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655-705 Level|   Overlapping Sets|                                       
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The table above shows the results of a survey of 100 voters who each 02 Oct 2020, 01:45
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Hi VeritasKarishma,

Thank you for your explanations on this question. Request you to help with the below solution

1) This is my matrix and I am able to solve the question putting 40 (above 70) in this matrix.
For Option 1
However the question is "who did not respond favorable to either" should mean who responded favourable to both or who did not respond favourable to both. My understanding is it should be worded as who did not respond favourable to both.

Not for either A or B = For Both A & B + Not for Both A & B.

So i feel the option is worded wrong ?

Also is the above method correct.

TIA
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The table above shows the results of a survey of 100 voters who each 06 Oct 2020, 02:55
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kagrawal16
Hi VeritasKarishma,

Thank you for your explanations on this question. Request you to help with the below solution

1) This is my matrix and I am able to solve the question putting 40 (above 70) in this matrix.
For Option 1
However the question is "who did not respond favorable to either" should mean who responded favourable to both or who did not respond favourable to both. My understanding is it should be worded as who did not respond favourable to both.

Not for either A or B = For Both A & B + Not for Both A & B.

So i feel the option is worded wrong ?

Also is the above method correct.
TIA
Show more

What you are suggesting is ambiguous.
Does 'not favourable to both' mean 'not favourable to A and not favourable to B' or does it mean 'not favourable to both but could be favourable to one of them'

'not favourable to either' clearly means 'not favourable to A and not favourable to B'

'I don't like either' means I like neither.

I don't understand how you arrived at 10 so unable to say if the method is correct.
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The table above shows the results of a survey of 100 voters who each 30 Nov 2020, 07:32
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Video solution from Quant Reasoning:
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The table above shows the results of a survey of 100 voters who each 22 Feb 2021, 12:33
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samrand

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

(1) The number of voters who did not respond "Favorable" for either candidate was 40.
(2) The number of voters who responded "Unfavorable" for both candidates was 10.

Show Spoiler
Attachment:
Candidates.png
Show more

Let's solve this question using the Double Matrix method. This technique can be used for questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
For this particular question, we only care about the people who responded Favorable for both candidates
So, we can combine the people who responded Unfavorable and Not Sure to get the following:

I've placed that star in the box representing the people who responded Favorable for both candidates.

Target question: What is the value in the top left box (denoted by a star)?


Statement 1: The number of voters who did not respond "Favorable" for either candidate was 40.
We can add this information to our matrix to get:


We now have enough information to complete the matrix:


So, the answer to the target question is 10 people responded responded Favorable for both candidates
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The number of voters who responded "Unfavorable" for both candidates was 10.
This information belongs in the bottom right box of the double matrix.

However, since we have combined the people who responded Unfavorable and Not Sure, we still don't know how many people responded Not Sure
As such, we don't have enough information to determine the value in the bottom right box, which means there's no way to complete the rest of the matrix.
Statement 2 is NOT SUFFICIENT

Answer: A

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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The table above shows the results of a survey of 100 voters who each 20 May 2021, 10:08
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samrand

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

(1) The number of voters who did not respond "Favorable" for either candidate was 40.
(2) The number of voters who responded "Unfavorable" for both candidates was 10.

Show Spoiler
Attachment:
Candidates.png
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Solution:

Question Stem Analysis:

We need to determine the number of voters who responded "Favorable" for both candidates.

Statement One Alone:

Since the number of voters who did not respond "Favorable" for either candidate was 40, the number of voters who responded "Favorable" for either candidate was 100 - 40 = 60. That is, the number of votes who responded "Favorable" for candidate M or N was 60. We can use the following formula (notice we need to determine the value of #(M and N)):

#(M or N) = #(M) + #(N) - #(M and N)

60 = 40 + 30 - #(M and N)

60 = 70 - #(M and N)

#(M and N) = 10

Statement one alone is sufficient.

Statement Two Alone:

Knowing the number of voters who responded "Unfavorable" for both candidates is not sufficient to determine the number of voters who responded "Favorable" for both candidates.

