- Translate the problem requirements: We have 32 total students, with some in music club (15), some in dance club (20), some in both clubs, and some in neither club (10). We need to find how many students are in exactly one club (not both).
- Calculate students in at least one club: Since 10 students are in neither club, determine how many are in at least one club by subtracting from the total.
- Find the overlap between clubs: Use the fact that when we add individual club memberships, we double-count students in both clubs, so compare this sum to actual students in clubs.
- Determine students in exactly one club: Subtract the overlap (students in both clubs) from the total students in at least one club to get those in only one club.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- We have a class of 32 students total
- 15 students are in the music club
- 20 students are in the dance club
- 10 students are not in either club
- We need to find how many students are in exactly one club (meaning they're in music OR dance, but not both)
Think of this like a classroom where some students have multiple interests and some have none at all. We want to count only those students who have exactly one interest - either music or dance, but not both.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
2. Calculate students in at least one club
If 10 students are in neither club, then the remaining students must be in at least one club.
Students in at least one club = Total students - Students in neither club
Students in at least one club = \(32 - 10 = 22\) students
So 22 students are involved in at least one of these clubs.
3. Find the overlap between clubs
Here's where we need to think carefully. When we add up music club members (15) and dance club members (20), we get \(15 + 20 = 35\). But we just calculated that only 22 students are actually in clubs!
This means we've counted some students twice - specifically, the students who are in both clubs.
Students counted twice = (Music + Dance) - Students actually in clubs
Students in both clubs = \(35 - 22 = 13\) students
So 13 students are in both the music club and the dance club.
Process Skill: INFER - Drawing the non-obvious conclusion about double-counting
4. Determine students in exactly one club
Now we can find students in exactly one club by taking all students in clubs and removing those who are in both clubs.
Students in exactly one club = Students in at least one club - Students in both clubs
Students in exactly one club = \(22 - 13 = 9\) students
Let's verify this makes sense:
- Students in music only = \(15 - 13 = 2\) students
- Students in dance only = \(20 - 13 = 7\) students
- Total in exactly one club = \(2 + 7 = 9\) students ✓
4. Final Answer
The number of students in only one of the two clubs is 9.
This matches answer choice B. 9.
Common Faltering Points
Errors while devising the approach
- Misinterpreting "only one club": Students often confuse "only one club" (exactly one) with "at least one club." They might try to find students in music OR dance (including both) rather than students in exactly one club but not both.
- Not recognizing the overlap scenario: Students may fail to realize that some students can be in both clubs simultaneously, leading them to treat the music and dance club memberships as mutually exclusive sets from the start.
- Incorrect constraint interpretation: Students might misread "10 students are not in either club" and incorrectly assume these students are in exactly one club, rather than understanding they are in neither club.
Errors while executing the approach
- Double-counting error: When adding music club members (15) and dance club members (20) to get 35, students often don't recognize that this exceeds the possible number of students in clubs (22), missing the critical insight about overlap.
- Arithmetic mistakes in overlap calculation: Students may incorrectly calculate the number of students in both clubs as \((15 + 20) - 32 = 3\) instead of \((15 + 20) - 22 = 13\), using total students rather than students actually in clubs.
- Verification calculation errors: When checking their work by calculating music-only and dance-only students separately, students may make arithmetic errors like \(15 - 13 = 3\) or \(20 - 13 = 6\), leading to incorrect totals.
Errors while selecting the answer
- Selecting the overlap value: Students who correctly calculate that 13 students are in both clubs might mistakenly select this as their final answer, confusing the intermediate step with what the question actually asks.
- Choosing students in at least one club: Students might select 22 (students in at least one club) instead of 9 (students in exactly one club), not properly distinguishing between these two different quantities.
