Problem:
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?
Answer Choices:
A. 4
B. 6
C. 8
D. 10
E. 12
Solution:
Since the median and mode are both 8 and the range is 18, the list must take on one of these two forms:
or (I):(II):a,b,8,8,a+18 where a≤b≤8≤a+18c,8,8,d,c+18 where c≤8≤d≤c+18.
The sum of the five integers must be 60, since their mean is 12. In case (I), the requirement that 2a+b+34=60 contradicts a,b≤8. In case (II), 2c+d+34=60 and c≤8≤d≤c+18 lead to these six pairs, (c,d):
(8,10),(7,12),(6,14),(5,16),(4,18),(3,20).
Thus, the second largest entry in the list can be any of the six numbers d=10,12,14,16,18,20.