A total of 800 students were asked whether they found two subjects, M

Since 200 students answered "yes" only for subject M, then the remaining 300 students who answered "yes" for subject M, also answered "yes" for subject R. So, 300 students answered "yes" for both subjects.

If 300 students answered "yes" for both subjects, then 400-300=100 students answered "yes" only for subject R.

So, we have that:
200 students answered "yes" only for subject M;
100 students answered "yes" only for subject R;
300 students answered "yes" for both subjects;

Therefore 800-(200+100+300)=200 students did not answer "yes" for either subject.

Answer: B.

Hope it's clear.
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This is another solution to this GMAT question. I hope it helps.

how did you deduce that the remaining 300 students must have said Yes to Subject R. There can be students who might have said Yes to Subject M and No to R and still be counted towards 500 who said Yes to M. Am I reading the premise wrongly?

Good question.

We know that 200 students answered YES only for subject M, so these 200 students answered something else for subject R. Now, what could have answered the remaining 300 students for subject R? If anyone of them answered either NO or UNSURE for subject R, then there would be more than 200 students who answered YES for only subject M. Therefore those 300 must have been answered YES for subject R.

Hope it's clear.
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----------------YES---------NO----UNSURE
Subject M----500--------200-----100
Subject R----400--------100-----300


A total of 800 students were asked whether they found two subjects, M and R, interesting. Each answer was either "yes" or "no" or "unsure", and the numbers of students who gave these answers are listed in the table above. If 200 students answered "yes" only for subject M, how many of the students did not answer "yes" for either subject?

A. 100
B. 200
C. 300
D. 400
E. 500


Total students = Yes Answers for M+ Yes answers for N - Yes intersection +Neither
800 = 500+400-300+Neither
Neither = 200


Archit

1. There are 800 students and each should give responses on 2 subjects. So there are totally 1600 responses
2. Y(m) + Y(mr)=500 --- (i) Y(r) + Y(mr)=400 ---- (ii)
where Y(m) and Y(r) are the number of yes responses to M only and R only resp, and Y(mr) is the number of yes responses to both M and R..
3. Number of yes responses to M only is 200 i.e., Y(m) =200. Therefore from (i) Y(mr)=300
4. Also from (ii) Y(r)= 400-Y(mr) =100.
5. We have Y(m) + Y(mr) + Y(r) + Y(neither) = 800, as each student should fall into one of these
6. Therefore Y(neither) i.e., those who did not answer yes to either of the subjects is 200.
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This can be solved easily with a venn diagram we are given for subject M yes only 200 and we can fill the other spots

The total is 800 = 200 + 100 + 300 + X

Neither Yes for both subject = 200

For people that are having a hard time understand this question, hopefully this table will help:

Hi, There is a typo in your table. The total no. of students who did not say yes to M should be 300. Thanks!

Given that 200 students answered YES only for subject M.
There are 500-200 = 300 more students who answered YES for subject M.
So, remaining 300 students who answered YES for subject M, also answered YES for subject R.
Hence, 300 students answered YES for both subjects.
This implies that 400-300 = 100 students answered YES only for subject R.

So, 800-(200+100+300)=200 students did NOT answer YES for either subject.

Hence option (B).

--
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I think the difficult part of this problem is organizing the information. The question gives you three different categories for answers and two different groups, which is a deviation from the normal overlapping sets problems that give you either three groups with overlap or two groups that overlap on two different binary factors.

However, notice that the question asks you about "YES" and (essentially) "NOT YES". This means that you can treat this like an overlapping set question in a 2x2 matrix (as has been described above).

If we re-write the matrix in the question for just "YES" and "NOT YES" and the two subjects, it looks like this:

YES NOT YES
Subject M 500 300
Subject R 400 400

We want to find the people who didn't vote yes for either category. More specifically, we want:

Total = (Yes for M) + (Yes for R) - (Yes for BOTH) + NEITHER

NEITHER is what we are looking for and we are given (Yes for M) and (Yes for R) and we need to find (Yes for BOTH) then we will have one variable to solve for. So how many people are in the (Yes for BOTH) category?

Ok so then the question tells you that 200 people answered YES for only Subject M. That means there must be (500-200) = 300 people who voted yes for both of them and so are getting double-counted (I think fozzzy's venn diagram shows this pretty well).

Aside: As others have mentioned you can also use this information to find out how many people voted YES for just Subject R (if that was the question). That figure would be 100.

