A class survey found that 50 students watched the Olympics on television on Saturday and 54 watched on Sunday. Of those who watched the Olympics on only one of these days, 8 chose Saturday and 12 chose Sunday. If every student watched at least one of these days, how many students are in the class?

Let's break this down using a Venn diagram approach with sets.

Let:
- \( S \) represent the set of students who watched the Olympics on Saturday.
- \( U \) represent the set of students who watched on Sunday.

We are given:
- \( |S| = 50 \) (students watched on Saturday),
- \( |U| = 54 \) (students watched on Sunday),
- 8 students watched only on Saturday,
- 12 students watched only on Sunday.

Let \( x \) be the number of students who watched the Olympics on both Saturday and Sunday. From this, we can set up the following expressions:

- Students who watched only on Saturday: \( |S \setminus U| = 8 \),
- Students who watched only on Sunday: \( |U \setminus S| = 12 \),
- The total number of students who watched on both days: \( x \).

Now, we know the total number of students who watched on Saturday includes those who watched only on Saturday and those who watched on both days, so:
\[
|S| = (S \setminus U) + x = 8 + x = 50
\]
Solving for \( x \):
\[
x = 50 - 8 = 42
\]
Thus, 42 students watched on both Saturday and Sunday.

Similarly, the total number of students who watched on Sunday includes those who watched only on Sunday and those who watched on both days:
\[
|U| = (U \setminus S) + x = 12 + x = 54
\]
Substituting \( x = 42 \):
\[
12 + 42 = 54
\]
This is consistent, confirming that \( x = 42 \) is correct.

Finally, the total number of students in the class is the sum of those who watched only on Saturday, only on Sunday, and on both days:
\[
\text{Total number of students} = (S \setminus U) + (U \setminus S) + x = 8 + 12 + 42 = 62
\]

Therefore, the total number of students in the class is **62**.

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Would you like a further breakdown or have any questions about this?

Here are 5 related questions to consider:
1. How would the solution change if 5 students watched on both days instead of 42?
2. What if 10 students didn’t watch on either day?
3. How can you represent this situation using set notation and operations?
4. Can you find the number of students who watched exactly one day?
5. What happens if some students watched neither day?

**Tip:** When solving set problems, try to clearly define overlapping sections, as it often simplifies calculations.

Solve a set theory problem where 50 students watched the Olympics on Saturday, 54 on Sunday, and some watched both days. Using Venn diagrams and algebraic methods, find how many students are in the class. This problem is ideal for students in Grades 8-10.

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