Question

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? [Official GMAT-2018]

Hint:

When dealing with overlapping sets, use the principle of inclusion-exclusion to avoid double-counting.

Answer 1

Step 1: Define the sets.
Let:
- \( A \) be the set of subjects who experienced sweaty palms.
- \( B \) be the set of subjects who experienced vomiting.
- \( C \) be the set of subjects who experienced dizziness.
We are given the following:
- \( |A| = 40% \times 300 = 120 \) subjects
- \( |B| = 30% \times 300 = 90 \) subjects
- \( |C| = 75% \times 300 = 225 \) subjects
Also, 35 percent of the subjects experienced exactly two of these effects. Thus: \[ 0.35 \times 300 = 105 \quad \text{subjects experienced exactly two effects.} \] Step 2: Use the inclusion-exclusion principle.
The total number of subjects who experienced at least one effect is 300. The total number of subjects who experienced exactly two effects is 105. Let \( x \) be the number of subjects who experienced only one effect. The equation for the total number of subjects is: \[ x + 105 + \text{subjects who experienced all three effects} = 300 \] Let \( y \) be the number of subjects who experienced all three effects. We also know that: \[ x + 105 + y = 300 \] Step 3: Solve for \( x \).
We are told that \( x \) is the number of subjects who experienced only one effect. To find this, we use the principle of inclusion-exclusion: \[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \] We can solve the equation step by step (and from the data, find that \( x = 150 \)). Step 4: Conclusion.
Thus, 150 subjects experienced exactly one of these effects.