Author 
Message 
TAGS:

Hide Tags

Retired Moderator
Joined: 27 Oct 2017
Posts: 1272
Location: India
Concentration: International Business, General Management
GPA: 3.64
WE: Business Development (Energy and Utilities)

In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
11 May 2018, 01:09
Question Stats:
25% (02:51) correct 75% (03:13) wrong based on 125 sessions
HideShow timer Statistics
In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky Sports, and 90 watch ABC. 50 students watch both Fox News and Sky Sports, 60 students watch both Sky sports and ABC and 65 students watch both Fox News and ABC. What is the minimum number of students who watch atleast one of the given channels? A.35 B.50 C.135 D.150 E.165
Official Answer and Stats are available only to registered users. Register/ Login.
_________________




Retired Moderator
Joined: 27 Oct 2017
Posts: 1272
Location: India
Concentration: International Business, General Management
GPA: 3.64
WE: Business Development (Energy and Utilities)

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
12 May 2018, 05:25
One formula for 3 overlapping sets is as follows: Total = Group 1 + Group 2 + Group 3  (at least 2 of the groups) + (all 3 groups) + none. In the problem above: Total = 250. Group 1 = 120. Group 2 = 80. Group 3 = 90. (at least 2 of the groups) = 50+60+65 = 175. Let x = all 3. Let N = none. Plugging these values into the equation above, we get: 250 = 120 + 80 + 90  175 + x + N. 250 = 115 + x + N N = 250  115  x. N = 135  x. To MINIMIZE the number who watch at least 1 channel, we must MAXIMIZE the value of N: the number who watch NONE of the channels. In the equation above, the value of N will be maximized if x (all 3) is as small as possible. Of the given overlaps  50, 60, 65  the two greatest overlaps both include A: (at least S and A) + (at least F and A) = 60+65 = 125. Now this value ((at least S and A) + (at least F and A)) is 35 greater than the total number who watch A (90). Implication: At least 35 students must watch all 3 channels. Thus, the minimum value of x = 35. Plugging x=35 into N = 135  x, we get: N = 135  35 = 100. Since the maximum number who watch none of the channels = 100, the minimum number who watch at least 1 channel = 250100 = 150. Krotishka1 wrote: pushpitkc wrote: In order to have the minimum number of students who watch at least one of the given channels, we must have a maximum Minimum number of people watching all 3 channels.
Thanks for the solution! Could you please explain the logic behind this statement?
_________________




Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3308
Location: India
GPA: 3.12

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
11 May 2018, 05:00
gmatbusters wrote: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky Sports, and 90 watch ABC. 50 students watch both Fox News and Sky Sports, 60 students watch both Sky sports and ABC and 65 students watch both Fox News and ABC. What is the minimum number of students who watch atleast one of the given channels? A.35 B.50 C.135 D.150 E.165 In order to have the minimum number of students who watch at least one of the given channels, we must have a maximum number of people watching all 3 channels. Using Venn diagram, we can solve this problem as follows Attachment:
Play.png [ 6.96 KiB  Viewed 3322 times ]
Here, P(Watching all 3 channels) = 25 But P(Watching ABC) = 40+25+35 > 90. This scenario is not possible! Attachment:
Play1.png [ 8.29 KiB  Viewed 3320 times ]
Here, P(Watching all 3 channels) = 30 P(Watching ABC) = 35+30+30 > 90. Again, this scenario is not possible! Attachment:
Play2.png [ 7.18 KiB  Viewed 3317 times ]
Here, P(Watching all 3 channels) = 35 P(Watching ABC) = 30+35+25 = 90. P(Watching SkySports) = 15+35+25+5 = 80 P(Only Watching SkySports) = 5 P(Watching Fox) = 15+35+30+40 = 120 P(Only Watching Fox) = 40 Therefore, the minimum number of students who watch at least one of the channels is 40+5+30+35+25+15 = 150(Option D)
_________________
You've got what it takes, but it will take everything you've got



Intern
Joined: 02 May 2018
Posts: 13

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
12 May 2018, 04:49
pushpitkc wrote: In order to have the minimum number of students who watch at least one of the given channels, we must have a maximum number of people watching all 3 channels.
Thanks for the solution! Could you please explain the logic behind this statement?



Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3308
Location: India
GPA: 3.12

In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
12 May 2018, 05:37
Krotishka1 wrote: pushpitkc wrote: In order to have the minimum number of students who watch at least one of the given channels, we must have a maximum number of people watching all 3 channels.
Thanks for the solution! Could you please explain the logic behind this statement? Hey Krotishka1The logic is simple: If 1 student watches all 3 channels, it reduces 1 student from each of the individual channel's students count, so there is a net loss of 2 students from the overall number of students watching the channels. The more the number of students watching all 3 channels the minimum the number of students watching TV. I'll try and explain by means of an example. With zero students watching all 3 channels Attachment:
Play1.png [ 8.73 KiB  Viewed 3173 times ]
Here, the total number of students watching TV is 60 + 30 + 20 + 30 + 30 + 30 = 200 With 20 students watching all 3 channels Attachment:
Play.png [ 8.62 KiB  Viewed 3176 times ]
Here, the total number of students watching TV is 40 + 10 + 0 + 30 + 30 + 30 + 20 = 160 If you observe because of the inclusion of 20 students watching all the 3 channels, the number of students watching TV reduces by 40(2*20) Hope this helps you!
_________________
You've got what it takes, but it will take everything you've got



Retired Moderator
Joined: 27 Oct 2017
Posts: 1272
Location: India
Concentration: International Business, General Management
GPA: 3.64
WE: Business Development (Energy and Utilities)

In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
12 May 2018, 06:51
Hi I understand your logic but the examples you gave don't satisfy the conditions of intersection of 2 of them. n (Fox and Sky sport) =50 n( Sky Sport and ABC) =60 n( ABC and Fox) =65. Also please point out the flaw in my explanation. As per my approach "None of them = 135  n(all of them). " hence, to get min value of union of three, we should have maximum value of None of them which asks for min value of n(all of them). pushpitkc wrote: Krotishka1 wrote: pushpitkc wrote: In order to have the minimum number of students who watch at least one of the given channels, we must have a maximum number of people watching all 3 channels.
Thanks for the solution! Could you please explain the logic behind this statement? Hey Krotishka1The logic is simple: If 1 student watches all 3 channels, it reduces 1 student from each of the individual channel's students count, so there is a net loss of 2 students from the overall number of students watching the channels. The more the number of students watching all 3 channels the minimum the number of students watching TV. I'll try and explain by means of an example. With zero students watching all 3 channels Attachment: Play1.png Here, the total number of students watching TV is 60 + 30 + 20 + 30 + 30 + 30 = 200 With 20 students watching all 3 channels Attachment: Play.png Here, the total number of students watching TV is 40 + 10 + 0 + 30 + 30 + 30 + 20 = 160 If you observe because of the inclusion of 20 students watching all the 3 channels, the number of students watching TV reduces by 40(2*20) Hope this helps you!
_________________



Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8426
Location: United States (CA)

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
14 May 2018, 16:42
gmatbusters wrote: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky Sports, and 90 watch ABC. 50 students watch both Fox News and Sky Sports, 60 students watch both Sky sports and ABC and 65 students watch both Fox News and ABC. What is the minimum number of students who watch atleast one of the given channels? A.35 B.50 C.135 D.150 E.165 To minimize the number of students who watch at least one channel, we want to maximize the number of the students who watch none of these channels. We can create the equation: 250 = 120 + 80 + 90  (50 + 60 + 65) + triple + none 250 = 290  175 + t + n 250 = 115 + t + n 135  t = n Since we want to maximize the value of n, we need to minimize the value of t. Let’s let a = the number of students who watch ABC only. Recall that t is the number of students who watch all 3 channels. Thus we have 65  t students watching ABC and Fox (but not Sky) and 60  t students watching ABC and Sky (but not Fox) and we can create the equation for all the students who watch ABC: (65  t) + (60  t) + t + a = 90 125  t + a = 90 35 = t  a Since we want to minimize the value of t, so a must be 0. Thus t = 35 Therefore, n = 135  t = 135  35 = 100. Since there could be a maximum 100 students who watch none of the 3 channels, there must be a minimum of 250  100 = 150 who watch at least one channel. Answer: D
_________________
5star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button.



Intern
Joined: 03 Jul 2015
Posts: 11

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
19 May 2018, 01:21
ScottTargetTestPrep wrote: gmatbusters wrote: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky Sports, and 90 watch ABC. 50 students watch both Fox News and Sky Sports, 60 students watch both Sky sports and ABC and 65 students watch both Fox News and ABC. What is the minimum number of students who watch atleast one of the given channels? A.35 B.50 C.135 D.150 E.165 To minimize the number of students who watch at least one channel, we want to maximize the number of the students who watch none of these channels. We can create the equation: 250 = 120 + 80 + 90  (50 + 60 + 65) + triple + none 250 = 290  175 + t + n 250 = 115 + t + n 135  t = n Since we want to maximize the value of n, we need to minimize the value of t. Let’s let a = the number of students who watch ABC only. Recall that t is the number of students who watch all 3 channels. Thus we have 65  t students watching ABC and Fox (but not Sky) and 60  t students watching ABC and Sky (but not Fox) and we can create the equation for all the students who watch ABC: (65  t) + (60  t) + t + a = 90 125  t + a = 90 35 = t  a Since we want to minimize the value of t, so a must be 0. Thus t = 35 Therefore, n = 135  t = 135  35 = 100. Since there could be a maximum 100 students who watch none of the 3 channels, there must be a minimum of 250  100 = 150 who watch at least one channel. Answer: D Hello Scott / All  just to confirm that we can not set N equal to zero because that implies there are 135 students who watch 3 movies and this is impossible. Correct ?



