644 students were surveyed total. 283 of the students drink tea. 291 drink coffee. 104 drink both tea and coffee. 168 drink both tea and soda. 143 drink both coffee and soda. 37 drink all three types of drinks. 53 don't drink any of these. How many drink only soda and why

We are given that:

- Total number of students surveyed = 644
- Number of students who drink tea = 283
- Number of students who drink coffee = 291
- Number of students who drink both tea and coffee = 104
- Number of students who drink both tea and soda = 168
- Number of students who drink both coffee and soda = 143
- Number of students who drink all three drinks (tea, coffee, soda) = 37
- Number of students who don't drink any of these drinks = 53

We need to find how many students drink only soda.

### Step-by-Step Breakdown:

1. **Total students involved in drinking any drink**:  
   Since 53 students don't drink any of these drinks, the remaining students (644 - 53) do drink at least one of the drinks.  
   \[
   644 - 53 = 591 \quad \text{students drink at least one of the drinks}.
   \]

2. **Setting up a formula for the number of students who drink soda**:  
   We need to calculate the total number of students who drink soda. Let's denote the number of students who drink only soda as \(x\). To calculate this, we need to break down the other groups first.

3. **Using inclusion-exclusion principle**:
   - Students who drink **tea** and **coffee** = 104
   - Students who drink **tea** and **soda** = 168
   - Students who drink **coffee** and **soda** = 143
   - Students who drink **all three drinks** = 37
   
   The overlapping groups have to be subtracted accordingly to avoid double counting.

4. **Students who drink only tea or only coffee (subtracting overlapping groups)**:
   - **Only tea**: \( 283 - (104 + 168 - 37) \) 
   - **Only coffee**: \( 291 - (104 + 143 - 37) \)
   
5. The remaining number of students should give us the number who drink **only soda**. This will involve solving \(x\) based on the exclusions from previous totals. 

I'll calculate this next.The number of students who drink only soda is **121**.

This result comes from subtracting the number of students who drink tea, coffee, and their various overlaps from the total students who drink any of the beverages. 

Would you like more details on how this was calculated, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How many students drink only tea?
2. How many students drink only coffee?
3. How many students drink both tea and soda, but not coffee?
4. How can the inclusion-exclusion principle be applied to similar problems?
5. What would change if an additional category of drink was introduced?

**Tip:** In problems involving overlapping sets, the inclusion-exclusion principle helps avoid double-counting overlapping groups.

644 students were surveyed total. 283 of the students drink tea. 291 drink coffee. 104 drink both tea and coffee. 168 drink both tea and soda. 143 drink both coffee and soda. 37 drink all three types of drinks. 53 don't drink any of these. How many drink only soda and why?

Explore a step-by-step solution to determine how many students drink only soda using the inclusion-exclusion principle. This problem includes overlapping sets for tea, coffee, and soda, and calculates the number of students drinking only soda. Suitable for Grades 10-12.

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