M7MBA wrote: ↑Wed Jun 24, 2020 7:00 am
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
[spoiler]OA=D[/spoiler]
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