Answer: A
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The table above shows the results of a survey of 100 voters who each 23 Sep 2021, 16:49
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THIS QUESTIONS IS WRONG!!!! there can't be 100 voters, there must be 115 voters for the statements to be logical.

It is possible to a gmat question to be wrong? I thought the statements in Data sufficiency must be true in order to answer the question. But in this case, what the statements are saying can't be true:

There are only 100 voters and they can only vote Favorable, Unfavorable or not sure for the 2 candidates.Let's say X is people who voted favourable for both candidates, Y Unfavorable for both candidates and Z Not sure for Both Candidates.. You can use the chart or a venn diagram, i think the latter is easier to see.

i) The first statement says that there are 40 who did not vote Favorable for either candidate, or 60 people voted Favorable. We can conclude that (70-x)+40=100 => x=10 (10 people voted favourable for both candidates) (this is also the explanation the GMAT Official guide gives, which is correct)

Only with this statement the question is not rational, why?: In order to have 100 voters and knowing that 40 people voted not favorable, it must be true that all the people who voted unfavorable for at least one plus all the people who voted not sure for at least one must be 40. Using the same equation it would be (55-y)+(75-z)+60=100 (people who voted Unfavorable for at least one) + (People who voted Not Sure for at least one) + (people who voted Favourable for at least one)=100. This equations end up like this: 90=y+z. But this is impossible, because y≤20 and Z≤35. So y+z ≤55, it can't be 90.

Think it this way, at most there are 20 people who voted Unfavorable for both candidates (the are only 20 who voted unfavorable for candidate M and 35 for candidate N, there can't be more than 20 votes for people who voted for both, in a venn diagram you would see the smaller circle entirely inside the bigger circle) If you said 21 people voted unfavorable for both, it means there are at least 21 who voted unfavorable for candidate M, which we know is not true. Happens the same for the voters of "not sure". At most there are 35 who voted not sure for both, because there are only 35 who voted not sure for candidate N and 40 for candidate M. Also, the smallest number of people who voted for each section is the maximum votes between candidate M and N for each option: Favorable, Unfavorable and Not sure is 40, 35 and 40 respectively, so there must be 115 voters, not 100.

ii) The second statement further demonstrate the inconsistency of this question: If there are 10 who voted unfavorable for both candidates (y=10), this means there are 45 people who voted in the "unfavorable section" making it inconsistent with the first statement (40 people did not respond favorable) unless there are -5 people who voted not sure (which is illogical)
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The table above shows the results of a survey of 100 voters who each 23 Sep 2021, 21:40
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ccontesse
THIS QUESTIONS IS WRONG!!!! there can't be 100 voters, there must be 115 voters for the statements to be logical.

It is possible to a gmat question to be wrong? I thought the statements in Data sufficiency must be true in order to answer the question. But in this case, what the statements are saying can't be true:

There are only 100 voters and they can only vote Favorable, Unfavorable or not sure for the 2 candidates.Let's say X is people who voted favourable for both candidates, Y Unfavorable for both candidates and Z Not sure for Both Candidates.. You can use the chart or a venn diagram, i think the latter is easier to see.

i) The first statement says that there are 40 who did not vote Favorable for either candidate, or 60 people voted Favorable. We can conclude that (70-x)+40=100 => x=10 (10 people voted favourable for both candidates) (this is also the explanation the GMAT Official guide gives, which is correct)

Only with this statement the question is not rational, why?: In order to have 100 voters and knowing that 40 people voted not favorable, it must be true that all the people who voted unfavorable for at least one plus all the people who voted not sure for at least one must be 40. Using the same equation it would be (55-y)+(75-z)+60=100 (people who voted Unfavorable for at least one) + (People who voted Not Sure for at least one) + (people who voted Favourable for at least one)=100. This equations end up like this: 90=y+z. But this is impossible, because y≤20 and Z≤35. So y+z ≤55, it can't be 90.

