OG13th Q 178

[spoiler]{D}[/spoiler]

anurag_7 wrote:Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?
(A) 105
(B) 125
(C) 130
(D) 180
(E) 195

How to solve such questions by using venn diagrams?

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - (2-group overlaps) * 2 - (3-group overlaps) * 3
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

Answer: Option D

Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?

A) 105

B) 125

C) 130

D) 180

E) 195

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 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 overcounted.

In the problem above:
Let T = 100%.
Sweaty palms = 40%.
Vomiting = 30%.
Dizziness = 75%.
Exactly 2 of the groups = 35%.
Let x = the percentage in all 3 groups.

Plugging these values into the formula, we get:
100 = 40 + 30 + 75 - 35 - 2x
100 = 110 - 2x
x=5.

Since 35% are in 2 of the groups and 5% are in all 3 groups, the percentage in exactly one of the groups = 100-35-5 = 60.
Number in exactly one of the groups = .6(300) = 180.

The correct answer is D.