The table shows the number of people who responded yes or no or d

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Fixed the image.

There are 2 ways to know the number of of people who did not respond “yes” to implementing either of the two programs

1. 1000 - [X(yes) U Y(yes)], where 'U' represents union.


2. [X(No) ⋂ Y(No)] + [X(No) ⋂ Y(Dont)] + [X(Dont) ⋂ Y(No)] + [X(Dont) ⋂ Y(Dont)], where '⋂' represents the intersection.


Statement 1-

It gives value of [X(yes) ⋂ Y(yes)] = 400 -300 =100

We know X(yes) , Y(yes) and [X(yes) ⋂ Y(yes)]. We can find [X(yes) U Y(yes)]. (Now use 1st way)

Sufficient.

Statement 2-

It gives the value of [X(No) ⋂ Y(No)]. But [X(No) ⋂ Y(Dont)], [X(Dont) ⋂ Y(No)] and [X(Dont) ⋂ Y(Dont)] are still unknown.

Insufficient

Straight to point: Given (Yes)U(No)U(Dk)=1000;

Let's use denotion ' for negation
i.e., Who did not respond yes =(YES)'= Total- (Yes)

1) Only yes to X = 300
----> X(yes) n Y(yes)=400-300=100
----> X(yes) U Y(yes)=X+Y-(X n Y)= 400+300-100=600=TOTAL (YES)
---->(YES)'=1000-(yes)=400. SUFFICIENT

2)No (XnY)=100; what about don't know case ? INSUFFICIENT


Therefore A is the answer.

HOPE this HELPS

THANKS

Where was X[yes] = 300 used?

The question asks for number of people who didn't answer "yes" to neither program at all. And the table tells us 400 answered Yes to program 1 and 300 to program 2, and likely there are some who answered "yes" to both questions.

So number of people who didn't answer "yes" at all = 1000 - 400 -300 + number of people who answered "yes" to both programs, which is all we need to find out.

1) if the number of answering yes to X only is 300, then we know the number of whom answered "yes" to both = 400-300 = 100, sufficient - AD
2) this conditions tells similar data in the "no" category, however which is insufficient given we don't know how many is overlapped in the "don't know" category.

the final answer is A

I have a question

In

----> X(yes) n Y(yes)=400-300=100

How can we assume that those 100 people have said Yes to both to X and Y? Couldn't be a case that 50 of those said Yes to X and Y and 50 said Yes to X but No to Y ?

Rules:
Total = X + Y - Both + Neither
X = OnlyX + Both

Given:
1000 = 400 + 300 - Both + Neither
Neither = No + Don't know

Mission: Find Neither

(1) --> OnlyX = 300 --> Both = 100 --> 1000 = 400 + 300 - 100 + Neither

Sufficient.


(2) --> No to Both = 100 -->

1000 = NoX + NoY - NoBoth + Other = 200 + 350 - 100 + Other
Other = Did not answer No to any category
Other = Don't know to Both + OnlyX + OnlyY + Both

What we really need to know here is the number who answered "Don't know to Both".


To me, statement 2 was tricky because my method of solving these kind of questions was turned upside down. However, I realise that we need to isolate the Yes-category. This is what we usually are asked to do in these questions and this is what statement 1 helped us to do. Now with 2, we instead was able to isolate the No-category.

If:

Yes + Dont Know + No = All

(1) gave us Yes + (Dont know + No)
(2) gave us No + (Dont know + Yes)

Interesting question!

This is a tricky one, and you'll find that a Venn diagram won't help.

In this question, the same 1,000 people were asked two different questions, and we are looking for the number of people who did not say yes to either question—in other words, the number of people who said “No” or “Don’t Know” to both Programs X and Y.

Obviously, this number has to be equal to or below 600, because only 600 people said “No” or “Don’t Know” to Program X alone, and presumably, at least some of these 600 people said “Yes” to Program Y—since it's unreasonable to expect the answers of all 1,000 respondents to be exactly the same with regard to both questions.

However, we don’t know exactly how many of these 600 people said “yes” to program Y only (this fact would also be sufficient), which is the number we would need to subtract from 600 to obtain the answer's value.

So, instead, apply the same concept in the opposite direction. Take the 700 people who said “No” or “Don’t Know” to Program Y, and subtract the 300 people who said “Yes” to Program X only (condition #1), since those 300 people don’t count, and they must have come from the “No/Don’t Know” category for Program Y—since they said yes ONLY to X. 700-300 = 400. Sufficient

Condition #2 is not sufficient, because it does not clarify the overlap between the "No/Don't Know" and "Yes" categories for each question, which is exactly what's holding us back from obtaining an answer.

Looking at the big picture, though, don’t let questions like this one mess with your head on your G-Day. If a question is taking too long, then take an educated guess and buckle down on the next one.

MartyTargetTestPrep

I would be so appreciative to learn your view as to how you would best solve this problem if you have time.

Hi woohoo921
Thanks for your post.


