Question: There are 35 students in art class and 57 students in dance class and 12 students are enrolled in both activities . Find the number of students who are either in art class or in dance class, a) when two classes meet at different hours b) when two classes meet at the same hour.
1. **Problem Statement:**
Find the number of students who are either in art class or in dance class given:
- Number of students in art class $= 35$
- Number of students in dance class $= 57$
- Number of students in both classes $= 12$
2. **Formula Used:**
The number of students in either art or dance (union) is given by the formula:
$$|A \cup D| = |A| + |D| - |A \cap D|$$
where $|A|$ is the number in art, $|D|$ is the number in dance, and $|A \cap D|$ is the number in both.
3. **Case a) When classes meet at different hours:**
Since classes meet at different hours, students can attend both, so the formula applies directly.
Calculate:
$$|A \cup D| = 35 + 57 - 12 = 80$$
4. **Case b) When classes meet at the same hour:**
Students cannot attend both classes simultaneously, so the number in both is effectively zero.
Calculate:
$$|A \cup D| = 35 + 57 - 0 = 92$$
**Final answers:**
- a) Number of students in either class when classes meet at different hours: $80$
- b) Number of students in either class when classes meet at the same hour: $92$