Gmat_mission wrote: ↑Sun Dec 27, 2020 3:39 pm
3 groups of 7 numbers each are made from positive integers from 1 to 21, inclusive. What is the highest possible median of the medians of these 3 groups?
A. 10
B. 11
C. 12
D. 14
E. 17.5
Answer:
D
Solution:
Let A, B, C be the 3 sets and a, b and c be their medians, respectively. For example, we could have A = {1, 2, 3, 4, 5, 6, 7}, B = {8, 9, 10, 11, 12, 13, 14}, and C = {15, 16, 17, 18, 19, 20, 21}. In this case, a = 4, b = 11, and c = 18 and we see that the median of the 3 medians is 11. Of course, the question is: can the median of the 3 medians be higher than 11?
Let’s analyze the given answer choices. We can eliminate choice E since the median of the 3 medians must be an integer from 1 to 21, inclusive. Now, let’s see if it can be 14. Before we check whether it can be 14, we can let a be the smallest median, b be the second smallest (or second largest) median and c be the largest median. Therefore, we see that in this case, b will be the median of the 3 medians. In other words we are checking whether b can be 14. If b can be 14, there must be 3 numbers in set B that are greater than 14 and since c is greater than 14, there must be 4 numbers in set C that are greater than 14. In other words, there are 7 numbers greater than 14. Since there are 7 integers from 1 to 21, inclusive, that are indeed greater than 14, we see that b could be 14, and it is the highest possible median of the 3 medians. (For example, we could have:A = {1, 2, 3, 4, 5, 6, 7}, B = {8, 9, 10, 14, 15, 16, 17} and C = {11, 12, 13, 18, 19, 20, 21}. In this case, a = 4, b = 14 and c = 18, and we see that the median of the 3 medians is 14.)
Answer: D 
