in a survey of 100 students, like pizza, 50 kike burgers, 30 like both. how many students like neither pizza nor burgers appply the steps of polya

To solve this problem using Polya's four-step process, let's go through each step:

### 1. **Understand the problem:**
- **Total number of students**: 100
- **Students who like pizza**: 60
- **Students who like burgers**: 50
- **Students who like both pizza and burgers**: 30
- We are asked to find how many students **like neither pizza nor burgers**.

### 2. **Devise a plan:**
We'll use the principle of set theory (specifically the inclusion-exclusion principle) to solve this. The steps include:
1. Add the number of students who like pizza and the number who like burgers.
2. Subtract the number of students who like both to avoid double-counting them.
3. Subtract this result from the total number of students to find how many like neither.

### 3. **Carry out the plan:**
Let's apply the steps.

- **Students who like either pizza or burgers** (or both):
  \[
  |P \cup B| = |P| + |B| - |P \cap B| = 60 + 50 - 30 = 80
  \]

This means 80 students like pizza, burgers, or both.

- **Students who like neither**:
  \[
  100 - 80 = 20
  \]

So, 20 students like neither pizza nor burgers.

### 4. **Look back and check:**
The problem asks for the number of students who like neither pizza nor burgers. We computed that 80 students like pizza, burgers, or both, so it follows that the remaining 20 students must like neither. This solution is consistent and correct.

### Final Answer: 
**20 students like neither pizza nor burgers.**

Would you like more details or clarifications?

Here are 5 related questions to expand this topic:
1. What if 40 students liked both instead of 30? How would the result change?
2. How would the formula adjust if there were more than two categories of food preferences?
3. Can this problem be solved using a Venn diagram approach?
4. What percentage of students like only pizza but not burgers?
5. How does the inclusion-exclusion principle apply to larger sets?

**Tip**: When dealing with set problems like these, always remember to subtract the intersection (common elements) to avoid double-counting!

In a survey of 100 students, 60 like pizza, 50 like burgers, and 30 like both. How many students like neither pizza nor burgers? Apply the steps of Polya.

Solve a set theory problem using Polya's steps. In a survey of 100 students, 60 like pizza, 50 like burgers, and 30 like both. Learn how to find how many students like neither using the inclusion-exclusion principle.

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