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Of the 600 residents of Clermontville, 35% watch the

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Re: Set theory [#permalink] New post 15 Jan 2010, 15:11
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: Set theory [#permalink] New post 14 Feb 2010, 23:28
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 [#permalink] New post 24 Oct 2012, 01:43
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


*****************************************************************************
Hello,

i see it's an old topic, but here is my solution (mostly done by instinctively, logically)

35% + 40% + 50% = 125%
125% - 18% = 107%

so we see that 7% is OUT, then calculating 7% from 600 (100%)

My answer 42.
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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 24 Oct 2012, 02:02
Rimara 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


*****************************************************************************
Hello,

i see it's an old topic, but here is my solution (mostly done by instinctively, logically)

35% + 40% + 50% = 125%
125% - 18% = 107%

so we see that 7% is OUT, then calculating 7% from 600 (100%)

My answer 42.


The 7% is the number of people watching all three counted twice. So we have to divide the 7% by 2 to give 3.5% to give 21 as the answer.

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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 24 Oct 2012, 02:51
MacFauz wrote:
Rimara 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


*****************************************************************************
Hello,

i see it's an old topic, but here is my solution (mostly done by instinctively, logically)

35% + 40% + 50% = 125%
125% - 18% = 107%

so we see that 7% is OUT, then calculating 7% from 600 (100%)

My answer 42.


The 7% is the number of people watching all three counted twice. So we have to divide the 7% by 2 to give 3.5% to give 21 as the answer.

Kudos Please... If my post helped.


**************************************************************

Dear MacFauz,

I know I look stupid :) but could you please explain - why we should divide 7% by 2??????

Thank you!
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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 24 Oct 2012, 03:40
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Lets take your solution and treat percents as numbers for simplicity.

35 + 40 + 50 = 125.

Here the 35 includes ppl in the 1st group alone, ppl in 1st & 2nd group, ppl in 1st & 3rd groups alone, & ppl in all 3 groups.

Similarly 40 includes ppl in the 2nd group alone, ppl in 2nd & 1st group, ppl in 2nd & 3rd groups alone, & ppl in all 3 groups.

Similarly 50 includes ppl in the 3rd group alone, ppl in 3rd & 1st group, ppl in 3rd & 2nd groups alone, & ppl in all 3 groups.

So we can see that as of now people in one group alone have been accounted for once, ppl in two groups have been accounted for twice. ie There are two copies each of 1st & 2nd only, 1st & 3rd only, 2nd & 3rd only. So what we want to do now is deduct one copy of each. So we get

125 - ((1st & 2nd only)+(1st & 3rd only)+(2nd & 3rd only)) i.e 125 - (No. of ppl in exactly two groups)., So we get

125 - 18 = 107

Now., we can also see that ppl in all three groups have been accounted for thrice. i.e there are three copies of ppl in all 3 groups. So in this 107, the number of ppl in all 3 groups has been represented 3 times. So we have accounted for this group two times more than what is necessary.

So the excess of 7 corresponds to that representation which is twice the number of ppl in 3 groups.

Hope it is clear.

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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 24 Oct 2012, 03:43
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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


I solved it within 30 seconds. Here is the calculation I did:

100% = 35% + 40% + 50% - 18% - 2x% (x denotes the percentage of people who watch all 3 shows)

=> x=3.5%
So, number of people who watch all 3 shows = 3.5% of 600 = 21, which is the answer.

Now, lets see why I did this way:
We are given percentages for all the categories of people (who watch one show, who watch two shows etc), so it is easier to deal in percentages only. If we are given some amounts in numbers, some in percentages, then we have to convert all these amounts into numbers (or percentages, if one is comfortable).

We are given that all people watch at least one of the shows. So, 100% people watch at least one of the shows.

Now, lets start counting:
35% watch the television show Island Survival
40% watch Lovelost Lawyers
50% watch Medical Emergency

So, adding all these, we get to 125%. So, 125% people watch these shows. But we are given 100% people watch these shows. (in any case, it could not be more than 100% but it can be less) So, we must have double counted people. Lets see in which all cases can we double count or triple count or count a person multiple times.

