PS: Sets

I posted a solution here:
https://www.beatthegmat.com/og-13-178-vi ... 11188.html

For a triple Venn diagram, assuming everybody is in at least one of the three groups A, B, and C:

Total = A + B + C - (exactly two) - 2 * (all three)

so

300 = 120 + 90 + 225 - 105 - 2 * (all three)

and

all three = 15

From there

Total = (exactly one) + (exactly two) + (all three)

300 = (exactly one) + 105 + 15

and (exactly one) = 180.

Arsenal wrote:of 300 subjects who participated in an exp using thearapy to reduce their fear of heigts. 40% experienced sweaty palms, 30% vomit and 75% dizziness.if all subjects experienced atleast one of these effects and 35% of 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

This is a 3-circle Venn Diagram problem. Because we do not know the number of unique items in this particular set, we can use the following formula:

Total # of Unique Elements = # in (Group A) + # in (Group B) + # in (Group C) – # in (Groups of Exactly Two) – 2 [#in (Group of Exactly Three)] + # in (Neither)

Next we can label our groups with the information presented.

# in Group A = # who experienced sweaty palms

# in Group B = # who experienced vomiting

# in Group C = # who experienced dizziness

We are given that 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.

We can solve for the number in each group:

# who experienced sweaty palms = 300 x 0.4 = 120

# who experienced vomiting = 300 x 0.3 = 90

# who experienced dizziness = 300 x 0.75 = 225

We are also given that all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects.

This means the following:

# in Groups of Exactly Two = 300 x 0.35 = 105

Since all the subjects experienced at least one of the effects it means that the # in (Neither) is equal to zero. We can now plug in all the information we have into our formula, in which T represents # in (Group of Exactly Three).

300 = 120 + 90 + 225 – 105 – 2T + 0

300 = 330 – 2T

30 = 2T

15 = T

Now that we have determined a value for T, we are very close to finishing the problem. The question asks how many of the subjects experienced only one of these effects.

To determine this, we can set up one final formula.

Total = # who experienced only 1 effect + # who experienced two effects + # who experienced all 3 effects + # who experienced no effects

We can let x represent the # who experienced only 1 effect.

300 = x + 105 + 15 + 0

300 = x + 120

180 = x

Answer is D