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Of the 600 residents of Clermontville, 35% watch the television show

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Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 20:56
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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
[Reveal] Spoiler: OA
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 21:07
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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?
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 21:11
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.
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 21:15
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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 :oops:


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.
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 21:29
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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

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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 09 Oct 2005, 23:49
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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 00:46
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.
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 06:59
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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......
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 07:34
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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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 07:48
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.
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 09:01
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?
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 10 Oct 2005, 20:31
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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 27 Apr 2007, 12:35
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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? :)
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 28 Apr 2007, 07:09
Indeed, the answer is 21:

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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 28 Apr 2007, 16:30
It´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.
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 27 Nov 2007, 07:50
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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 :oops:


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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 20 Sep 2009, 04:03
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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 24 Oct 2009, 23:38
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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
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 15 Jan 2010, 16:11
Prometoh wrote:
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


please let me know if my logic is sound if we introduce a change to the original question: instead of 18% watch exactly 2 of these shows, let's change it to: 18% watch at least 2 of these shows

would the answer then be

Total = A + B + C - N2 - N3 => 42 answer C?

I'm using the set theory that:
# in exactly 2 sets = N2 - 3*N3
# in two or more sets = N2 - 2*N3

So for the total:
Total = A + B + C - N2 + N3 now becomes:
Total = A + B + C - (18 + 2*N3) + N3
Total = A + B + C - 18 - N3
Total = 35 + 40 + 50 - 18 - N3 = 100
N3 = 7% = 42
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Re: Of the 600 residents of Clermontville, 35% watch the television show [#permalink]

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New post 15 Feb 2010, 00:28
xiao85yu wrote:
Prometoh wrote:
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


please let me know if my logic is sound if we introduce a change to the original question: instead of 18% watch exactly 2 of these shows, let's change it to: 18% watch at least 2 of these shows

would the answer then be

Total = A + B + C - N2 - N3 => 42 answer C?

I'm using the set theory that:
# in exactly 2 sets = N2 - 3*N3
# in two or more sets = N2 - 2*N3

So for the total:
Total = A + B + C - N2 + N3 now becomes:
Total = A + B + C - (18 + 2*N3) + N3
Total = A + B + C - 18 - N3
Total = 35 + 40 + 50 - 18 - N3 = 100
N3 = 7% = 42


HI xiao85yu,

As per the set theory formula
Total = g1+g2+g3 - (sum of all two groups) -2(sum of all three) + neither
600 = 210+240+300 - (108) - 2(sum of all three) + 0
2(sum of all three) = 750 - 600 - 108
2(sum of all three) = 42
(sum of all three) = 21

that's why its D.

I think you missed to divide the answer with 2.

Cheers!
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Re: Of the 600 residents of Clermontville, 35% watch the television show   [#permalink] 15 Feb 2010, 00:28

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