If say a person X watches one show only, he would be counted only once, among the people watching that particular show, say Island Survival. We don't have a problem with this. We need to count every person once.
If X watches two shows, he would appear in the counting of both shows. But we want him to be counted only once. What do we do? Suppose 10 people watch two shows Island Survival and Lovelost Lawyers, then we would have counted those people two times (once for each show). So, to eliminate this double counting. We need to subtract this 10 from the sum of the people watching shows, to get at the total number of people watching these shows. Thus, we subtract from the sum, all the people who watch two shows.

For example: If I say that there are 10 people who watch show A and 8 people who watch show B and 3 people watch both show A & B. What are total number of people (who watch any of these shows)?
We have A-10
B-8
total number - A + B = 18
Is it? No. Because 3 people who watch both shows have been counted twice.
So, total number of people = 10 + 8 - 3 = 15

Let's verify this number. We know out of 15, 3 (let's say X,Y,Z) people see both shows. So, 12 others remaining, who watch either show A or show B.

10 people see show A. Now, out of these 10, three are X, Y & Z. There are 7 remaining who watch show A. So, out of 12 people above, we take out 7 people here. Now, we have 5 people left to watch show B.

So, people who watch show B = 5 + X,Y & Z = 8

So, our calculations are correct.

Now, we come to third category of people, who watch all three shows. These people will be counted thrice. We want to count them once. So, we subtract 2 times their number from the sum.

Thus, we arrive at the equation I mentioned in the beginning:

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

Remember x is in percentage here. Also, remember that we have used 100% because we are given that everyone watches one of these shows. If we were given that 5% population don't watch any of these shows, we would have used 95% as the sum.

If anyone has any queries, please feel free to ask.

Cheers,
CJ

PS: Probably some people may not be comfortable with the percentages as I have used above, they may use numbers instead. The logic holds for numbers also.
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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 17 Apr 2013, 12:12
Answer is D, but it's very tricky.

We're trying to find how many people watch ALL three shows.

We have 600 people, but the shows in aggregate have 750 viewers.

We know that 108 people watch both shows.

Now let's look at what the 750 number includes:

The aggregate viewer number, or 750, counts the people who watch all three shows 3 times.
The population number, or 600, counts the people who watch all three shows only once.
The aggregate viewer number, or 750, counts the total of people who watch exactly two shows TWICE since it is all inclusive.
The population number, or 600, counts it only once.


To understand why, draw a ven diagram with three overlapping circles. If 108 is the total of people who watch exactly two shows, then we can represent 1/3 of 108, or 36, for any given two-way overlap. Each TV show has two of these two-way overlaps, and since there are three shows total, we have a total of SIX two-way overlaps. If each two way overlap represents 1/3 of 108, and there are six of them, then we simplify to the 108 being counted twice in the 750 number as opposed to once in the 600 number.

Now look at the differences: 600 counts the people who watch exactly three shows only once, and the people who watch exactly two shows only once. 750 counts the people who watch exactly three shows THREE times, and the people who watch exactly two shows TWICE. We then need to look at the differences:

x= number of people who watch three shows.

600=750-(108)-(2x)
2x=42
x=21

People who got 42 for the answer are not counting the fact that the people who watch all three shows occurs once naturally in the 600 population size if you break it down to the same segments to compare it to the 750 number.
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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 29 Jun 2013, 03:08
two formulas to remember
total=A+B+C-(sum of exactly 2)- 2*(all three) + neither
total=A+B+C -(SUM OF(AUB+AUC+BUC))+neither
these were originally posted by bunnel.
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Re: Of the 600 residents of Clermontville, 35% watch the [#permalink] New post 09 Jul 2013, 02:31
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I would go with D.
Total = A + B + C - (exactly 2-shows) - 2 (3 shows)
100 = 35 + 40 + 50 - 18 - 2x
--> x= 3.5%
<--> 21 people = D
Re: Of the 600 residents of Clermontville, 35% watch the   [#permalink] 09 Jul 2013, 02:31
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