IMO C

gmatmachoman 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) 66

its 21

residents watch IS+residents watch LL+residents watch MS=35+40+50=125

so watch exactly 2 of these shows+2*(residents watch all of the shows)=25
or residents watch all of the shows=(25-18)/2=3.5%=600*35/1000=21

Ans option D

my approach:

overlapping sets formula

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

x= number watching all 3 shows

600=210+240+300-108-2x
x=21

I am a little confused here. Where is the 2 *(all three) coming from? I keep getting 42 because I am not dividing by 2. I am not seeing where the 2 is coming from.

arvindiyer wrote:I am a little confused here. Where is the 2 *(all three) coming from? I keep getting 42 because I am not dividing by 2. I am not seeing where the 2 is coming from.

Picture a guy in some focus group or something who watches all three shows, let's call him Bill. When the guy conducting the survey says "raise your hand if you watch Island Survivor", Bill raises his hand and gets counted. Then the guy says, "raise your hand if you watch Lovelost Lawyers", Bill again raises his hand and gets counted. Then the guy says, "raise your hand if you watch Medical Emergency", and Bill raises his hand and gets counted. So Bill has been counted three times, even though he's only one guy. And more generally, when we do Group 1 + Group 2 + Group 3, every person who watches all three shows gets counted three times, because they are in all three groups. But we only want to count each person one time, regardless of how many shows they watch, so in this case we can counteract that initial overcounting by subtracting two times the number of people who watch all three shows. So subtracting 2*(all three) at the end is just because (all three) got counted 3 times at the beginning, so we make up for that at the end by subtracting them twice.

I see- so if in theory if I had 4 shows, I would subtract out 3 times the number. Thank you very much for your help.

arvindiyer wrote:I see- so if in theory if I had 4 shows, I would subtract out 3 times the number. Thank you very much for your help.

Yeah, assuming everything else worked out similarly. But these things can work out in different ways depending on what is given, so you really have to pay close attention to who's getting overcounted and how many times they're being over counted, and who's included in the groups you're subtracting out. Sometimes you have to add the people in all 3 groups back in. If you haven't had much exposure to these types of problems, you should find a few different types to practice on so you can get used to the most common setups.

In any case, a problem like this with 4 groups would be horrendously complicated, so you shouldn't have to worry about it on the GMAT.