What is the source of this problem?
It is almost identical to -- but harder than -- PS178 in the OG13:
https://www.beatthegmat.com/og-13-178-vi ... 11188.html
Architj wrote:In a certain experiment conducted to reduce fear of heights, a total of 700 subjects took part. The experiment was conducted using virtual - reality therapy. Of the 700 subjects, 32% experienced vomiting, 76% experienced dizziness and 42% experienced sweaty palms. Each subject experienced at least of the three effects. 56% of the subjects experienced exactly one of the three effects.How many subjects experienced exactly two of the three effects?
a. 198
b. 208
c. 224
d. 244
e. 266
Here is a 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 the 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 over counted.
In the problem above:
Let T = 100%.
Vomiting = 32%.
Dizziness = 76%.
Sweaty Palms = 42%.
Let the percentage in exactly 2 groups = x.
Let the percentage in all 3 groups = y.
Plugging these values into the formula above, we get:
100 = 32 + 76 + 42 - x - 2y
x + 2y = 50%.
56% of the subjects experienced exactly one of the three effects.
Since 56% are in exactly 1 group, 44% are in two or three of the groups.
Thus:
x + y = 44%.
Subtracting the blue equation from the red equation, we get:
(x + 2y) - (x+y) = 50-44
y = 6%.
Since x+y = 44% and y=6%, we get:
x = 44-6 = 38%.
Since 38% are in exactly 2 groups, we get:
38% of 700 = 266.
The correct answer is
E.
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