Think it this way, at most there are 20 people who voted Unfavorable for both candidates (the are only 20 who voted unfavorable for candidate M and 35 for candidate N, there can't be more than 20 votes for people who voted for both, in a venn diagram you would see the smaller circle entirely inside the bigger circle) If you said 21 people voted unfavorable for both, it means there are at least 21 who voted unfavorable for candidate M, which we know is not true. Happens the same for the voters of "not sure". At most there are 35 who voted not sure for both, because there are only 35 who voted not sure for candidate N and 40 for candidate M. Also, the smallest number of people who voted for each section is the maximum votes between candidate M and N for each option: Favorable, Unfavorable and Not sure is 40, 35 and 40 respectively, so there must be 115 voters, not 100.

ii) The second statement further demonstrate the inconsistency of this question: If there are 10 who voted unfavorable for both candidates (y=10), this means there are 45 people who voted in the "unfavorable section" making it inconsistent with the first statement (40 people did not respond favorable) unless there are -5 people who voted not sure (which is illogical)
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ccontesse

One person could have voted "Favourable" for one candidate and "Unfavourable" for the other.

(people who voted Unfavorable for at least one) + (People who voted Not Sure for at least one) + (people who voted Favourable for at least one)=100

in your explanation above is not correct. This would be more than 100. If A voted Favourable for one and Unfavorable for the other, he would be included in both first and third groups.

Number of people is 100. Number of instances of votes is 200 (each person puts one vote each for the two candidates)
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The table above shows the results of a survey of 100 voters who each 29 Oct 2022, 18:39
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samrand

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

(1) The number of voters who did not respond "Favorable" for either candidate was 40.
(2) The number of voters who responded "Unfavorable" for both candidates was 10.

Show Spoiler
Attachment:
Candidates.png
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I realize that this question is discussed at nauseum, but to clarify, this question cannot be solved using a triple overlapping set to address the problem with divesting from the "not favorable"=unfavorable + unsure? Just wanted to make sure I understand
the point of this question was the ambiguity in data for statement 2 in deciding between what categories to select.

Triple overlapping sets are my weakness!
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The table above shows the results of a survey of 100 voters who each 30 Oct 2022, 23:06
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woohoo921

We can't do this as a triple set, because there aren't three things that can overlap. Each person made 2 choices--one about M and one about N--so there's no triple case here. Rather, we have 2 different choices, each of which has 3 mutually exclusive (non-overlapping) options. If we lump together "unfavorable" and "not sure," we can make a Venn diagram to show the overlap between "favorable M" and "favorable M," but that's still just two overlapping choices, not three.
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The table above shows the results of a survey of 100 voters who each 31 Oct 2022, 09:12
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woohoo921

We can't do this as a triple set, because there aren't three things that can overlap. Each person made 2 choices--one about M and one about N--so there's no triple case here. Rather, we have 2 different choices, each of which has 3 mutually exclusive (non-overlapping) options. If we lump together "unfavorable" and "not sure," we can make a Venn diagram to show the overlap between "favorable M" and "favorable M," but that's still just two overlapping choices, not three.
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Thank you DmitryFarber!

Have you come across similar problem types where you need to make this sort of lump? I would love to practice more of this unique type. Thank you again.
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The table above shows the results of a survey of 100 voters who each 31 Oct 2022, 22:47
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Hmm, not that I can think of. I think the trick to mastering Overlapping Sets is to get the core techniques down (double-set matrix, triple Venn diagram, and formulas), and then replay each question you see a few times. You can expect the GMAT to run some kind of variation, but they may not pull the same one you've prepared for, so you just have to be ready to adjust! If you do find another like this, maybe link it here.
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The table above shows the results of a survey of 100 voters who each 11 Mar 2023, 17:43
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Bunuel

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

Voters responded favorable for at least one candidates = 40+30-x = 70-x (x represent the # of voters who responded favorable for both candidates)

(1) The number of voters who did not respond “favorable” for either candidate was 40 --> The voters responded favorable for at least one 100-40=60=70-x --> x=10. Sufficient.

(2) The number of voters who responded “unfavorable” for both candidates was 10. Clearly not sufficient.

Answer A.
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Doesn't the term 'at least one' include M intersection N too, i.e. 10?
Can you please provide more explanation about inferring complements of sets? How the complement of 'number of voters who did not respond “favorable” for either candidate' would be 'favourable for either but excluding the intersection'?

