Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 60687

In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 02:40
Question Stats:
61% (02:00) correct 39% (01:56) wrong based on 56 sessions
HideShow timer Statistics
Competition Mode Question In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000 visited the multiplex and watched at least one of 3 shows. If 420 people watched A, 250 watched B, 450 watched C, what is the maximum number of people who watched all the 3 shows ? (A) 0 (B) 50 (C) 60 (D) 100 (E) 250 Are You Up For the Challenge: 700 Level Questions
Official Answer and Stats are available only to registered users. Register/ Login.
_________________



CrackVerbal Quant Expert
Joined: 12 Apr 2019
Posts: 365

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 05:00
This is a fairly straightforward question on maximization of regions of a Venn diagram. In all maximization and minimization questions, remember that it’s not about taking the largest or the smallest value. It’s about taking the largest of the smallest value that does not make any region of the Venn diagram negative. Let us go ahead and draw the Venn diagram now: Attachment:
22nd Nov 2019  Reply 4.jpg [ 50.16 KiB  Viewed 730 times ]
Since 1000 people visited the multiplex, the rectangle has 1000 people in total. The question also says that everyone watched at least one of the shows. This means everyone who visited the multiplex will be inside one of the circles. The area inside the rectangle but outside the circles will be ZERO. In order to maximize the number of people who watched all 3, we will have to minimize the other regions. The minimum possible value of any region in a Venn diagram is 0. From the diagram, we see that the sum of the sets = 420 + 250 + 450 = 1120. But, the sum of the sets contains the regions a, b and c twice and the region x thrice. If we remove a, b and c once and x twice from 1120, we get 1000 since this is the actual number of people who visited the multiplex (without repetition of any region). This means, 1120 –(a+b+c+2x) = 1000 which, on simplification, gives us, a + b + c + 2x = 120. To maximize x, we take a+b+c = 0 and so, 2x = 120 or x = 60. The correct answer option is C. Hope this helps!
_________________



GMAT Club Legend
Joined: 18 Aug 2017
Posts: 5738
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 03:24
1000  0 = 420 + 250 + 450  In Exactly two  2(all 3) 120 = Exactly two + 2(all 3) exactly three we have to take exactly two as Zero 120 = 2(all 3) 60 = all 3 IMO C
In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000 visited the multiplex and watched at least one of 3 shows. If 420 people watched A, 250 watched B, 450 watched C, what is the maximum number of people who watched all the 3 shows ?
(A) 0 (B) 50 (C) 60 (D) 100 (E) 250



VP
Joined: 20 Jul 2017
Posts: 1265
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
Updated on: 23 Nov 2019, 04:40
Let the number of people who watched all the 3 shows = X and those who watched exactly 2 shows = Y
Formula: Total  Neither = A + B + C  (Exactly 2 categories)  2(Exactly 3 categories)
—> 1000  0 = 450 + 250 + 420  Y  2X —> 2X = 1120  1000  Y —> 2X = 120  Y —> X = 60  Y/2
For X to be maximum, Y has to be minimum = 0
—> Maximum value if X = 60  0 = 60
IMO Option C
Originally posted by Dillesh4096 on 22 Nov 2019, 04:11.
Last edited by Dillesh4096 on 23 Nov 2019, 04:40, edited 2 times in total.



VP
Joined: 24 Nov 2016
Posts: 1124
Location: United States

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 04:27
Quote: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000 visited the multiplex and watched at least one of 3 shows. If 420 people watched A, 250 watched B, 450 watched C, what is the maximum number of people who watched all the 3 shows ?
(A) 0 (B) 50 (C) 60 (D) 100 (E) 250 T=A+B+Cboth2(mid)+neither 1000=420+250+450both2mid+0 120=both2mid 2mid=120both mid=60both/2 both=0; mid=60 Ans (C)



Senior Manager
Joined: 01 Mar 2019
Posts: 406
Location: India
Concentration: Strategy, Social Entrepreneurship
GPA: 4

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 06:04
From image: A=a+b+d+e=420 B=c+b+e+f=250 C=d+e+f+g=450 Total=a+b+c+d+e+f+g=1000=a+b+d+e+c+b+e+f+d+e+f+g  (d+b+f)  2e 1000 = 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) Exactly 3=60 OA:C
Attachments
Capture.PNG [ 8.74 KiB  Viewed 837 times ]



Senior Manager
Joined: 25 Jul 2018
Posts: 486

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
22 Nov 2019, 14:10
Total= 1000 A= 420. B= 250. C= 450 Neither —0(zero)
The maximum number of people—Z(3 shows) —???
A= a+ a1+b1 +Z= 420 B= b+b1+ c1+Z= 250 C= c+c1+ a1+ Z= 450 —————————— a+b+c+a1+b1+c1+Z= 1000 a+b+c+ 2(a1+b1+c1)+ 3Z= 1120 —> (a1+b1+c1)+ 2Z= 120
—> if (a1+b1+c1)= 0, then 2Z= 120 —> Z= 60
The answer is C
Posted from my mobile device



Director
Joined: 18 May 2019
Posts: 665

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
23 Nov 2019, 06:36
We are given that a total of 1000 people patronized at least one of three shows, A, B, and C. This implies none =0. We are also given that the number that attended show A=420, the number that attended show B=250, and the number that attended show C=450.
Let m, n, and z be the number who attend both A and B only, both A and C only, and both B and C only respectively. Likewise let a, b, and c be the number who attend only show A, only B, and only C respectively. Let x be the number who attend all three shows. a+b+c+m+n+z+x=1000 but a=420mnx, b=250mzx, and c=450nzx so, (420mnx)+(250mzx)+(450nzx)+m+n+z+x=1000 1120(m+n+z)2x=1000 120(m+n+z)=2x hence x=60(m+n+z)/2 For maximum x, then (m+n+z)=0 Therefore the maximum possible people who attended all three shows is 60.
The answer is therefore option C.



Intern
Joined: 26 Apr 2019
Posts: 42
Location: India
Concentration: Finance, Leadership
GMAT 1: 690 Q49 V34 GMAT 2: 700 Q49 V36 GMAT 3: 720 Q50 V37
GPA: 3.99

Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
Show Tags
23 Nov 2019, 06:53
answer should be d, we get (450+420+250)=1120 , now get maximum ppl who watched all show substitute value from given answer . it gets solved in 30 sec max .




Re: In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000
[#permalink]
23 Nov 2019, 06:53






