How to Solve Venn Diagrams with 3 Circles - mathsathome.com - Free Printable

Problem Analysis:

We are given information about 30 students who play different sports: basketball (B), football (F), and tennis (T). The task is to fill in the Venn diagram based on the provided data and determine how many students play each combination of sports.

#### Given Data:
1. Total students = 30
2. Students who play basketball (B) = 20
3. Students who play football (F) = 16
4. Students who play tennis (T) = 15
5. Students who play both basketball and tennis = 10
6. Students who play both basketball and football = 11
7. Students who play both football and tennis = 9
8. Students who play all three sports = 7

Step-by-Step Solution:

#### Step 1: Understand the Venn Diagram
The Venn diagram has three overlapping circles representing basketball (B), football (F), and tennis (T). The regions are as follows:
- Central region (B ∩ F ∩ T): Students who play all three sports.
- Overlapping regions: Students who play exactly two sports (e.g., B ∩ F, B ∩ T, F ∩ T).
- Non-overlapping regions: Students who play only one sport.

#### Step 2: Fill in the Central Region
From the data, we know that 7 students play all three sports. This value goes in the central region of the Venn diagram.

$$
\text{B ∩ F ∩ T} = 7
$$

#### Step 3: Calculate Students Playing Exactly Two Sports
We need to find the number of students who play exactly two sports. To do this, we subtract the number of students who play all three sports from the intersections of two sports.

1. Basketball and Football (B ∩ F):
- Total playing both basketball and football = 11
- Subtract those who play all three: \( 11 - 7 = 4 \)
- So, 4 students play only basketball and football.

2. Basketball and Tennis (B ∩ T):
- Total playing both basketball and tennis = 10
- Subtract those who play all three: \( 10 - 7 = 3 \)
- So, 3 students play only basketball and tennis.

3. Football and Tennis (F ∩ T):
- Total playing both football and tennis = 9
- Subtract those who play all three: \( 9 - 7 = 2 \)
- So, 2 students play only football and tennis.

#### Step 4: Calculate Students Playing Only One Sport
Next, we calculate the number of students who play only one sport. We use the total number of students for each sport and subtract those who play multiple sports.

1. Only Basketball (B):
- Total playing basketball = 20
- Subtract those who play basketball and football: 4
- Subtract those who play basketball and tennis: 3
- Subtract those who play all three: 7
- So, \( 20 - 4 - 3 - 7 = 6 \)
- Therefore, 6 students play only basketball.

2. Only Football (F):
- Total playing football = 16
- Subtract those who play basketball and football: 4
- Subtract those who play football and tennis: 2
- Subtract those who play all three: 7
- So, \( 16 - 4 - 2 - 7 = 3 \)
- Therefore, 3 students play only football.

3. Only Tennis (T):
- Total playing tennis = 15
- Subtract those who play basketball and tennis: 3
- Subtract those who play football and tennis: 2
- Subtract those who play all three: 7
- So, \( 15 - 3 - 2 - 7 = 3 \)
- Therefore, 3 students play only tennis.

#### Step 5: Verify the Total
Finally, we verify that the total number of students adds up to 30.

- Only basketball: 6
- Only football: 3
- Only tennis: 3
- Basketball and football: 4
- Basketball and tennis: 3
- Football and tennis: 2
- All three: 7

Summing these:
$$
6 + 3 + 3 + 4 + 3 + 2 + 7 = 30
$$

The total is correct.

Final Answer:

The completed Venn diagram values are:
- Only basketball: 6
- Only football: 3
- Only tennis: 3
- Basketball and football: 4
- Basketball and tennis: 3
- Football and tennis: 2
- All three: 7

$$
\boxed{6, 3, 3, 4, 3, 2, 7}
$$