3 groups of 7 numbers each are made from positive integers from 1 to 2

Question

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

chetan2u · Accepted Answer

globaldesi 
There are 3 groups and a median of each of these 3 groups.
Now let them be  M_1<M_2<M_3 , so we are looking for the MAX possible value of  M_2 .

What would surely be more than M_2?

a) The group containing  M_2  will have three numbers more and three numbers less than  M_2 , so 3 numbers are greater than  M_2 .

b) The group containing  M_3  will have three numbers greater than  M_3 , and  M_2<M_3 so at least 4 numbers are greater than  M_2 .

Total = 3+4=7, and hence the largest possible value of  M_2  is  21-7=14

D

globaldesi · Answer

Can you please explain the solution for this question.

LevanKhukhunashvili · Answer

IMO we were asked to find the median of the 3 medians

1) 1, 2, 3,  4 , 11, 12, 13
2) 8, 9, 10,  14 , 15, 16, 17
3) 5, 6, 7,  18 , 19, 20, 21

Median of 4, 14, 18 is 14

I dont like this question honestly
I don't think it's GMAT-like

Regards
L

unraveled · Answer

LevanKhukhunashvili You may like this one .. https://gmatclub.com/forum/positive-int ... 15010.html Of course better one.

unraveled · Answer

3 groups having 7 numbers each are to be made in such manner that we get largest possible median for G3 and least possible for G1. Since three medians are to be found, that constraint would lead to arranging numbers in ascending order(at least the colored part). Rest of the part of the groups would be interchangeable which does not affect our case anyhow.

G1: 1, 2, 3, 4, 5, 6, 7 
G2: 8, 9, 10,  14, 15, 16, 17 
G3: 11, 12, 13,  18, 19, 20, 21

For G1 4 ≤ median ≤ 10, any value would not matter. The colored part is what matters most giving us 14 as maximum median possible among the three medians.

Hope this helps...

ScottTargetTestPrep · Answer

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

Michele4 · Answer

NOWHERE in the question is stated that these 3 sets CANNOT have the same integer values, so the highest possible median is 21 if:
A - (21,21,21,21,21,21,21)
B - (21,21,21,21,21,21,21)
C - (21,21,21,21,21,21,21)

jack5397 · Answer

Hi  unraveled ,
Can you pls explain why you chose 14 as the median to be maximum and arranged the numbers ?

Also it is no where mentioned that each group need to have the unique 7 values ?

Thanks!

unraveled · Answer

Hey jack5397 , Please take this question with a pinch of salt as someone already said so in earlier post. I posted this question way back, early in my gmat journey and not knowing that such question don't match GMAC's level. Anyway, it looks absurd kind of a question, however, it does tingle your brain enough to brush and brainstorm so that it's more of refreshing exercise than to bang you head trying to solve it. For this reason itself i posted to seek what other fellow members would do to such a question. Why I solved it like that(2 1/2 years ago) is because i found it easier to explain someone and nothing else. I was considering minimum possible and maximum possible values and thus solved it that way. It was sufficient enough but not necessary that such a solution is there. Do note that all 3 sets having 21 as the sole element is also a possibility though questionable if GMAC's standards are to be followed as no question can have such a simple solution OR if not then no question can have two solutions. Hope you don't mind seeing this response.

Dooperman · Answer

we have to select 3 medians (lets say a,b,c) and maximize the Median of medians (here b).

to max b, we need to minimize a,c.

to minimize a
First set 1) 1,2,3,4,5,6,7 : a=4.

to minimize c, we need to make it closest to b. and since we have 14 numbers remaining we should think of dividing the numbers such that medians of the 2 sets are adjacent numbers.

Divide the sets on the basis of Odd and even Numbers

Set 2) 8,10,12,14,16,18,20 : b = 14

Set 3) 9,11,13,15,17,19,21 : c = 15

Median of a,b,c = 14.

Kinshook · Answer

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?

Let us consume all smallest numbers in first group.

Group 1 = {1,2,3,4,5,6,7} : Median = 4

Group 2 = {8,10,12,14,16,18,20}: Median = 14

Group 3 = {9, 11, 13, 15, 17, 19, 21} : Median = 15

Highest possible median of the medians of these 3 groups = 14

IMO D

3 groups of 7 numbers each are made from positive integers from 1 to 2

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

3 groups of 7 numbers each are made from positive integers from 1 to 2

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards