Problem Solving

The mean of six Positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is

A. 26
B. 28
C. 30
D. 32
E. 34
Official Answer: D

coolhabhi wrote:The mean of six Positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is

A. 26
B. 28
C. 30
D. 32
E. 34
Official Answer: D

Sum of the 6 integers = (number)(average) = 6*15 = 90.
Let the 6 integers be -- in ascending order -- a, b, c, d, e, f.

To MAXIMIZE the value of f, we must MINIMIZE the rest.

The MODE is the value that appears the MOST.
Since the mode must be less than 18, the smallest possible value for the mode is 1.
Let a=1 and b=1, yielding the following:
1, 1, c, d, e, f.

Since the median is 18, the average of c and d is 18:
(c+d)/2 = 18
c+d = 36.

To minimize the value of e, we must minimize the value of d.
Since there is only ONE MODE, there cannot be another integer that appears twice.
Thus, c and d must be different integers, implying that the smallest possible combination for c and d is c=17 and d=19:
1, 1, 17, 19, e, f.

Since no integer other than 1 can appear twice, the smallest possible value for e is 20, yielding the following:
1, 1, 17, 19, 20, f.

Thus:
f = 90-1-1-17-19-20 = 32.

The correct answer is D.