However, we can now solve for the variable we are looking for:

Total = (Yes for M) + (Yes for R) - (Yes for BOTH) + NEITHER
800 = 500 + 400 - 300 + NEITHER
200 = NEITHER

(B)

Since 200 students answered "yes" for only subject M, the remaining 300 students who answered “yes” for subject M must also have answered “yes” for subject R. That means only 100 students answer “yes” for only subject R. In other words, the total number of students who answered “yes” to one or both subjects is 200 + 300 + 100 = 600, which means 800 - 600 = 200 students did not answer “yes” for either subject.

Answer: B
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Another approach is to use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of students, and the two characteristics are:
- Said "yes" to liking subject M or didn't say "yes" to liking subject M
- Said "yes" to liking subject R or didn't say "yes" to liking subject R

IMPORTANT: Notice that I just lumped the "unsure" respondents in with those who answered "no." It's okay to do this since the question is only interested in those who did not answer "yes"
So, we can CONDENSE our table to get:
Subject M: 500 answered "yes," and 300 did NOT answer "yes"
Subject R: 400 answered "yes," and 400 did NOT answer "yes"

We can now set up our diagram as follows:


The question tells us 200 students answered "yes" only for subject M
So, we know that 200 students can be placed in the bottom-left box:


From here, we can find the other values in the empty boxes:


The question asks: How many of the students did not answer "yes" for either subject?
The bottom-right box represents those students:


So, the correct answer is B

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:



EXTRA PRACTICE QUESTION

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----------------YES---------NO----UNSURE
Subject M----500--------200-----100
Subject R----400--------100-----300


Given: A total of 800 students were asked whether they found two subjects, M and R, interesting. Each answer was either "yes" or "no" or "unsure", and the numbers of students who gave these answers are listed in the table above.

Asked: If 200 students answered "yes" only for subject M, how many of the students did not answer "yes" for either subject?

If 200 students answered "Yes" only for subject M, then 500-200=300 students answered "Yes" for both subjects and 400-300=100 students answered "Yes" only for subject R.

Number of students who did not answer "Yes" to any subject = 800 - 200 - 100 - 300 = 200 students

IMO B
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Attached is a visual that should help.

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GIVEN:

• A total of 800 students responded to a survey about their interest in two subjects – M and R.
  
  • For each subject, each respondent answered with a “Yes”, “No” or “Unsure”.
• 200 students answered “yes” only for subject M.

TO FIND:

• Number of students who DID NOT answer “Yes” for either subject.

Let’s first understand what exactly the asked question means. Once you’re clear with this, the final working out will be straightforward.


UNDERSTAND THE QUESTION:
We want the number of students who did NOT answer “Yes” to either subject. To understand this clearly, let’s break the statement into two small pieces.

Part 1:
First, we’ll understand what it means when we say a student does NOT answer “Yes” to a subject. Well, not answering “Yes” means answering either “No” or “Unsure” – that is everything except “Yes”.
For any subject, we can find these people by subtracting those who did say “Yes” from the total people who responded, that is, from 800.

• Subject M: #Students not answering “Yes” for subject M
  
  • = 800 - #students answering “Yes” to M
• Subject R: #Students not answering “Yes” for subject R
  
  • = 800 - #students answering “Yes” to R

Part 2:
Next, we need to understand that “not saying yes to either” means “not saying yes to ANY of M and R- not a yes to M and not a yes to R.”
So, if we combine everything above, we get that our required number is:
800 – (# of students who answered “Yes” to M or R) ----(I)


WORKING OUT:
From (I), we understood that to answer our question, we need to find the #students who answer “Yes” to M or R.
Now, since there are chances of some students having said “Yes” to both subjects, the #students who said “Yes” to either M or R is NOT just the sum of those who said “yes” to M and those who said “yes” to R.
The correct formula to calculate the #students who answered “Yes” to M or R is:

(#Answered “Yes” to M) UNION (#Answered “Yes” to R).

To comfortably find this value, let’s visualize the situation on a Venn diagram.


Venn Diagram:
We already know the following from the given information:

• #Students who answered “Yes” to M = 500
• #Students who answered “Yes” to R = 400
• #Students who answered “Yes” to ONLY M = 200 ----(II)

Here, ‘x’, ‘y’ and ‘z’ represent those who answered “Yes” to only M, to both M and R, and to only R, respectively.
So, from (II), we can say that x = 200, x + y = 500, and y + z = 400.
Solving, we get y = 300 and z = 100.

So, the number of students who answered “Yes” to either of the subjects, that is, (Yes to M) UNION (Yes to R) is given by x + y + z.

- So, x + y + z = 200 + 300 + 100 = 600

Using (I), the required answer = 800 – 600 = 200


Correct Answer: Choice B


Best,
Aditi Gupta
Quant expert, e-GMAT
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