The diagram shows the relationships between 3 groups of stockholders

Question

https://gmatclub.com/forum/download/file.php?id=81584

The diagram shows the relationships between 3 groups of stockholders of Company HQS and the number of stockholders in each group

Group P consists of those stockholders who receive a paper copy of the annual report. 
 Group M consists of the majority stockholders. 
 Group T consists of those stockholders who are traders

Exactly 8 stockholders belong to all three groups. It is possible that one or more of the regions in the diagram do not contain any members.

Select from each drop-down menu the option that creates the most accurate statement based on the information provided.

If all majority stockholders receive paper reports, then  GRAPH_1_PLACEHOLDER  is the maximum number of traders who could receive paper reports.

If exactly 10 majority stockholders receive paper reports, then  GRAPH_2_PLACEHOLDER  is the maximum number of traders who could also be majority stockholders.

egmat · Accepted Answer

A brilliant question that is truly difficult for the following students:
 
 Students who cannot visualize Venn diagrams and movements in the same to maximize or minimize the regions on the Venn diagram. 
 Students who are not in the habit of visualizing and infering as they read question statements.

So, watch this video solution and if you found this question difficult, identify the reason. Is it because you belong to both categories above or just one of the categories?

https://www.youtube.com/watch?v=kqlngdpYtIg

Another thing of interest in this question is that the two blanks are independent of each other. They both require separate processing but if you know how to process one, you should be able to process the other one too.

In fact, if students practice the minimizing and maximizing of regions on a Venn diagram, then this question can become one of the time saver questions!

shivanisomani02 · Answer

Solution using Venn Diagram.

chetan2u · Answer

We know what each group contains and what does the region of overlap of all three contains. 1. All majority stockholders recieve paper reports, so 8 are in the overlap region of all three(P&M&T), while there are 4 majority stockholders in Only( P and M). We are looking for the maximum number of traders who could receive paper reports: The majority stockholdres(M) do not have any influence in the region shared by P & T, so all 245 can be recieving paper reports. 2. Exactly 10 majority stockholders receive paper reports, so the breakdown would be Only(P&M) - 10-8 or 2, while All three = 8. Thus, we are left with only 2 from M group who can be placed in the overlap region of M&T. SO, maximum number of traders who could also be majority stockholders.= 8(all three) + 2(M&T) = 10 GMAT-Club-Forum-ftohflsq.png

ruis · Answer

Official answer :
Let a, b, and c be the numbers, respectively, of members in Groups P and M but not T, Groups P and T but not M, and Groups M and T but not P. The labeled diagram below shows for each of the 7 regions in the original diagram the number of stockholders in terms of a, b, and c. For example, since there are 12 members in Group M, it follows that the number of members in Group M only is 12 − (a + 8 + c) = 4 − a − c.
 https://s3.amazonaws.com/wmx-api-production/courses/50731/images/Graphic%201%20(000915).gif

We are to assume that each member of Group M belongs to Group P and determine the maximum possible number of members belonging to both Group T and Group P. From our assumption, it follows that 4 −  a  −  c  = 0 and  c  = 0, or  a  = 4 and  c  = 0. Using this information, and suppressing information about Group M (which is not relevant to answer the question), the labeled diagram above can be replaced with the labeled diagram below:
 https://s3.amazonaws.com/wmx-api-production/courses/50731/images/Graphic%202%20(000915).gif 
Because each of the three regions represents a nonnegative number of members, it follows that 237 −  b  ≥ 0, or  b  ≤ 237. Therefore, the number of members belonging to both Group T and Group P, namely 8 +  b , is no more than 8 + 237 = 245. Moreover, letting  a  = 4,  b  = 237, and  c  = 0, it is easy to see from the first labeled diagram above that it is possible to have 245 members belonging to both Group T and Group P.

The correct answer is  245 .

