Very Tough Venn Diagram Ques....Experts Please Help !!!

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A Statement 1 enough
B Statement 2 enough
C Both Statements required
D Each Statement enough
E Both Statements not enough

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by MartyMurray » Fri Jan 01, 2016 11:00 pm
A car manufacturer organized a test drive of two of its prototypes. Each driver tested both prototypes and was asked whether he or she liked each prototype. If a driver did not like a prototype, he or she was asked whether its performance was the main reason. A driver did not like a prototype because of its performance 180 times, whereas only half as often a driver did not like a prototype because of some other major reason.

How many test drivers were there?

(1) 120 people liked both prototypes.
(2) As many people liked neither prototype as liked both of them.
We know that 180 + 180/2 times a driver did not like a prototype.

Statement 1 tells us how many liked both, but we can't tell how many liked just one of them, and so we can't tell how many drivers there were.

Insufficient.

Statement 2 does not give us any numbers to work with beyond what we already have. So while from the question we know how many times drivers didn't like cars, that's all we know.

Insufficient.

From the question we know that 180 + 180/2 = 270 times drivers did not like cars. That gives us 270 Not Likes.

From Statement 1 we know that 120 drivers liked both cars. That gives us 240 Likes.

From Statement 2 and Statement 1 we know that 120 drivers liked neither car. That takes care of 240 Not Likes.

270 total Not Likes - 240 Not Likes by drivers who liked neither = 30 Not Likes by drivers who liked one car and not the other.

So we must also have 30 Likes by drivers who liked one and not the other.

[240 likes(by drivers who liked both) + 270 not likes + 30 likes(by drivers who liked one)]/2 cars = 540/2 = 270 drivers.

So combined the statements are sufficient.

The correct answer is C.
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A car manufacturer organized a test drive of two of its prototypes. Each driver tested both prototypes and was asked if he or she liked the prototype. If a driver did not like a prototype, he or she was asked if the prototype's performance was the main reason. A driver did not like a prototype because of its performance 180 times, whereas only half as often a driver did not like a prototype because of some other major reason. How many test-drivers were there?

(1) 120 people liked both prototypes.
(2) As many people liked neither prototype as liked both of them.
A driver did not like a prototype because of its performance 180 times, whereas only half as often a driver did not like a prototype because of some other major reason.
Implication:
Total number of negative reviews = 180 + (1/2)(180) = 270

Let the two prototypes be A and B.
Total number of people = (people who like A but not B) + (people who like B but not A) + (people who like both A and B) + (people who like neither A nor B)

Statement 1: 120 people liked both prototypes.
Case 1: 120 like both A and B, 120 people like neither A nor B
Resulting equation:
Total number of people = (people who like A but not B) + (people who like B but not A) + 120 + 120

Since the 120 people who like neither A nor B give a negative review to A and a negative review to B -- for a total of 240 negative reviews -- 30 people must give a negative review only to A or only to B, bringing the total number of negative reviews to 270.
If 10 people like A but not B and 20 people like B but not A, we get:
Total number of people = 10 + 20 + 120 + 120 = 270

Case 2: 120 like both A and B, 100 people like neither A nor B
Resulting equation:
Total number of people = (people who like A but not B) + (people who like B but not A) + 120 + 100

Since the 100 people who like neither A nor B give a negative review to A and a negative review to B -- for a total of 200 negative reviews -- 70 people must give a negative review only to A or only to B, bringing the total number of negative reviews to 270.
If 20 people like A but not B and 50 people like B but not A, we get:
Total number of people = 20 + 50 + 120 + 100 = 290

Since the total number of people can be different values, INSUFFICIENT.

Statement 2: As many people liked neither prototype as liked both of them.
Case 1 also satisfies Statement 2.
In Case 1, the total number of people = 270

Case 3: 100 people like both A and B, 100 people like neither A nor B
Resulting equation:
Total number of people = (people who like A but not B) + (people who like B but not A) + 100 + 100

Since the 100 people who like neither A nor B give a negative review to A and a negative review to B -- for a total of 200 negative reviews -- 70 people must give a negative review only to A or only to B, bringing the total number of negative reviews to 270.
If 20 people like A but not B and 50 people like B but not A, we get:
Total number of people = 20 + 50 + 100 + 100 = 270

Since the total number of people in both Case 1 and Case 3 is THE SAME -- 270 -- Statement 2 is SUFFICIENT.

B
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Below is the algebra behind Statement 2.

Let:
T = the total number of people
A = people who like A but not B
B = people who like B but not A
AB = people who like both A and B
N = people who like neither A nor B
Resulting equation:
T = A + B + AB + N

Everyone in the A group gives exactly 1 negative review (against B).
Everyone in the B group gives exactly 1 negative review (against A).
Everyone in the N group gives 2 negative reviews (one against A, the other against B).
Since the prompt indicates a total of 270 negative reviews, we get:
A + B + 2N = 270

Statement 2: As many people liked neither prototype as liked both of them.
Substituting AB = N into T = A + B + AB + N, we get:
T = A + B + N + N
T = A + B + 2N

Since A + B + 2N = 270, there are 270 people.
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