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In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000

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Director
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In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000  [#permalink]

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New post 17 Jul 2019, 04:51
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A
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C
D
E

Difficulty:

  55% (hard)

Question Stats:

59% (02:28) correct 41% (01:40) wrong based on 29 sessions

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In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000 people visited the multiplex and watched at least one of the 3 shows. If 420 people watched A, 250 watched B, 450 watched C, what is the maximum number of people who watched all of the 3 shows?

A. 0
B. 50
C. 60
D. 100
E. 250
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Re: In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000  [#permalink]

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New post 18 Jul 2019, 04:18
1
This is a very good question on Venn diagrams, wherein some regions of the diagram have to be maximized.
In questions on maximization in Venn diagrams, remember that it is not about taking the biggest value all the time; it is about taking the biggest possible value.
Also remember that investing time on getting the Venn diagram right is equivalent to half the job done.

So, let us now draw the Venn diagram first:

Attachment:
18th July - Reply 4 - 1.JPG
18th July - Reply 4 - 1.JPG [ 46.79 KiB | Viewed 240 times ]



Since the question mentions that all of the 1000 people watched at least one of the shows, n = 0.
To find the maximum number of people watching all the 3 shows, we need to maximize the value of g.

As we see in the diagram, the maximum value of g = 60. A smart way of checking if this is indeed the biggest POSSIBLE value, is by plugging it back in the Venn diagram and seeing if it satisfies all conditions.

Substituting the value of g=60, we get a Venn diagram that looks like this:

Attachment:
18th July - Reply 4 - 2.JPG
18th July - Reply 4 - 2.JPG [ 25.97 KiB | Viewed 239 times ]


So, 60 is the maximum value of g.
The correct answer option is C.

Hope this helps!
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Re: In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000  [#permalink]

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New post 18 Jul 2019, 05:54
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For sets consists of three elements we know the formulae

Total - None = A + B + C - In Exactly two - 2(In Exactly 3)

as we have given in stem that 1000 people visited the multiplex and watched at least one of the 3 shows so None part in above formula will be Zero

1000 - 0 = 420 + 250 + 450 - In Exactly two - 2(In Exactly 3)
120 = Exactly two + 2(In Exactly 3)
to maximize exactly three we have to take exactly two as Zero
120 = 2(In Exactly 3)
60 = Exactly 3

Option C is the correct answer
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Re: In a multiplex 3 shows A, B, and C run in 3 auditoriums. One day, 1000   [#permalink] 18 Jul 2019, 05:54
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