Also, would be great if you could direct me to resources/ videos which can help me better understand how to infer complements for such difficult cases for more conceptual clarity?

Thanks!
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The table above shows the results of a survey of 100 voters who each 12 Mar 2023, 05:31
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Bunuel

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

Voters responded favorable for at least one candidates = 40+30-x = 70-x (x represent the # of voters who responded favorable for both candidates)

(1) The number of voters who did not respond “favorable” for either candidate was 40 --> The voters responded favorable for at least one 100-40=60=70-x --> x=10. Sufficient.

(2) The number of voters who responded “unfavorable” for both candidates was 10. Clearly not sufficient.

Answer A.

Doesn't the term 'at least one' include M intersection N too, i.e. 10?
Can you please provide more explanation about inferring complements of sets? How the complement of 'number of voters who did not respond “favorable” for either candidate' would be 'favourable for either but excluding the intersection'?

Also, would be great if you could direct me to resources/ videos which can help me better understand how to infer complements for such difficult cases for more conceptual clarity?

Thanks!
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(1) The number of voters who did not respond “favorable” for either candidate was 40:

Since out of 100 voters 40 did NOT respond “favorable” for either candidate (40 did not like any of M and N), the remaining 60 voters must have responded favorable for at least one candidate (the remining 60 mush have liked either M or N or both), how else?

So, 60 = (all who liked M) + (all who liked N) - (all who liked both) (subtracting to get rid of doublecounting both)

19. Overlapping Sets



For more:
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Hope it helps.
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The table above shows the results of a survey of 100 voters who each 13 Mar 2023, 01:33
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Bunuel

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

Voters responded favorable for at least one candidates = 40+30-x = 70-x (x represent the # of voters who responded favorable for both candidates)

(1) The number of voters who did not respond “favorable” for either candidate was 40 --> The voters responded favorable for at least one 100-40=60=70-x --> x=10. Sufficient.

(2) The number of voters who responded “unfavorable” for both candidates was 10. Clearly not sufficient.

Answer A.

Doesn't the term 'at least one' include M intersection N too, i.e. 10?
Can you please provide more explanation about inferring complements of sets? How the complement of 'number of voters who did not respond “favorable” for either candidate' would be 'favourable for either but excluding the intersection'?

Also, would be great if you could direct me to resources/ videos which can help me better understand how to infer complements for such difficult cases for more conceptual clarity?

Thanks!
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Check out this blog post: https://anaprep.com/sets-statistics-nuances-of-sets/
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The table above shows the results of a survey of 100 voters who each 01 Jun 2024, 23:46
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Bunuel, are there similar tricky Overlapping Sets questions? It will be of great help in solving Two-Part Analysis Questions as well.
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The table above shows the results of a survey of 100 voters who each 02 Jun 2024, 02:35
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Bunuel, are there similar tricky Overlapping Sets questions? It will be of great help in solving Two-Part Analysis Questions as well.
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­Check here:

https://gmatclub.com/forum/search.php?s ... &tag_id=45
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The table above shows the results of a survey of 100 voters who each 01 Nov 2024, 12:48
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Bunuel

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

The number of voters who responded favorably to at least one candidate is equal to M + N - Both = 40 + 30 - x = 70 - x (where x represents the number of voters who responded favorable for both candidates)

(1) The number of voters who did not respond “favorable” for either candidate was 40:

Since out of 100 voters 40 did NOT respond “favorable” for either candidate, the remaining 60 voters must have responded favorable for at least one candidate. Hence, 70 - x = 60, which gives x = 10.

Sufficient.

(2) The number of voters who responded “unfavorable” for both candidates was 10.

The above implies that there are 90 people who responded “favorable” for either one or "not sure" for either one. Clearly not sufficient to fin the number of voters who responded favorable for both candidates.

Answer A.


