In a survey of 300 people, 180 like burgers, 160 like pizza, and 100 like both. Find: 

The number of people who like only burgers. 

The number of people who like only pizza. 

The number of people who like neither. 

This problem involves set theory and can be solved using the principles of Venn diagrams. Let’s define the sets:

- \(B\): People who like **burgers**.
- \(P\): People who like **pizza**.
- \(B \cap P\): People who like **both** burgers and pizza.

### Given:
- \( |B| = 180 \) (people who like burgers)
- \( |P| = 160 \) (people who like pizza)
- \( |B \cap P| = 100 \) (people who like both)
- Total people in the survey = 300

### Step 1: Number of people who like only burgers
This means people who like burgers but **not pizza**, i.e., \( B \setminus P \):
\[
|B \setminus P| = |B| - |B \cap P| = 180 - 100 = 80
\]

### Step 2: Number of people who like only pizza
This means people who like pizza but **not burgers**, i.e., \( P \setminus B \):
\[
|P \setminus B| = |P| - |B \cap P| = 160 - 100 = 60
\]

### Step 3: Number of people who like neither burgers nor pizza
The total number of people who like at least one (burgers or pizza) is given by:
\[
|B \cup P| = |B| + |P| - |B \cap P| = 180 + 160 - 100 = 240
\]
Thus, the number of people who like neither is:
\[
300 - |B \cup P| = 300 - 240 = 60
\]

### Final Answers:
1. **Number of people who like only burgers**: 80  
2. **Number of people who like only pizza**: 60  
3. **Number of people who like neither**: 60  

Would you like to see a Venn diagram for better understanding?

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Here are 5 related questions:
1. What is the total number of people who like at least one of the two items?
2. How many people like exactly one item (either burger or pizza, but not both)?
3. If another 50 people join the survey, how would it affect the results?
4. What percentage of people in the survey like both pizza and burgers?
5. How would these numbers change if everyone in the survey liked at least one item?

**Tip:** When solving Venn diagram problems, always check for overlapping sets carefully to avoid counting duplicates.

In a survey of 300 people, 180 like burgers, 160 like pizza, and 100 like both. Find: The number of people who like only burgers, the number of people who like only pizza, and the number of people who like neither.

Discover how to analyze survey results with set theory, including finding the number of people who like only burgers, only pizza, or neither. This problem uses Venn diagrams and the principle of inclusion-exclusion, with step-by-step calculations based on a 300-person survey.

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