The mean of a set

Question

The mean and median of a set of 9 distinct positive integers is 13. What is the maximum possible value for the largest integer in the set?

 

Please help

Please add the steps to how you got there if you have time

Thanks!

Answer 1

Since the median is 13, the middle (5th) value must be 13.  Numbers to the left must be 13 or smaller; numbers to the right must be 13 or greater.  Call the last number in the sequence x.  Since we want x to be as large as possible, we want all of the other numbers to be as small as possible:

 

0  0  0  0  13  13  13  13  x

 

The median = 13

 

The mean = 13 = (4·13 + x)/9

 

Solve for x.  If the numbers in the data set cannot repeat (distinct?), then the sequence would be:

 

0  1  2  3  13  14  15  16  x

 

The median is still 13.  The mean is:

 

(0+1+2+3+13+14+15+16+x)/9 = 13

 

Solve for x.