I did it this way!
Is it right?? only thing i was not sure was is there any overlap between the unfavourable and not sure votes. because actual total for unfav without overlap is 35, then how will not sure which is so huge account for it??
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The table above shows the results of a survey of 100 voters who each 06 Feb 2025, 02:20
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@[color=#683d3d]KarishmaB[/color] Ma'am, can you please simplify Statement 1. The rest of the 60 people should be both favourable to M&N, why will it be atleast?
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TheNona
still cannot understand ... any body can present a matrix please ? thanks in advance


I am assuming you are unable to figure out why statement 1 is sufficient. Think of it this way:

(1) The number of voters who did not respond "Favorable" for either candidate was 40.

This means that 60 voters responded 'Favorable' for at least one candidate, right? Now we need to find how many responded Favorable to both.

Now forget this question and think of another sets question:

There are 60 voters in a constituency. Each voter has to vote 'favorable' for at least one of the two candidates - M and N. Candidate M gets 40 favorable votes and candidate N gets 30 favorable votes. How many voters voted 'favorable' for both the candidates?

It's an easy enough question with 2 sets. You will just use
60 = 30 + 40 - Both
Both = 10

This is exactly what is required of you in this question. Just that there is a lot of other data to distract you. You know that 60 voters voted favorable for atleast one candidate. You also know that M got 40 favorable votes and N got 30 favorable votes (from the table int he question). you just need to find the value of 'both'. Focus on what you have to find, and the given relevant info.

Check out the video discussing Overlapping Sets here: https://youtu.be/HRnuURqGhmg
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The table above shows the results of a survey of 100 voters who each 08 Feb 2025, 03:00
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"Did not respond 'Favorable' for either candidate" means that they were either Unfavorable or Not Sure for both M and N. But that doesn't mean that everyone else was Favorable for both M and N. The other 60 might be Unfavorable or Not Sure for one, but Favorable for the other. In other words, we can break the 100 people into 2 groups: 40 = favorable for 0 of the 2 measures, 60 = favorable for 1-2 of the 2 measures.
ragnarok13
@[color=#683d3d]KarishmaB[/color] Ma'am, can you please simplify Statement 1. The rest of the 60 people should be both favourable to M&N, why will it be atleast?
KarishmaB
TheNona
still cannot understand ... any body can present a matrix please ? thanks in advance


I am assuming you are unable to figure out why statement 1 is sufficient. Think of it this way:

(1) The number of voters who did not respond "Favorable" for either candidate was 40.

This means that 60 voters responded 'Favorable' for at least one candidate, right? Now we need to find how many responded Favorable to both.

Now forget this question and think of another sets question:

There are 60 voters in a constituency. Each voter has to vote 'favorable' for at least one of the two candidates - M and N. Candidate M gets 40 favorable votes and candidate N gets 30 favorable votes. How many voters voted 'favorable' for both the candidates?

It's an easy enough question with 2 sets. You will just use
60 = 30 + 40 - Both
Both = 10

This is exactly what is required of you in this question. Just that there is a lot of other data to distract you. You know that 60 voters voted favorable for atleast one candidate. You also know that M got 40 favorable votes and N got 30 favorable votes (from the table int he question). you just need to find the value of 'both'. Focus on what you have to find, and the given relevant info.

Check out the video discussing Overlapping Sets here: https://youtu.be/HRnuURqGhmg
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The table above shows the results of a survey of 100 voters who each 21 Jul 2025, 05:20
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samrand

The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

(1) The number of voters who did not respond "Favorable" for either candidate was 40.
(2) The number of voters who responded "Unfavorable" for both candidates was 10.

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You can solve this question using an approach I learned in GMATNinja video (https://www.youtube.com/watch?v=TRlVgm7RdmE&t=2579s)

First convert the table into a regular 2*2 grid as follows
We are asked to find x.

Fav MNot Fav MTotal
Fav N x ? 30
Not Fav N 70
Total4060100

Now go to statement 1 - We can get x as below. Sufficient.


Fav MNot Fav MTotal
Fav N 40-30=10 30
Not Fav N 70-40=304070
Total4060100

Statement 2 - This is a tricky word play and its different from statement 1 in the sense that it is a subset of the information given in statement 1. However, by knowing how many viewed both unfavourably is not sufficient to find x.

Answer = A

Hope this helps.
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