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

1. Translate the problem requirements: We have 300 subjects experiencing three effects (sweaty palms, vomiting, dizziness) with given percentages for each effect. We know all subjects experienced at least one effect, 35% experienced exactly two effects, and we need to find how many experienced exactly one effect.
2. Set up the three-way overlap structure: Organize the given information and identify what we know versus what we need to find, recognizing this as a problem about categorizing subjects by how many effects they experienced.
3. Apply the constraint that all subjects experienced at least one effect: Use the fact that subjects fall into exactly three categories (one effect only, two effects only, three effects only) and these must sum to 300 total subjects.
4. Calculate the missing quantities using logical relationships: Work backwards from the totals to find how many subjects experienced all three effects, then determine how many experienced exactly one effect.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know in plain English. We have 300 people who tried virtual-reality therapy for fear of heights. Each person experienced at least one of three side effects: sweaty palms, vomiting, or dizziness.

Here's what the problem tells us:

- 40% experienced sweaty palms = \(0.40 \times 300 = 120\) people

- 30% experienced vomiting = \(0.30 \times 300 = 90\) people

- 75% experienced dizziness = \(0.75 \times 300 = 225\) people

- 100% experienced at least one effect = all 300 people

- 35% experienced exactly two effects = \(0.35 \times 300 = 105\) people

We need to find: How many people experienced exactly one effect?

Process Skill: TRANSLATE - Converting percentages to actual numbers of people makes the problem more concrete

2. Set up the three-way overlap structure

Think of this like sorting people into buckets based on how many effects they experienced. Since everyone experienced at least one effect, we have three types of people:

- People who experienced exactly 1 effect (what we're looking for)

- People who experienced exactly 2 effects = 105 people (given)

- People who experienced exactly 3 effects = unknown (let's call this x)

These three groups must add up to all 300 people:

\((\text{Exactly 1 effect}) + (\text{Exactly 2 effects}) + (\text{Exactly 3 effects}) = 300\)

\((\text{Exactly 1 effect}) + 105 + x = 300\)

So: \((\text{Exactly 1 effect}) = 300 - 105 - x = 195 - x\)

3. Apply the constraint that all subjects experienced at least one effect

Now we need to find x (people with all three effects). Here's the key insight: when we count the total occurrences of each effect, we're counting some people multiple times.

- People with exactly 1 effect are counted once in the totals

- People with exactly 2 effects are counted twice in the totals

- People with exactly 3 effects are counted three times in the totals

Total effect occurrences = \(120 + 90 + 225 = 435\)

This total can also be calculated as:

\(435 = (\text{People with 1 effect}) \times 1 + (\text{People with 2 effects}) \times 2 + (\text{People with 3 effects}) \times 3\)

\(435 = (195 - x) \times 1 + 105 \times 2 + x \times 3\)

\(435 = 195 - x + 210 + 3x\)

\(435 = 405 + 2x\)

Process Skill: INFER - Recognizing that people are counted multiple times in the effect totals is the key insight

4. Calculate the missing quantities using logical relationships

Solving for x:

\(435 = 405 + 2x\)

\(30 = 2x\)

\(x = 15\)

So 15 people experienced all three effects.

Therefore, people with exactly one effect = \(195 - x = 195 - 15 = 180\)

Let's verify:

- Exactly 1 effect: 180 people

- Exactly 2 effects: 105 people

- Exactly 3 effects: 15 people

- Total: \(180 + 105 + 15 = 300\) ✓

Effect count check:

- Total effect occurrences: \((180 \times 1) + (105 \times 2) + (15 \times 3) = 180 + 210 + 45 = 435\) ✓

Final Answer

180 subjects experienced only one of these effects. The answer is D.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly two effects" vs "at least two effects"

Students often confuse the phrase "35% experienced exactly two effects" and think it means "at least two effects." This leads them to incorrectly categorize people who experienced all three effects as part of the "exactly two" group, fundamentally changing the problem setup.

2. Failing to recognize the three-way categorization structure

Many students jump straight into Venn diagram formulas without first understanding that every person falls into exactly one of three categories: exactly 1 effect, exactly 2 effects, or exactly 3 effects. Missing this key insight makes it difficult to set up the constraint equations properly.

3. Not understanding the "counting multiple times" concept

Students often struggle to realize that when we add up the individual effect totals (120 + 90 + 225 = 435), we're counting people multiple times based on how many effects they experienced. This is the critical insight needed to create the equation that solves for the unknown.

Errors while executing the approach

1. Incorrect coefficient assignment in the counting equation

When setting up the equation 435 = (people with 1 effect) × 1 + (people with 2 effects) × 2 + (people with 3 effects) × 3, students sometimes assign wrong coefficients, such as using 1, 1, 1 instead of 1, 2, 3, not understanding that the coefficient represents how many times each person is counted.

2. Substitution errors when replacing variables

After establishing that "people with exactly 1 effect = 195 - x," students often make substitution mistakes in the counting equation, either forgetting to substitute or substituting incorrectly, leading to wrong algebraic expressions.

Errors while selecting the answer

1. Selecting the intermediate value instead of the final answer

Students might calculate x = 15 (people with all three effects) correctly but then mistakenly select 15 or 195 as their final answer instead of computing 195 - 15 = 180. They lose track of what the question is actually asking for.