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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