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655-705 (Hard)|   Overlapping Sets|                                          
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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.


(A) Statement 1 alone is sufficient, but statement 2 alone is not.
(B) Statement 2 alone is sufficient, but statement 1 alone is not.
(C) Both statements together are sufficient, but neither alone is sufficient.
(D) Each statement alone is sufficient.
(E) Neither statement is sufficient, even when combined.



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The table above shows the results of a survey of 100 voters who each 14 Jan 2010, 16:03
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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.
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The table above shows the results of a survey of 100 voters who each Updated on: 06 Sep 2023, 23:08
Originally posted by KarishmaB on 22 Mar 2013, 02:12.
Last edited by KarishmaB on 06 Sep 2023, 23:08, edited 1 time in total.
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TheNona
still cannot understand ... any body can present a matrix please ? thanks in advance
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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 03 Jul 2011, 05:17
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siddhans

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 more

A lot of the information in the table just serves as a distraction here. The first column of the table gives us a standard 2-circle Venn diagram: those people who like M, those people who like N, and those people who like both. We want to know how many like both. If we know exactly how many people are in our diagram (i.e. how many people responded 'favourable' to at least one candidate), we can answer that question. Statement 1 tells us that 40 of the 100 people are *not* in our diagram, so 60 people must be in the diagram, and that info is sufficient. I can't draw a Venn diagram here, but I'd fill it in as follows:

Favorable for M only: 40 - x
Favorable for both M and N: x
Favorable for N only: 30 - x

From Statement 1 I know these three quantities add to 60, which gives me one equation in one unknown.

Statement 2 doesn't tell you how many people are in the diagram, so is not sufficient.

As for your question about using a 'matrix' or 'formula' here, I wouldn't consider doing either.
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The table above shows the results of a survey of 100 voters who each 24 Apr 2015, 23:19
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TheNona
still cannot understand ... any body can present a matrix please ? thanks in advance
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For all those who are trying to solve this question with matrix approach.

First, as always make a 3x3 matrix with rows and columns corresponding to mutually exclusive options. Notice that there are 3 unknown values in each row/column. So you need 2 or sum of 2 values in a row/column to find the third/desired value.

Attachment:
File comment: 3x3 matrix
3x3matrix.PNG
3x3matrix.PNG [ 4.43 KiB | Viewed 91089 times ]

Now, Statement 1 tells you the no. of voters who did not respond favorable for either of the candidates. This means, Not Fav = Unfavorable + Not Sure.
For simplicity, convert the 3x3 matrix to a 2x2 matrix with choices as "Fav" and "Not Fav".
Attachment:
File comment: 2x2 matrix
2x2matrix.PNG
2x2matrix.PNG [ 3.64 KiB | Viewed 89994 times ]
Now, given the no. of Not Favorable responses = 40, you can easily fill up the rest of the matrix to get to the answer.

Statement 2, gives you only the no. of unfavorable responses = 10. Remember, you need at least 2 sum of 2 other values to find the desired value. You can put it in the 3x3 matrix, to find out that the statement 2 is clearly insufficient.
Attachment:
File comment: 3x3 matrix for statement 2
3x3matrix statement 2.PNG
3x3matrix statement 2.PNG [ 4.47 KiB | Viewed 89194 times ]

Hope this helps.
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The table above shows the results of a survey of 100 voters who each 22 Mar 2013, 21:09
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TheNona

Thanks Karishma for the great explanation . In fact what I was struggling with is the insufficiency of B . Cannot understand why ?
Show more

Let's consider statement 2:

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

So this means that 90 people replied "favorable" or "not sure" to at least one of the candidates.

Also, going back to the table in the question,
Unfavorable for candidate M = 20
Unfavorable for candidate N = 35
Unfavorable for both = 10
Unfavorable for at least one = 20 + 35 - 10 = 45

So, all we know from here is the following:
1. 10 people responded "unfavorable" to both candidates
2. 35 (= 45 - 10) people responded "unfavorable to one candidate and "favorable" or "not sure" to the other.
3. 55 people responded with "favorable" or "not sure" to both the candidates.

Now we don't know the split between "favorable" and "not sure".
It is possible that 10 people responded "favorable" to both and it is also possible that that number is 15.
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still cannot understand ... any body can present a matrix please ? thanks in advance
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I didn't see this explicitly mentioned on this thread or the OA but....

For Statement 1, is this the formula that was used??

A + B - both + neither = total

40 + 30 - X + 40 = 100

X = 10


If so, I'm confused why the total is 100 and not 70, any help would be greatly appreciated thanks!
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The table above shows the results of a survey of 100 voters who each 24 Mar 2014, 20:55
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kaji
I didn't see this explicitly mentioned on this thread or the OA but....

For Statement 1, is this the formula that was used??

A + B - both + neither = total

40 + 30 - X + 40 = 100

X = 10


If so, I'm confused why the total is 100 and not 70, any help would be greatly appreciated thanks!
Show more

When you are considering the set of 'neither', you must take the total 'total'.

'total' would be either 60 or 100 (and not 70 because 70 is not the number of people. It is the number of instances which includes double counting of people who favor both candidates) depending on whether you include neither or not.

A + B - Both = 60 (the total number of people who favor at least one)
30 + 40 - Both = 60
Both = 10

OR

A + B - Both + Neither = 100 (total number of people including those who favored neither candidate)
30 + 40 - Both + 40 = 100
Both = 10

Remember, in this formula, 'total' is the total number of people without any double counting.
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The table above shows the results of a survey of 100 voters who each 11 Aug 2014, 02:07
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I don't understand why we assume that the 90 responded "favorable" or "not sure" but concerning the first statement we say that 60 people responded "favorable" ? What about the fraction of the 60 people who responded "not sure"? Are the "not sure" people included in the 40?
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The table above shows the results of a survey of 100 voters who each 11 Aug 2014, 22:31
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lou34

I don't understand why we assume that the 90 responded "favorable" or "not sure" but concerning the first statement we say that 60 people responded "favorable" ? What about the fraction of the 60 people who responded "not sure"? Are the "not sure" people included in the 40?
Show more

The structure of the two statements is quite different:

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

40 people did not give "Favorable" to either. It means the rest of the 60 people gave Favorable to at least one or both the candidates.

(2) The number of voters who responded "Unfavorable" for both candidates was 10.

10 people responded Unfavorable for both. It means the rest of the 90 people gave either "Favorable" or "Not Sure" to at least one of the two candidates.
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The table above shows the results of a survey of 100 voters who each 21 Oct 2014, 09:25
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Hi Karishma,

Thanks for the detailed explanation. In your response, 'Unfavorable for at least one = 20 + 35 - 10 = 45'. Shouldn't it be "exactly one" instead of "atleast one"

Thanks,
Venkitaraman.S


VeritasPrepKarishma
TheNona

Thanks Karishma for the great explanation . In fact what I was struggling with is the insufficiency of B . Cannot understand why ?

Let's consider statement 2:

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

So this means that 90 people replied "favorable" or "not sure" to at least one of the candidates.

Also, going back to the table in the question,
Unfavorable for candidate M = 20
Unfavorable for candidate N = 35
Unfavorable for both = 10
Unfavorable for at least one = 20 + 35 - 10 = 45

So, all we know from here is the following:
1. 10 people responded "unfavorable" to both candidates
2. 35 (= 45 - 10) people responded "unfavorable to one candidate and "favorable" or "not sure" to the other.
3. 55 people responded with "favorable" or "not sure" to both the candidates.

Now we don't know the split between "favorable" and "not sure".
It is possible that 10 people responded "favorable" to both and it is also possible that that number is 15.
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Venkitaraman
Hi Karishma,

Thanks for the detailed explanation. In your response, 'Unfavorable for at least one = 20 + 35 - 10 = 45'. Shouldn't it be "exactly one" instead of "atleast one"

Thanks,
Venkitaraman.S

Show more

No. It is unfavorable to at least one.

The 10 you subtract is because 10 is double counted.

20 people voted unfavorable for M. This includes 10 people who voted unfavorable for both candidates.
35 people voted unfavorable for N. This also includes 10 people who voted unfavorable for both candidates.
The 10 people are counted twice when we add 20 and 35. So we subtract it out once and we get 45. This includes the people who voted unfavorable for both. Hence "atleast one" is the right term to use here.
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The table above shows the results of a survey of 100 voters who each 06 Jul 2016, 06:06
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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.
Show more

Why we are not considering the option of "NOT SURE" to vote i.e. 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. -- Over here why 60 = voters who responded either favorable or NOT SURE.

I am sure I am missing something silly but I unable to figure it out. Please help.
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The table above shows the results of a survey of 100 voters who each 06 Jul 2016, 21:38
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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.

Why we are not considering the option of "NOT SURE" to vote i.e. 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. -- Over here why 60 = voters who responded either favorable or NOT SURE.

I am sure I am missing something silly but I unable to figure it out. Please help.
Show more

40 => People who did not respond "favourable" to either candidate. Then what did they respond? Either "Unfavourable" or "Not sure" to both candidates. 40 is not the number of people who respond "unfavourable". They are the ones who did not respond "favourable".
So 100 - 40 = 60 people responded "favourable" to at least one candidate. To the other candidate, they could have responded with "unfavourable" or "not sure" but to one at least they did respond with "favourable".
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The table above shows the results of a survey of 100 voters who each 23 Nov 2016, 19:25
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Tough one! Attached is a visual (with Venn diagrams) that should help. They key concept being tested here is realizing that the opposite of "neither of" is "at least one of," and that those two categories combine for 100% of the total.
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Screen Shot 2016-11-23 at 6.25.08 PM.png
Screen Shot 2016-11-23 at 6.25.08 PM.png [ 119.81 KiB | Viewed 72919 times ]

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Bunuel , IanStewart, VeritasPrepKarishma for statement 1 alone - why is it necessary that out of the 100 voters, the remaining 60 voters necessarily voted "favorable" for M/N or both. Some of them could have merely voted "not sure" or "unfavorable" for both of them but not all of them may have compulsorily voted "favorable" for either one of them. As this is not explicitly mentioned, that was what threw me off while solving the question. Please help
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Aditi10
Bunuel , IanStewart, VeritasPrepKarishma for statement 1 alone - why is it necessary that out of the 100 voters, the remaining 60 voters necessarily voted "favorable" for M/N or both. Some of them could have merely voted "not sure" or "unfavorable" for both of them but not all of them may have compulsorily voted "favorable" for either one of them. As this is not explicitly mentioned, that was what threw me off while solving the question. Please help
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(1) The number of voters who did not respond "Favorable" for either candidate was 40.

Out of 100 people, 40 did not vote favourable for either. They voted either unfavourable or 'not sure' for both. There were exactly 40 such people. So what about the other 60 people? It means the rest of the 60 people gave Favorable to at least one or both the candidates.
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Hi Bunuel, VeritasKarishma
This cannot be done via Venn diagrams, right?

Thanks!
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himanshu11pathak
Hi Bunuel, VeritasKarishma
This cannot be done via Venn diagrams, right?

Thanks!
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Such questions are far more suited to 2 by 2 matrices but can be done using venn diagrams too. There are some solutions on page 2 using venn diagrams. Check this:
https://gmatclub.com/forum/the-table-ab ... l#p1767001
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