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Plz explain Official Guide PS Q.178

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Plz explain Official Guide PS Q.178

by utkalnayak » Mon Jan 05, 2015 6:55 pm
178. 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

I could not understand the official explanation at all. Plz help. TIA
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by GMATGuruNY » Mon Jan 05, 2015 7:01 pm
178. 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 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 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.
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by ceilidh.erickson » Wed Jan 07, 2015 9:19 am
Another option when you have 3 groups that overlap is to draw a Venn diagram:

Image

The total for sweaty palms would be: a + d + e + g

vomiting: b + e + f + g

dizziness: c + d + f + g

If we added all those together, we would have a + b + c + 2d + 2e + 2f + 3g

Our actual total for everyone should be: a + b + c + d + e + f + g
So, we've over-counted by d + e + f + 2g, and we'll need to subtract these out to get the real total.

In other words, we've double-counted the people in 2 out of 3 categories (d, e, and f), and triple-counted those in all 3 (g) when we added up the sub-totals of all the groups. Since 40% + 30% + 75% = 145%, we know that the extra 45% represents the people we've double- and triple-counted.

Since this problems treat the people in 2 categories as a single group, then d + e + f = 35%
If we subtract 35% from the total, we're no longer double-counting these people:
145% - 35% = 110%

The leftover 10% must represent the people we triple-counted. In other words, 2g out of the 3g. That must mean that g = 5%.

all three effect: 5% of subjects
2 effects: 35% of subjects
That must mean that 60% of subjects experience just one effect.

(0.6)300 = 180

The answer is D.
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