I will share with you a methodical approach to solve this question. For this, I will use some basic inferences (all inferred from the information given in the question) and the Venn Diagram representation of the given data.


So, following the basic principles of solving any Data Sufficiency question, let’s start with the question stem analysis.


QUESTION STEM ANALYSIS:
The table shows the number of people who responded “yes” or “no” or “don't know” when asked whether their city council should implement environmental programs X and Y. If a total of 1,000 people responded to the question about both programs, what was the number of people who did not respond “yes” to implementing either of the two programs?

Let’s first understand what is given here.

A total of 1000 people responded to some survey about two programs – X and Y. And all of them responded for each program in one of three ways - “yes”, “no” or “don’t know”. Now, the question wants us to find the number of people who DID NOT respond ‘yes’ to either of the two programs.

That is, we want the number of people who did NOT respond ‘yes’ to program X and did NOT respond “yes” to program Y. To understand this whole thing, let’s first understand what it means for a person who does NOT respond “yes” to a program.

• For program X, those who do not respond “Yes” implies they responded either “no” or “don’t know” – this is everything except “yes”. So, if we want the number of people not responding “yes”, we can find them by subtracting those who do respond “yes” from the total of 1000.
  
  So, # people not responding Yes = 1000 - # of people responding “yes”
  
• Similarly, for program Y, those who do not respond “yes” can be found as 1000 - # of people responding “yes”.

Overall, we can say that to find the number of people who DID NOT respond ‘yes’ to either of the two programs, we MUST find those who said “Yes” to either of the two programs. And then we’ll just subtract that number from 1000.

So, our required number = {1000 - # of people who responded “yes” to either X or Y}.

NOTE: Now, there are chances of some people being common - those who said “yes” to both programs. That is why the number of people who said “yes” to either X or Y is NOT just the sum of those who said “yes” to X and those who said “yes” to Y. The correct formula for #people who responded “yes” to either X or Y is below:

(#Responded “yes” to X) UNION (#Responded “yes” to Y).

Let’s now see what the statements give us, starting with the analysis of statement 1.


STATEMENT 1 ANALYSIS:
The number of people who responded “yes” to implementing only Program X was 300.

So, this statement tells us about those who responded “yes” to only program X. For using this effectively, let’s try to make the Venn Diagram for those who responded “yes” to either program.


Here, ‘a’ and ‘c’ represent those who responded “yes” to ONLY program X and to ONLY program Y, respectively, and ‘b’ represents those who responded “yes” to both programs.

Now, statement 1 gives us the value for ‘a’ as 300.

• This implies b = 400 – a = 400 – 300 = 100
• And since we now know the value of b, the value of c = 300 – b = 200

So, the number of people who said “yes” to either of the two programs = a + b + c = 300 + 100 + 200 = 600.

And that is exactly what is needed. So, we can say the required number = 1000 - (a + b + c) = 1000 – 600 = 400.

Hence, a UNIQUE answer, which makes statement 1 alone SUFFICIENT.


STATEMENT 2 ANALYSIS:
The number of people who responded “no” to implementing Program X and “no” to implementing Program Y was 100.

This statement tells us about the number of people who responded “NO” to both programs.

Note: Without any analysis, we can see that this statement gives us nothing about the people who respond “yes” to one or both of X and Y. So, it looks like this statement will not be of much use. Still, we will do the same analysis as we did in statement 1 and be sure of our answer.



Per statement 2, q = 100. These are the people who responded “no” to both programs.

• Now, p + q = 200, and q = 100. This implies p = 100.
• Similarly, q + r = 350 and q = 100. This implies r = 250.
• Combining the two inferences above, we get:
  
  • #people who responded “no” to either of the two programs = p + q + r = 450.

But even after this analysis, does this provide us with any information about the number of people who responded “yes” to either of the two programs? NO!

That means the information in statement 2 alone is INSUFFICIENT to provide a unique answer to the main question asked.


So, our correct answer choice is option A.


Hope this helps!


Best,
Aditi Gupta
Quant expert, e-GMAT

­Think of Sets here.

Number of people who said yes to X = 400
Number of people who said yes to Y = 300

Question: How many people did not say yes to either of the 2 programs?

(1) The number of people who responded “yes” to implementing only Program X was 300.

So of 400 who said Yes to program X, 300 said yes to only X. This means 100 people said yes to Y too. 
Hence 100 people said yes to both X and Y.

X or Y = X + Y - Both = 400 + 300 - 100 = 600

So 600 people said yes to either X or Y or both. So 400 people did not say yes to either of the programs.

Sufficient alone.

(2) The number of people who responded “no” to implementing Program X and “no” to implementing Program Y was 100.

Using the same concept as above, we can find how many people did not say NO to either of the 2 programs but we cannot say how many people did not say yes to either of the 2 programs. 

Not sufficient alone.

Answer (A)

Check a similar, interesting question here: 
https://youtu.be/1RZdJTKCDYs