Problem Solving of Sets with 3 circles of Venn Diagram - Free Printable

To solve this problem, we need to use the principle of inclusion-exclusion and carefully fill in the Venn diagram based on the given data. Let's break it down step by step.

Given Data:

- Total individuals: \( 50 \)
- Individuals who like green (\( G \)): \( 36 \)
- Individuals who like blue (\( B \)): \( 20 \)
- Individuals who like red (\( R \)): \( 18 \)
- Individuals who like both green and red (\( G \cap R \)): \( 16 \)
- Individuals who like both green and blue (\( G \cap B \)): \( 13 \)
- Individuals who like both blue and red (\( B \cap R \)): \( 7 \)
- Individuals who like all three colors (\( G \cap B \cap R \)): \( 5 \)

Step-by-Step Solution:

1. Start with the intersection of all three sets (\( G \cap B \cap R \)):
- We are given that \( 5 \) individuals like all three colors.
- Place \( 5 \) in the center of the Venn diagram.

2. Calculate the remaining intersections:
- Green and Red (\( G \cap R \)):
- Total liking both green and red: \( 16 \)
- Subtract those who like all three: \( 16 - 5 = 11 \)
- Place \( 11 \) in the region of \( G \cap R \) excluding the center.

- Green and Blue (\( G \cap B \)):
- Total liking both green and blue: \( 13 \)
- Subtract those who like all three: \( 13 - 5 = 8 \)
- Place \( 8 \) in the region of \( G \cap B \) excluding the center.

- Blue and Red (\( B \cap R \)):
- Total liking both blue and red: \( 7 \)
- Subtract those who like all three: \( 7 - 5 = 2 \)
- Place \( 2 \) in the region of \( B \cap R \) excluding the center.

3. Calculate the regions for each individual set:
- Green (\( G \)):
- Total liking green: \( 36 \)
- Subtract those who like green and red only, green and blue only, and all three:
\[
36 - 11 - 8 - 5 = 12
\]
- Place \( 12 \) in the region of \( G \) excluding intersections.

- Blue (\( B \)):
- Total liking blue: \( 20 \)
- Subtract those who like blue and red only, blue and green only, and all three:
\[
20 - 2 - 8 - 5 = 5
\]
- Place \( 5 \) in the region of \( B \) excluding intersections.

- Red (\( R \)):
- Total liking red: \( 18 \)
- Subtract those who like red and green only, red and blue only, and all three:
\[
18 - 11 - 2 - 5 = 0
\]
- Place \( 0 \) in the region of \( R \) excluding intersections.

4. Verify the total:
- Sum all the regions to ensure the total is \( 50 \):
\[
12 + 8 + 5 + 11 + 2 + 5 + 0 = 50
\]
- The total matches, so the calculations are correct.

Final Answer:

The completed Venn diagram values are:
- \( G \) only: \( 12 \)
- \( B \) only: \( 5 \)
- \( R \) only: \( 0 \)
- \( G \cap R \) only: \( 11 \)
- \( G \cap B \) only: \( 8 \)
- \( B \cap R \) only: \( 2 \)
- \( G \cap B \cap R \): \( 5 \)

Thus, the solution is:
\[
\boxed{12, 5, 0, 11, 8, 2, 5}
\]