Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40

[G Gmat_mission]

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

Answer: D

Source: Official Guide

[Brent@GMATPrepNow]

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

Answer: D

Source: Official Guide

40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness.
We get:


35 percent of the subjects experienced exactly two of these effects
35% of 300 = 105
So, a + b + c = 105

If 120 experienced experienced sweaty palms, we can write: X + a + b + d = 120
If 90 experienced experienced vomiting, we can write: Y + a + c + d = 90
If 225 experienced experienced dizziness, we can write: Z + b + c + d = 225

ADD all three equations to get: X + Y + Z + 2a + 2b + 2c + 3d = 435
Rewrite as follows: X + Y + Z + 2(a + b + c) + 3d = 435

Since a + b + c = 105, we can substitute to get: X + Y + Z + 2(105) + 3d = 435
Simplify to get: X + Y + Z + 210 + 3d = 435
Subtract 210 from both sides to get: X + Y + Z + 3d = 225

300 subjects participated in the experiment
We can write: X + Y + Z + a + b + c + d = 300
Substitute: X + Y + Z + 105 + d = 300
Subtract 105 from both sides: X + Y + Z + d = 195

We now have two different equations:
X + Y + Z + 3d = 225
X + Y + Z + d = 195

Subtract the bottom equation from the top equation to get: 2d = 30, which means d = 15
Replace d with 15 to get: X + Y + Z + 15 = 195
Subtract 15 from both sides: X + Y + Z = 180

So, 180 subjects experienced only one effect.

Answer: D

[Scott@TargetTestPrep]

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

Answer: D

Source: Official Guide

Solution:

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).

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)

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


Alternate Solution:

Alternatively, we can use the following formula for percentages:

100= % of (Group A) + % of (Group B) + % of (Group C) – % of (Groups of Exactly Two) – 2 [% of (Group of Exactly Three)] + % of (Neither)

Next we can label our groups with the information presented.

% of Group A = % who experienced sweaty palms

% of Group B = % who experienced vomiting

% of Group C = % who experienced dizziness

Thus, we have:

100 = 40 + 30 + 75 - 35 - 2[% of (Group of Exactly Three)] + 0

100 = 110 - 2[% of (Group of Exactly Three)]

2[% of (Group of Exactly Three)] = 10

% of (Group of Exactly Three) = 5

Notice that we have:

100 = % of (Group of Exactly One) + % of (Group of Exactly Two) + % of (Group of All Three)

100 = % of (Group of Exactly One) + 35 + 5 + 0

100 = % of (Group of Exactly One) + 40

% of (Group of Exactly One) = 60

Thus, there are 300 * 0.6 = 180 people belonging to exactly one of the groups.

Answer: D