https://file.aiquickdraw.com/mathbot/file/1726282903229kadc616p.jpg

Let's break down the problem described in the image to create a Venn diagram that represents the students enrolled in math subjects under Professors Annie, Ennet, and Yano.

### Information Summary:
- **Total students** = 208
- **Students enrolled in all 3 professors' subjects** = 4
- **Students enrolled under Prof. Ennet** = 48
- **Students enrolled under both Prof. Ennet and Prof. Yano (but not Prof. Annie)** are twice as many as those enrolled under both Prof. Ennet and Prof. Annie (but not Prof. Yano), and 4 times as many as those enrolled under all three professors.
- **Students in Prof. Yano's classes** = 124
- **Students enrolled in no math subject** = 27
- **The group of students enrolled under both Prof. Ennet and Prof. Yano (but not Prof. Annie)** is equal to the group of students enrolled under both Prof. Annie and Prof. Yano (but not Prof. Ennet).

### Venn Diagram Variables:
Let’s define the following variables for the sections of the Venn diagram:
- \(x_A\) = Number of students enrolled only under Prof. Annie.
- \(x_E\) = Number of students enrolled only under Prof. Ennet.
- \(x_Y\) = Number of students enrolled only under Prof. Yano.
- \(x_{AE}\) = Number of students enrolled under both Prof. Annie and Ennet (but not Yano).
- \(x_{AY}\) = Number of students enrolled under both Prof. Annie and Yano (but not Ennet).
- \(x_{EY}\) = Number of students enrolled under both Prof. Ennet and Yano (but not Annie).
- \(x_{AEY} = 4\) (students enrolled under all three professors).

From this, we can begin forming equations.

### Step-by-Step Approach:

1. **Total number of students:**
   \[
   x_A + x_E + x_Y + x_{AE} + x_{AY} + x_{EY} + x_{AEY} + \text{students in no subject} = 208
   \]
   Which simplifies to:
   \[
   x_A + x_E + x_Y + x_{AE} + x_{AY} + x_{EY} + 4 + 27 = 208
   \]
   \[
   x_A + x_E + x_Y + x_{AE} + x_{AY} + x_{EY} = 177
   \]

2. **Prof. Ennet's total:**
   \[
   x_E + x_{AE} + x_{EY} + 4 = 48
   \]
   Thus,
   \[
   x_E + x_{AE} + x_{EY} = 44
   \]

3. **Prof. Yano's total:**
   \[
   x_Y + x_{AY} + x_{EY} + 4 = 124
   \]
   Thus,
   \[
   x_Y + x_{AY} + x_{EY} = 120
   \]

4. **Ennet-Yano relation**: Twice as many students are enrolled under both Prof. Ennet and Yano (but not Annie) compared to those enrolled under both Ennet and Annie (but not Yano):
   \[
   x_{EY} = 2x_{AE}
   \]

5. **The group of students under both Ennet and Yano (but not Annie)** is equal to those under both Annie and Yano (but not Ennet):
   \[
   x_{EY} = x_{AY}
   \]

We can now solve the system of equations using the relationships provided. Let’s calculate these values to complete the Venn diagram.The system of equations gives us the following relationships for the different groups of students:

1. \( x_A = x_{EY} + 13 \) (Students enrolled only under Prof. Annie)
2. \( x_E = 44 - \frac{3}{2}x_{EY} \) (Students enrolled only under Prof. Ennet)
3. \( x_Y = 120 - 2x_{EY} \) (Students enrolled only under Prof. Yano)
4. \( x_{AE} = \frac{x_{EY}}{2} \) (Students enrolled under both Prof. Annie and Ennet)
5. \( x_{AY} = x_{EY} \) (Students enrolled under both Prof. Annie and Yano)
6. \( x_{EY} \) is the number of students enrolled under both Prof. Ennet and Yano (but not Annie).

Now, to find exact values, we can try specific values for \(x_{EY}\). Since \(x_{EY}\) appears in several expressions, let’s test reasonable values for \(x_{EY}\) based on constraints, like total students and professor totals.

Do you want me to proceed with testing values for \(x_{EY}\) or solve further?

Complete a Venn diagram to represent the following: There are 208 students under the new BS Math course. The Mathematics-Physics-Statistics Department offers math subjects under Prof. Annie, Prof. Ennet, and Prof. Yano. The breakdown of student enrollment and relationships between their subjects are described.

This problem involves solving a Venn diagram of students enrolled in courses under three professors. Given relationships and totals, students need to determine enrollment numbers using algebraic systems. Suitable for college-level students.

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