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# Of the 600 residents of Clermontville, 35% watch the

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Intern
Joined: 14 Jun 2005
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Of the 600 residents of Clermontville, 35% watch the [#permalink]  09 Oct 2005, 19:56
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Question Stats:

82% (03:13) correct 18% (01:43) wrong based on 199 sessions
Of the 600 residents of Clermontville, 35% watch the television show Island Survival,
40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents
watch at least one of these three shows and 18% watch exactly 2 of these shows, then
how many Clermontville residents watch all of the shows?

(A) 150
(B) 108
(C) 42
(D) 21
(E) -21
Manager
Joined: 15 Jul 2005
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Kudos [?]: 5 [0], given: 0

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I get (C) 42

of the 600 residents, 210, 240 and 300 watch Island Survival,
Lovelost Lawyers and Medical Emergency respectively. this adds up to 750. we also know that 108 (18%) watch 2 shows, which leaves 642. therefore 42 watch all three shows.

Whats the OA?
Intern
Joined: 02 Oct 2005
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Hi,
I get 42 and here is my method

600 = ((35%*600+50%*600+40%*600) - (18%*600) - sum of people watching all 3

solving we get 42. Do let me know if this is the right answer.
SVP
Joined: 24 Sep 2005
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Re: Set theory [#permalink]  09 Oct 2005, 20:15
tingle wrote:
Of the 600 residents of Clermontville, 35% watch the television show Island Survival,
40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents
watch at least one of these three shows and 18% watch exactly 2 of these shows, then
how many Clermontville residents watch all of the shows?

(A) 150
(B) 108
(C) 42
(D) 21
(E) -21

Why am I getting E

210 watch IS
300 watch ME
240 watch LL
108 watch any two.

The number of viewers watch more than 1 programe is 210+300+240 -600= 150
------> the number of viewers watch all three programe is 150-180=42
C.
Manager
Joined: 06 Oct 2005
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Location: Beantown
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Just modify the formula A + B - both + neither = total:

A + B + C - two shows - three shows = total

35% + 40% + 50% - 18% - x% = 100%

x=7%

600 * .07 = 42

Brendan.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5063
Location: Singapore
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Kudos [?]: 237 [0], given: 0

The rules of a venn diagram is if you add each individual sections, you get the total, which is 600 in this case.

So 210 + 240 + 300 - 108 - X = 600 where x represents people who watch all three.

so x = 42
Senior Manager
Joined: 14 Apr 2005
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Location: India, Chennai
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Re: Set theory [#permalink]  09 Oct 2005, 23:46
It is
240 + 300 + 210 - 108 - X = 600,
Where 240 watch Island Survival
300 watch Lovelost lawyers
210 watch medical emergency
108 watch 2 of those shows
X # who watch all the 3 shows
600 - Total # of residents.
Solving the equn we get x = 42.
Intern
Joined: 14 Jun 2005
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OA is D....

Here's the explanation:-

I+L+M-(IL+LM+IM)-2ILM= 600
210+240+300-(108)-2ILM= 600
ILM = 21

Why are we deducting 2ILM from the total???? shouldnt we be adding it back to the total as we already have deducted ILM thrice while deducting 108 from the equation. Pls explain......
Director
Joined: 09 Jul 2005
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OA is D.

Lets see my reasoning.

For a three set problem we have the following formula:

100= A+B+C-AB-AC-BC+ABC, which is the same as the following formula

100= A+B+C+(-AB-AC-BC+ABC+ABC+ABC)-2ABC.

The term between parantheses value 18% so the equation to resolve is

100=35+40+50-18-2ABC

therefore the value of ABC is 3.5% of 600, is 21. D is the correct answer
Senior Manager
Joined: 11 May 2004
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I got 42.

I drew a venn diagram and split all the of shows. So, I got 210+240+300 = 750. Then reduced this by 18% ie 108 to get 642. Since there are only 600 residents, that must mean than 42 watch all the shows.
Manager
Joined: 03 Oct 2005
Posts: 87
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tingle wrote:
OA is D....

Why are we deducting 2ILM from the total???? shouldnt we be adding it back to the total as we already have deducted ILM thrice while deducting 108 from the equation. Pls explain......

18% is the list of people that watch *exactly 2 shows*.

I still don't get it. Why are we dividing 42 by 2?
Director
Joined: 21 Aug 2005
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By Set theory, for three sets intersecting each other,

Total = N(A) + N(B) + N(C) - (N(A n B) + N(A n C) + N(C n B)) +
N(A n B n C)
Where,
N = Number of -
n = 'Intersection'

We need to find out N(A n B n C)
The sum "N(A n B) + N(A n C) + N(C n B)" includes the value "N(A n B n C)" three times.

Per the question, "N(A n B) + N(A n C) + N(C n B)" - "3 * N(A n B n C)" = 18%
So, "N(A n B) + N(A n C) + N(C n B)" = 18% + "3 * N(A n B n C)"
Therefore,
Total = N(A) + N(B) + N(C) - "18% + [3 * N(A n B n C)]" + N(A n B n C)]"

600 = 210 + 240 + 300 - 108 - [2 *N(A n B n C)]

=> N(A n B n C) = 21

Hence D
Manager
Joined: 10 Oct 2005
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gsr wrote:
By Set theory, for three sets intersecting each other,

Total = N(A) + N(B) + N(C) - (N(A n B) + N(A n C) + N(C n B)) +
N(A n B n C)
Where,
N = Number of -
n = 'Intersection'

We need to find out N(A n B n C)
The sum "N(A n B) + N(A n C) + N(C n B)" includes the value "N(A n B n C)" three times.

Per the question, "N(A n B) + N(A n C) + N(C n B)" - "3 * N(A n B n C)" = 18%
So, "N(A n B) + N(A n C) + N(C n B)" = 18% + "3 * N(A n B n C)"
Therefore,
Total = N(A) + N(B) + N(C) - "18% + [3 * N(A n B n C)]" + N(A n B n C)]"

600 = 210 + 240 + 300 - 108 - [2 *N(A n B n C)]

=> N(A n B n C) = 21

Hence D

Thanks a bunch gsr..... Excellent Explanation!!!!
Manager
Joined: 18 Apr 2007
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I got 21 my first try too, but after so many 42s I thought I was wrong. Here is my reasoning:

IS=210
LL=240
ME=300
w=# shared between IS and LL
y=# shared between IS and ME
z=# shared between LL and ME
x=# shared between all three shows (this is the # we're trying to find)

We have to break the total # of residents into 3 groups:
1) those who only watch 1 show-->IS-x-w-y
LL-x-w-z
ME-x-y-z
2) those who watch only 2 shows-->w+y+z (which we know from the stem is 108)
3)those who watch all three shows-->x

So, add these three groups together and you get your 600 residents.

IS-x-w-y+LL-x-w-z+ME-x-y-z+w+y+x+x=600

Simplify: (IS+LL+ME)+(w+y+z)-2x-2w-2y-2z=600

Simplify: 750+108-2(x+w+y+z)=600
-2(x+w+y+z)=-258
x+(108)=129
x=21

I tried to make my explanation really detailed so that all my steps would be really understandable. How did I do?
Manager
Joined: 11 Nov 2006
Posts: 144
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Kudos [?]: 2 [0], given: 0

I = 35%
L = 40%
E = 50%
neither = 0
exactly 2 shows: IL+LE+IE - 3 * ILE = 18%

-> total = I+L+E-(IL+LE+IE)+ILE+neither
100% = 35%+40%+50%-(18%+3*ILE)+ILE+0%
--> ILE = 3.5%

3.5% of 600 people = 21 people
Senior Manager
Joined: 24 Nov 2006
Posts: 350
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Kudos [?]: 17 [0], given: 0

ItÂ´s D.

Considering that all 600 ppl watch at least one show:

AUBUC = 100% = A + B + C - (AnB + BnC + CnA) + AnBnC = 35 + 40 + 50 - (18 + 3X) - X (X = AnBnC).

Therefore: 100 = 107 - 2X => X = 3.5%. 600 = 21.

D.
CEO
Joined: 21 Jan 2007
Posts: 2756
Location: New York City
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Re: Set theory [#permalink]  27 Nov 2007, 06:50
1
KUDOS
tingle wrote:
Of the 600 residents of Clermontville, 35% watch the television show Island Survival,
40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents
watch at least one of these three shows and 18% watch exactly 2 of these shows, then
how many Clermontville residents watch all of the shows?

(A) 150
(B) 108
(C) 42
(D) 21
(E) -21

Why am I getting E

Triple Set Theory
100 = A + B + C – [AB + AC + BC] – [2*ALL]

100 = 35 + 40 + 50 – 18 – 2x
100 = 125 – 18 – 2x
100 = 107 – 2x
2x = 7
x = 3.5

3.5% of 600 residents = 600*.035 = 21

This method is MUCH faster than Venn Diagram Approach
Manager
Joined: 10 Jul 2009
Posts: 129
Location: Ukraine, Kyiv
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Kudos [?]: 95 [0], given: 60

Re: Set theory [#permalink]  20 Sep 2009, 03:03
used both methods. drew Venn diagram+used formula.

35% from 600=210
40% from 600=240
50% from 600=300
18% from 600=108 (108/3=36. 36 people by 3 groups. watching two of the shows)
x - no of people watching all three

thus

600=210+240+300-(36+x)-(36+x)-(36+x)+x
-42=-2x
x=21

D
_________________

Never, never, never give up

Manager
Joined: 15 Sep 2009
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Re: Set theory [#permalink]  20 Sep 2009, 04:04
After the deductions,I'm getting the ans as 21.

I wud go with option D
Manager
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Re: Set theory [#permalink]  24 Oct 2009, 22:38
1
KUDOS
tingle wrote:
Of the 600 residents of Clermontville, 35% watch the television show Island Survival,
40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents
watch at least one of these three shows and 18% watch exactly 2 of these shows, then
how many Clermontville residents watch all of the shows?

(A) 150
(B) 108
(C) 42
(D) 21
(E) -21

D for me

Just plug the numbers in this equation:

Total = A + B + C - N2 - 2*N3
Re: Set theory   [#permalink] 24 Oct 2009, 22:38

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