AIM TO CRACK GMAT 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.
Here is the formula for 3 overlapping groups:
T = A + B + C - (AB + AC + BC) - 2(ABC)
The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.
In the problem above:
Let T = 100%.
Island Survival = 35.
Lovelost Lawyers = 40.
Medical emergency = 50.
Exactly 2 of the shows = 18.
Let x = the percentage that watch all 3 shows.
Plugging these values into the formula, we get:
100 = 35 + 40 + 50 - 18 - 2x
100 = 107 - 2x
x=3.5.
Since 3.5% of the 600 residents watch all 3 shows, the number who watch all 3 shows = (3.5/100)(600) = 21.
The correct answer is
D.
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