Problem Solving

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 ,
n(A u B u C)= n(A) + n(B) + n(C) - n(A n B) - n(B n C) - n(C n A) + n(A n B n C)
A = 40% of 300 = 120
B = 30% of 300 = 90
C = 75% of 300 = 225
n(A n B) + n(B n C) + n(C n A) = 35% of 300 = 105
n(A u B u C) = 300
Putting values in above formula:
300 = 435 - 105 + n(A n B n C)
300 = 330 + n(A n B n C)
n(A n B n C) = -30
What would be the next step?

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eitijan 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

Here ,
n(A u B u C)= n(A) + n(B) + n(C) - n(A n B) - n(B n C) - n(C n A) + n(A n B n C)
A = 40% of 300 = 120
B = 30% of 300 = 90
C = 75% of 300 = 225
n(A n B) + n(B n C) + n(C n A) = 35% of 300 = 105
n(A u B u C) = 300
Putting values in above formula:
300 = 435 - 105 + n(A n B n C)
300 = 330 + n(A n B n C)
n(A n B n C) = -30
What would be the next step?

There are two versions of the three-set overlap equation. In the version you used, n(A u B u C)= n(A) + n(B) + n(C) - n(A n B) - n(B n C) - n(C n A) + n(A n B n C), the red portion refers to those in at least the two groups designated, not exactly those two groups. In other words, it's not correct to plug the 105 figure into that equation, because the 105 are in exactly two groups.

The alternative version is Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither. This one would have been the appropriate equation to plug the 105 into.

(And the fact that you ended up with a negative value is the clue that something is amiss.)

This is probably best done visually (with a Venn diagram): it's easy to lose track of the inclusion/exclusion equation if you try to do this sort of problem algebraically.