When the focus group of 24 people are asked to consider two possible a

Question

When the focus group of 24 people are asked to consider two possible ads, they are also asked about the talent used in those ads. One ad features an on-camera celebrity endorser, but the celebrity may not be recognizable to all demographics. The other ad is an animated treatment with a voice provided by a comedian whose voice may or may not be recognized by viewers. Members of the focus group are asked if they recognize the celebrity endorser and if they recognize the voice in the animation. A total of 16 group members said they recognized the celebrity in the live ad, and 12 said they recognized the voice in the animation. Find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.

Bismuth83 · Accepted Answer

1. We are given two characteristics that people either recognized or not. The question asks us to  find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.

2. Let's put the information in the text into a table:
    
 Recognized celebrity 
 Didn't recognize celebrity 
 Sum 
 
  Recognized voice 
 x 
 ? 
 12 
 
  Didn't recognize voice 
 ? 
 y 
 ? 
 
  Sum 
 16 
 ? 
 Total = 24

3. The question can be turned into finding the minimum of x and the maximum of y. Now, let's fill in some blanks:
    
 Recognized celebrity 
 Didn't recognize celebrity 
 Sum 
 
  Recognized voice 
 x 
 12 - x 
 12 
 
  Didn't recognize voice 
 16 - x 
 y 
 ? 
 
  Sum 
 16 
 ? 
 Total = 24

4. Since the total is 24,  x + (16 - x) + (12 - x) + y = 24 \rightarrow y = x - 4 .

5. All the values must be positive, so,  4 \leq x \leq 12 . The minimum of x is 4 and the maximum of y is when x is at its maximum, which is 12. That means y = 12 - 4 = 8.

6. Our answer will be:   the minimum number of respondents who recognized both - 4; the maximum possible number who recognized neither - 8  .

poojaarora1818 · Answer

Could you please provide the solution? I have tried but have been unable to find the correct solution. To my best understanding, it's a problem on overlapping sets where we need to minimize one value to maximize the other value.

SatvikVedala · Answer

Hi  poojaarora1818

Following is my explanation for the problem

24 (Focus group) = 16 (Group1) + 12 (Group2) - x (Both) + N (neither)

24 = 28-x+N

x-N = 4

For N to be max x-4 needs to be max. Also we know that maximum value x can take is 12 (i.e. min(16,12))

12-4 = Nmax

Therefore, Nmax = 8

For x to be min N needs to be min. The minimum value N can take is 0

xmin = 4+Nmin => xmin = 4

Therefore xmin = 4 & Nmax = 8

poojaarora1818 · Answer

Thank you so much for the detailed solution. Now, it brings more clarity to me and makes sense.

Raeesa · Answer

The easiest and the quickest way I found to do this:

Total # of people= 24
# that recognise celebritiy= 16 
# that doesn't recognise celibrities voice= 8

# that recognises Comedian's voice= 12
# that does not recognise comedian's voice= 12

Minimum number who recognised both:

This is basically asking us minimum number of overlapping members. 
So imagine there are 24 chairs (total people) and two groups, group A and group B. Group A has 16 people (recognising celebrities) and group B has 12 people, total of 28. 
One person on one chair would use 24 members, and the remaining 4 would have to sit on the laps of the ones who are already sitting (overlap). Others might choose to sit on laps as well if they want, but a minimum of 4 HAVE to sit.

Therefore, ans is 4.

The maximum possible number who recognised neither:

# that doesn't recognise the celebrity's voice= 8
# that does not recognise comedian's voice= 12

The maximum number will come from the maximum number that can sit over other groups' laps. 
Comedians are more than celebrities 
So comedians take the chairs and the celebrities sit on top of them 
The maximum number that can do that is the total # that don't recognise celibrities voice= 6

So the ans is 8.

pappal · Answer

Recognize actor in live add L=16
recognize voice of comedian C=12
1. to find the minimum no. surveyed who recognized the both, we should assume N of none(who didn't recognize both)=0
so total =L+C - minimum who recognized both =>24=16+12 - minimum both or minimum both =4
2. to find the maximum who recognized none we must consider that all of the 12 who recognized voice of the comedian also recognized the actor in the live add(maximize both category), which means that 04 belong to the category which  only  recognized the actor in the live add.
hence N of none (who didn't recognize both)=24-16=8

When the focus group of 24 people are asked to consider two possible a

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Minimum number who recognized both	Maximum possible number who recognized neither
		0
		2
		4
		6
		8
		10

	Recognized celebrity	Didn't recognize celebrity	Sum
Recognized voice	x	?	12
Didn't recognize voice	?	y	?
Sum	16	?	Total = 24

GMAT Two-part Analysis (TPA) Questions

When the focus group of 24 people are asked to consider two possible a

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