Priyaranjan wrote:Q. Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40% experienced sweaty palms, 30% experienced vomiting and 75% experienced dizziness. If all of the subjects experienced at least one of these effects and 35% 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
Can it be solved using Venn diagrams?
Solution:
This is a 3-circle Venn Diagram problem in which 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 only experienced 1 effect.
300 = x + 105 + 15 + 0
300 = x + 120
180 = x
The answer is
D
