At a certain school there are 83 students taking statistics and 89 students taking computer science. If 40 of these...

1. Translate the problem requirements: We need to find students taking "statistics OR computer science but NOT both" - this means students who are taking exactly one subject (either statistics only OR computer science only), excluding those taking both subjects.
2. Identify the overlapping groups: Recognize that some students are counted in both subject totals because they take both classes, so we need to separate the exclusive groups.
3. Calculate students taking only statistics: Subtract the overlap from the statistics total to find those taking statistics exclusively.
4. Calculate students taking only computer science: Subtract the overlap from the computer science total to find those taking computer science exclusively.
5. Sum the exclusive groups: Add the "statistics only" and "computer science only" groups to get our final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what the question is asking. We have students taking statistics and computer science classes, and some students are taking both subjects.

The key phrase is "taking statistics or computer science but not both." This means we want to count:

• Students taking ONLY statistics (not computer science)
• Students taking ONLY computer science (not statistics)

We do NOT want to count students who are taking both subjects.

Process Skill: TRANSLATE - Converting the phrase "or...but not both" into clear mathematical understanding

2. Identify the overlapping groups

Let's think about this like groups of people at a school event:

• 83 students are in the "statistics group"
• 89 students are in the "computer science group"
• 40 students appear in BOTH groups (they're taking both classes)

The problem is that when we count \(83 + 89 = 172\), we're double-counting those 40 students who appear in both groups. To find students taking exactly one subject, we need to separate out the exclusive groups.

3. Calculate students taking only statistics

To find students taking ONLY statistics, we take the total statistics students and subtract those who are also taking computer science:

Students taking only statistics = Total statistics students - Students taking both
Students taking only statistics = \(83 - 40 = 43\)

This makes sense: out of 83 statistics students, 40 of them are also taking computer science, so the remaining 43 are taking statistics exclusively.

4. Calculate students taking only computer science

Similarly, to find students taking ONLY computer science, we take the total computer science students and subtract those who are also taking statistics:

Students taking only computer science = Total computer science students - Students taking both
Students taking only computer science = \(89 - 40 = 49\)

This means out of 89 computer science students, 40 are also taking statistics, so the remaining 49 are taking computer science exclusively.

5. Sum the exclusive groups

Now we add up the students taking exactly one subject:

Students taking statistics OR computer science but NOT both = Students taking only statistics + Students taking only computer science
= \(43 + 49 = 92\)

Final Answer

The answer is 92 students are taking statistics or computer science but not both.

This matches answer choice B. 92.

To verify: We found 43 students taking only statistics and 49 students taking only computer science, giving us \(43 + 49 = 92\) total students taking exactly one of these subjects.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "or...but not both" as "and"

Students often confuse the phrase "statistics or computer science but not both" and think they need to find students taking both subjects. This leads them to simply answer 40 (the overlap), missing the actual requirement to find students taking exactly one subject.

2. Misunderstanding what "but not both" excludes

Some students interpret "statistics or computer science but not both" as the total number of students taking either subject (including those taking both), leading them to use the formula \(|\mathrm{S} \cup \mathrm{CS}| = |\mathrm{S}| + |\mathrm{CS}| - |\mathrm{S} \cap \mathrm{CS}| = 83 + 89 - 40 = 132\), which counts everyone including those taking both subjects.

3. Failing to break down the problem into exclusive groups

Students may not realize they need to separate the total counts into "only statistics," "only computer science," and "both" groups. Without this breakdown, they cannot systematically identify which groups to include or exclude from their final answer.

Errors while executing the approach

1. Arithmetic errors in subtraction

When calculating students taking only one subject, students may make simple arithmetic mistakes: incorrectly computing \(83 - 40 = 42\) instead of 43, or \(89 - 40 = 48\) instead of 49, leading to a final answer of 90 instead of 92.

2. Double-counting or incorrect addition

Some students might accidentally add the 40 students taking both subjects to their final calculation, computing something like \(43 + 49 + 40 = 132\), not realizing they should exclude the overlap group entirely.

Errors while selecting the answer

1. Selecting the wrong calculated value

Students may arrive at multiple intermediate values during their solution (like 43, 49, 40, 132) and mistakenly select one of these instead of their final answer of 92. For instance, they might choose 132 (answer choice C) thinking it represents the total students taking either subject.