A survey asks 236 people "What beverage do you drink in the morning?", and offers choices:
- Coffee only
- Tea only
- Both tea and coffee
- Neither tea nor coffee

Suppose 32 report tea only, 68 report coffee only, and 64 report both.
How many people drink tea in the morning?
How many people drink coffee in the morning?
How many people drink neither tea nor coffee?

We can solve this using set theory, treating the problem as a union of sets.

Let:
- \( T \) represent the set of people who drink tea.
- \( C \) represent the set of people who drink coffee.
- \( |T| \) is the total number of people who drink tea.
- \( |C| \) is the total number of people who drink coffee.
- \( |T \cap C| \) is the number of people who drink both tea and coffee.

### Given:
- \( 32 \) people drink only tea (i.e., \( |T| - |T \cap C| = 32 \)).
- \( 68 \) people drink only coffee (i.e., \( |C| - |T \cap C| = 68 \)).
- \( 64 \) people drink both tea and coffee (i.e., \( |T \cap C| = 64 \)).
- The total number of people surveyed is \( 236 \).

### Step 1: Find the number of people who drink tea.
The number of people who drink tea includes those who drink only tea and those who drink both tea and coffee:
\[
|T| = 32 + 64 = 96
\]
So, 96 people drink tea.

### Step 2: Find the number of people who drink coffee.
Similarly, the number of people who drink coffee includes those who drink only coffee and those who drink both tea and coffee:
\[
|C| = 68 + 64 = 132
\]
So, 132 people drink coffee.

### Step 3: Find the number of people who drink neither tea nor coffee.
The total number of people surveyed is 236, so the number of people who drink neither tea nor coffee is the difference between the total and those who drink tea or coffee:
\[
\text{People who drink neither} = 236 - (|T \cup C|)
\]
Now, \( |T \cup C| = |T| + |C| - |T \cap C| \), which is the number of people who drink tea or coffee (or both):
\[
|T \cup C| = 96 + 132 - 64 = 164
\]
Thus, the number of people who drink neither is:
\[
236 - 164 = 72
\]

### Final Answers:
- **96** people drink tea.
- **132** people drink coffee.
- **72** people drink neither tea nor coffee.

Would you like further clarification on any part? Here are some related questions:

1. How would the results change if the total number of surveyed people increased?
2. What is the probability that a randomly selected person drinks both tea and coffee?
3. Can you represent the results using a Venn diagram?
4. How would the problem change if a person could choose only one option?
5. What happens if we increase the number of beverage choices?

**Tip**: Set theory and Venn diagrams are excellent tools for solving problems involving overlapping groups.

A survey asks 236 people 'What beverage do you drink in the morning?', and offers choices: Coffee only, Tea only, Both tea and coffee, Neither tea nor coffee. Suppose 32 report tea only, 68 report coffee only, and 64 report both. How many people drink tea in the morning? How many people drink coffee in the morning? How many people drink neither tea nor coffee?

Solve a set theory problem involving 236 people who report their morning beverage preferences: tea, coffee, both, or neither. Learn how to calculate the number of people who drink each option using union and intersection of sets. This problem is suitable for students in Grades 9-11 and covers core concepts in set theory.

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