We are to assume that the number of members belonging to Group M and Group P is 10 and determine the maximum possible number of members belonging to both Group T and Group M. From our assumption, it follows that  a  + 8 = 10, or  a  = 2. Using this information, the first labeled diagram above can be replaced with the labeled diagram below:
 https://s3.amazonaws.com/wmx-api-production/courses/50731/images/Graphic%203%20(000915).gif

Because 2 −  c  cannot be negative, it follows that  c  ≤ 2. Therefore, the number of members belonging to both Group T and Group M, namely 8 +  c , is no more than 8 + 2 = 10. Moreover, letting  a  = 2,  b  = 0, and  c  = 2, it is easy to see from the first labeled diagram above that it is possible to have 10 members belonging to both Group T and Group M.

The correct answer is  10 .

Gemmie · Answer

1. If all majority stockholders receive paper reports, then   is the maximum number of traders who could receive paper reports.

All majority stockholders receive paper reports
=> Those belong to only 2 groups M-T = 0

Maximum is when all Traders (T) are also those who receive Paper reports (P) (no one belongs to group T only)
=> 245 Traders

2. If exactly 10 majority stockholders receive paper reports, then   is the maximum number of traders who could also be majority stockholders.

P-T-M (those belong to all 3 groups) = 8
P-M (those belong to 2 groups P & M) = 10
M (those belong to group M) = 12

=> Those belong to M but not P => 2, each of whom can be either (i) in group M only or (ii) M-T (those belong to 2 groups T & M)

=> Maximum is case (ii)

=> Total T-M (those belongs to 2 groups T&M) = 8 + 2 = 10

KarishmaB · Answer

Given info in the graph: Paper - 3100 Majority - 12 Trader - 245 All - 8 If all majority stockholders receive paper reports, then ____ is the maximum number of traders who could receive paper reports. This just means that the M circle has been pushed inside the P circle (shown below). 'All' still remains 8 only, of course. Now think about it - what stops us from pushing the T circle also inside the P circle? Nothing. Screenshot 2024-04-13 at 8.17.04 PM.png We can certainly do it since we have no constraints on any other region. This is what it will look like and it is acceptable. Hence all traders could get the paper reports. ANSWER: 245 If exactly 10 majority stockholders receive paper reports, then ____ is the maximum number of traders who could also be majority stockholders. The overlap of M and P is exactly 10. We already know that 8 people lie in the 'All' region hence these 8 overlap with M and P (along with T). Now we know that 2 overlap in the M and P but not T region (as shown). Then 2 majority shareholders do not get paper reports and must lie outside the P circle. Screenshot 2024-04-13 at 8.21.04 PM.png What could be the maximum overlap of M and T? M and T already have an overlap of 8 (in the 'All' region). I want to put more people in the hatched area in the diagram below. Well, we have only 2 majority shareholders left who have not been placed yet. We can put them in the hatched area so that 8 + 2 = 10 people overlap in M and T circles. ANSWER: 10 Check these links for discussion on overlapping sets: https://anaprep.com/sets-statistics-thr ... x-in-sets/ https://anaprep.com/sets-statistics-why ... re-tricky/ GMAT-Club-Forum-5oe2pjqd.png

anushree01 · Answer

Although it says it is 805+ difficulty, I believe its reasonably easy if you understand what`s given & know the concepts of majority in overlapping sets.

Bunuel · Answer

The difficulty level of a question on the site, after sufficient attempts, is determined  automatically  based on various parameters collected from users' attempts via timer, such as the percentage of correct answers and the time taken to answer the question. So, this is an 805+ level question based on our statistics.

0PS0 · Answer

Just wanted to clarify the answer for the second part;
If exactly 10 majority stockholders receive paper reports, then _ is the maximum number of traders who could also be majority stockholders.
The answer is 10 and not 2 because we are including stockholders part of all 3 groups?

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=88475 Yes. The answer is 10 because “traders who could also be majority stockholders” includes everyone who is both in Group T and Group M, including the 8 stockholders who are in all three groups. Those 8 are already majority stockholders and traders. Also, since 12 majority stockholders exist and exactly 10 receive paper reports, the remaining 2 majority stockholders could also be traders. So maximum = 8 + 2 = 10. GMAT-Club-Forum-46hrb2fx.png

The diagram shows the relationships between 3 groups of stockholders

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The diagram shows the relationships between 3 groups of stockholders

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