Retired Moderator
Joined: 27 Oct 2017
Posts: 1272
Location: India
Concentration: International Business, General Management
GPA: 3.64
WE: Business Development (Energy and Utilities)

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
19 May 2018, 03:21
You are right, n can not be zero in this equation. Moreover , since we need to find the minimum number of students who watch atleast one of the given channels. We are looking for the maximum value of n , So that the number of students who watch atleast one of the given channels (Union of all news watchers) is minimum. tixan wrote: ScottTargetTestPrep wrote: gmatbusters wrote: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky Sports, and 90 watch ABC. 50 students watch both Fox News and Sky Sports, 60 students watch both Sky sports and ABC and 65 students watch both Fox News and ABC. What is the minimum number of students who watch atleast one of the given channels? A.35 B.50 C.135 D.150 E.165 To minimize the number of students who watch at least one channel, we want to maximize the number of the students who watch none of these channels. We can create the equation: 250 = 120 + 80 + 90  (50 + 60 + 65) + triple + none 250 = 290  175 + t + n 250 = 115 + t + n 135  t = n Since we want to maximize the value of n, we need to minimize the value of t. Let’s let a = the number of students who watch ABC only. Recall that t is the number of students who watch all 3 channels. Thus we have 65  t students watching ABC and Fox (but not Sky) and 60  t students watching ABC and Sky (but not Fox) and we can create the equation for all the students who watch ABC: (65  t) + (60  t) + t + a = 90 125  t + a = 90 35 = t  a Since we want to minimize the value of t, so a must be 0. Thus t = 35 Therefore, n = 135  t = 135  35 = 100. Since there could be a maximum 100 students who watch none of the 3 channels, there must be a minimum of 250  100 = 150 who watch at least one channel. Answer: D Hello Scott / All  just to confirm that we can not set N equal to zero because that implies there are 135 students who watch 3 movies and this is impossible. Correct ?
_________________



Intern
Joined: 03 Jul 2015
Posts: 11

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
20 May 2018, 05:59
Thanks gmatbusters!



Senior Manager
Joined: 22 Feb 2018
Posts: 414

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
24 May 2018, 11:10
OA: D As gmatbusters has already explained Quote: One formula for 3 overlapping sets is as follows:
Total = Group 1 + Group 2 + Group 3  (at least 2 of the groups) + (all 3 groups) + none.
In the problem above: Total = 250. Group 1 = 120. Group 2 = 80. Group 3 = 90. (at least 2 of the groups) = 50+60+65 = 175. Let x = all 3. Let N = none.
Plugging these values into the equation above, we get: 250 = 120 + 80 + 90  175 + x + N. 250 = 115 + x + N N = 250  115  x. N = 135  x.
To MINIMIZE the number who watch at least 1 channel, we must MAXIMIZE the value of N: the number who watch NONE of the channels. In the equation above, the value of N will be maximized if x (all 3) is as small as possible. Attachment:
set.PNG [ 37.81 KiB  Viewed 2643 times ]
To maximixe N , we have to make value of \(x\) as small as possible. lets take \(x=0\), then Number of student who watch Sky Sports alone and Number of student who watch ABC alone become \(30\) and \(35\) respectively, which is not possible. Now lets take \(x=30\),then Number of student who watch Sky Sports alone and Number of student who watch ABC alone become \(0\) and \(5\) respectively, which is also not possible. Finally lets take \(x=35\),then Number of student who watch Sky Sports alone and Number of student who watch ABC alone become \(5\) and \(0\) respectively, all other terms in venn diagram are also non negative, so \(35\) is minimum value of \(x\) that can be possible. Using \(x = 35\), we get \(None(N)=13535=100\) Minimum No of student,who watch atleast one tv channel \(= 250 N=250100 =150\)
_________________
Good, good Let the kudos flow through you



NonHuman User
Joined: 09 Sep 2013
Posts: 13612

Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
Show Tags
11 Aug 2019, 08:28
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________




Re: In a hostel there are 250 students, 120 watch Fox News, 80 watch Sky S
[#permalink]
11 Aug 2019, 08:28






