https://file.aiquickdraw.com/mathbot/file/1726977750139qy0pomi0.jpg

Let's break down the information from the problem to draw a Venn diagram and calculate the number of students in each group.

### Given Information:
- Total number of students = 76
- Students studying French = 26
- Students studying Spanish = 25
- Students studying Mandarin = 25
- Students studying both French and Spanish = 6
- Students studying both French and Mandarin = 9
- Students studying both Spanish and Mandarin = 6
- Students studying all three languages = 1

### Steps to solve:
1. **Three-way overlap (F ∩ S ∩ M)**: There is 1 student studying all three languages.
2. **Two-way overlaps**:
   - For **French and Spanish** (F ∩ S): 6 students, but 1 is already included in the all-three group, so 6 - 1 = 5 study only French and Spanish.
   - For **French and Mandarin** (F ∩ M): 9 students, but 1 is already counted in all three, so 9 - 1 = 8 study only French and Mandarin.
   - For **Spanish and Mandarin** (S ∩ M): 6 students, but 1 is counted in all three, so 6 - 1 = 5 study only Spanish and Mandarin.
3. **Individual languages**:
   - For **French only**: Out of 26 students studying French, 5 study French and Spanish, 8 study French and Mandarin, and 1 studies all three. Therefore, 26 - (5 + 8 + 1) = 12 students study only French.
   - For **Spanish only**: Out of 25 students studying Spanish, 5 study French and Spanish, 5 study Spanish and Mandarin, and 1 studies all three. Therefore, 25 - (5 + 5 + 1) = 14 students study only Spanish.
   - For **Mandarin only**: Out of 25 students studying Mandarin, 8 study French and Mandarin, 5 study Spanish and Mandarin, and 1 studies all three. Therefore, 25 - (8 + 5 + 1) = 11 students study only Mandarin.

### Verification:
Adding up all groups should give us the total of 76 students.
- French only: 12
- Spanish only: 14
- Mandarin only: 11
- French and Spanish only: 5
- French and Mandarin only: 8
- Spanish and Mandarin only: 5
- All three languages: 1

Sum = 12 + 14 + 11 + 5 + 8 + 5 + 1 = 76, which matches the total.

The correct Venn diagram should have:
- 12 in the "French only" region.
- 14 in the "Spanish only" region.
- 11 in the "Mandarin only" region.
- 5 in the "French and Spanish" overlap.
- 8 in the "French and Mandarin" overlap.
- 5 in the "Spanish and Mandarin" overlap.
- 1 in the middle representing all three languages.

Would you like any further details or clarifications?

### Further questions:
1. How would the Venn diagram look if an additional language were added?
2. What would happen if the total number of students were 100 instead of 76?
3. Can you calculate how many students are studying only one language?
4. How would the overlap numbers change if no student studied all three languages?
5. Can you derive a general formula for solving such problems with 4 sets?

**Tip**: For more complex problems with many overlapping sets, using a systematic inclusion-exclusion method helps simplify the process.

In a class of 76 students, 26 are studying French, 25 are studying Spanish, and 25 are studying Mandarin. There are 6 students studying both French and Spanish, 9 are studying both French and Mandarin, and 6 are studying both Spanish and Mandarin. One student is taking all three foreign language subjects. Choose the correct Venn diagram to illustrate the given situation.

Learn how to create a Venn diagram for 76 students studying three languages: French, Spanish, and Mandarin. Understand how to calculate overlapping groups using the Inclusion-Exclusion principle. Perfect for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation