SOLUTION: If S is a set of 5 positive, distinct integers with an average (arithmetic mean) of 9 and a median of 7, what is the least possible value of the largest number in the set?

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Question 472883: If S is a set of 5 positive, distinct integers with an average (arithmetic mean) of 9 and a median of 7, what is the least possible value of the largest number in the set?
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Let the numbers be 

A, B, C, D, E 

from smallest to largest, i.e., A < B < C < D < E

We want E to be smallest possible.

%28A%2BB%2BC%2BD%2BE%29%2F5 = 9

Multiply through by 5

A + B + C + D + E = 45

Since the median is 7,  C = 7

A + B + 7 + D + E = 45

To make E the smallest, we have to make A and B
as large as possible. But they have to be less
than median C = 7. So we choose A = 5 and B = 6.

5 + 6 + 7 + D + E = 45

18 + D + E = 45

         E =  27 - D

E is smallest possible when D is the largest possible,
so that we subtract the most possible from 27:

But  E > D, or

27 - D > D

    27 > 2D

  13.5 > D

So D is the largest value posible when D = 13

So E is smallest when E = 27 - 13 = 14

The 5 numbers are

5, 6, 7, 13, 14.

and 14 is the least possible value for the 
largest number in the set.

Edwin