Z = 35
This would make the two numbers 17 and 18, but this makes
19 not the median.
Clearly any integer larger also would not work.
So our set is 13 13 17 18 19
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