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

1. Translate the problem requirements: We have 300 subjects, each experiencing at least one of three effects (sweaty palms, vomiting, dizziness). We know the percentages experiencing each effect and that 35% experienced exactly two effects. We need to find how many experienced exactly one effect.
2. Convert percentages to actual numbers: Transform all given percentages into concrete numbers of subjects to work with real quantities rather than abstract percentages.
3. Categorize subjects by number of effects experienced: Since every subject experienced at least one effect, we can group them into exactly one effect, exactly two effects, or all three effects.
4. Apply the total effects counting principle: The sum of all individual effects equals the number of subjects with one effect plus twice those with two effects plus three times those with all three effects.
5. Solve for subjects with exactly one effect: Use the relationship established in the previous step to find the number of subjects experiencing only one effect.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find. We have 300 subjects in a virtual-reality therapy experiment. Each person experienced at least one of three side effects: sweaty palms, vomiting, or dizziness. Think of this like sorting people into groups based on their symptoms - some people might have just one symptom, some might have two symptoms, and some might have all three symptoms. Our goal is to find how many people experienced exactly one symptom.

Given information:

• Total subjects: 300
• 40% experienced sweaty palms
• 30% experienced vomiting
• 75% experienced dizziness
• All subjects experienced at least one effect
• 35% experienced exactly two effects

Process Skill: TRANSLATE - Converting the problem's language into clear mathematical understanding

2. Convert percentages to actual numbers

Working with actual numbers of people is much clearer than working with percentages. Let's convert each percentage to the actual number of subjects:

• Sweaty palms: 40% of 300 = \(0.40 \times 300 = 120\) subjects
• Vomiting: 30% of 300 = \(0.30 \times 300 = 90\) subjects
• Dizziness: 75% of 300 = \(0.75 \times 300 = 225\) subjects
• Exactly two effects: 35% of 300 = \(0.35 \times 300 = 105\) subjects

Now we have concrete numbers to work with instead of abstract percentages.

3. Categorize subjects by number of effects experienced

Since every subject experienced at least one effect, we can divide all 300 subjects into three distinct groups:

• Group A: Subjects with exactly 1 effect
• Group B: Subjects with exactly 2 effects = 105 subjects (given)
• Group C: Subjects with exactly 3 effects (all three symptoms)

These three groups must add up to 300 total subjects:
\(\mathrm{Group\ A} + \mathrm{Group\ B} + \mathrm{Group\ C} = 300\)
\(\mathrm{Group\ A} + 105 + \mathrm{Group\ C} = 300\)
\(\mathrm{Group\ A} + \mathrm{Group\ C} = 195\)

So we need to find Group A (exactly 1 effect) and Group C (exactly 3 effects).

4. Apply the total effects counting principle

Here's the key insight: we can count the total number of individual effects in two different ways. When we add up all the individual symptoms (\(120 + 90 + 225 = 435\)), we're counting each person's effects. But we can also count by considering how many effects each group contributes:

• People with exactly 1 effect contribute 1 effect each
• People with exactly 2 effects contribute 2 effects each
• People with exactly 3 effects contribute 3 effects each

So: Total individual effects = \(1 \times (\mathrm{Group\ A}) + 2 \times (\mathrm{Group\ B}) + 3 \times (\mathrm{Group\ C})\)
\(435 = 1 \times (\mathrm{Group\ A}) + 2 \times (105) + 3 \times (\mathrm{Group\ C})\)
\(435 = \mathrm{Group\ A} + 210 + 3 \times (\mathrm{Group\ C})\)

Process Skill: INFER - Drawing the non-obvious conclusion that we can count effects in two equivalent ways

5. Solve for subjects with exactly one effect

Now we have two equations with two unknowns:

• Equation 1: \(\mathrm{Group\ A} + \mathrm{Group\ C} = 195\)
• Equation 2: \(\mathrm{Group\ A} + 210 + 3 \times (\mathrm{Group\ C}) = 435\)

From Equation 2: \(\mathrm{Group\ A} + 3 \times (\mathrm{Group\ C}) = 435 - 210 = 225\)

Now we can solve by substitution. From Equation 1: \(\mathrm{Group\ A} = 195 - \mathrm{Group\ C}\)
Substituting into our modified Equation 2:
\((195 - \mathrm{Group\ C}) + 3 \times (\mathrm{Group\ C}) = 225\)
\(195 - \mathrm{Group\ C} + 3 \times (\mathrm{Group\ C}) = 225\)
\(195 + 2 \times (\mathrm{Group\ C}) = 225\)
\(2 \times (\mathrm{Group\ C}) = 30\)
\(\mathrm{Group\ C} = 15\)

Therefore: \(\mathrm{Group\ A} = 195 - 15 = 180\)

6. Final Answer

180 subjects experienced exactly one effect.

Let's verify:

• 180 subjects with 1 effect each = 180 effects
• 105 subjects with 2 effects each = 210 effects
• 15 subjects with 3 effects each = 45 effects
• Total effects: \(180 + 210 + 45 = 435\) ✓
• Total subjects: \(180 + 105 + 15 = 300\) ✓

The answer is D. 180

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly two effects" as "at least two effects"
Students often confuse the constraint "35% experienced exactly two effects" with "at least two effects." This leads them to incorrectly include people with all three effects in this 35% group, fundamentally changing the problem structure and making it unsolvable with the given approach.

2. Failing to recognize the need for categorization by number of effects
Many students get overwhelmed by the multiple percentages and try to use Venn diagrams or inclusion-exclusion principle directly without first organizing subjects into clear categories (1 effect, 2 effects, 3 effects). This makes the problem much more complex than necessary and often leads to incorrect setups.

3. Not recognizing the "total effects counting" principle
Students may struggle to see that they can count the total individual effects in two equivalent ways: by summing up all the individual symptom totals (120 + 90 + 225) and by counting how many effects each group contributes (1×Group A + 2×Group B + 3×Group C). Missing this key insight prevents them from setting up the crucial second equation.

Errors while executing the approach

1. Calculation errors when converting percentages to actual numbers
Students frequently make arithmetic mistakes when calculating 40% of 300, 30% of 300, etc. For example, calculating 75% of 300 as 275 instead of 225, or 35% of 300 as 115 instead of 105. These errors propagate through the entire solution.

2. Algebraic errors when solving the system of equations
When solving the two equations (Group A + Group C = 195 and Group A + 3×Group C = 225), students often make substitution errors or arithmetic mistakes. Common errors include incorrectly simplifying (195 - Group C) + 3×Group C to 195 + 2×Group C, or making errors when solving 2×Group C = 30.

3. Setting up the effects counting equation incorrectly
Students may incorrectly write the total effects equation, such as using Group A + 2×Group B + 3×Group C = 435 instead of 1×Group A + 2×Group B + 3×Group C = 435, or forgetting to account for the fact that people with exactly 2 effects contribute 2 effects each.

Errors while selecting the answer

1. Confusing what the question is asking for
After correctly calculating Group A = 180, Group B = 105, and Group C = 15, some students may select 105 (people with exactly two effects) or 195 (people with one or three effects) instead of 180 (people with exactly one effect). This happens when they lose track of what the question specifically asks for.

2. Selecting an intermediate calculation result
During the solution process, students encounter several numbers like 195 (Group A + Group C), 225 (total effects minus effects from Group B), and 105 (exactly two effects). Some students may mistakenly select one of these intermediate results, particularly 195, thinking it represents the final answer.