Finding Students Who Like Both Music and Dancing

This problem involves finding the number of students who like both music and dancing, given the total number of students, those who like music, those who like dancing, and those who like neither. We can solve this using the principles of set theory or by visualizing it with a Venn diagram.

Understanding the Given Information

Let's break down the numbers provided in the question:

• Total number of students in the class: 60
• Number of students who like music: 45
• Number of students who like dancing: 50
• Number of students who like neither music nor dancing: 5

Calculating Students Liking at Least One Activity

First, we can find the number of students who like at least one of the activities (music or dancing). These are the students who are not in the group that likes neither. The total number of students is the sum of students who like at least one activity and students who like neither.

Number of students liking at least one activity = Total students - Number of students liking neither

Number of students liking at least one activity = $\text{60} - \text{5} = \text{55}$

So, 55 students like either music, or dancing, or both music and dancing.

Using the Principle of Inclusion-Exclusion

We can use the formula for the union of two sets. Let M be the set of students who like music, and D be the set of students who like dancing. The number of students who like at least one activity is the number of students in the union of sets M and D, denoted as $|M \cup D|$. The number of students who like both activities is the number of students in the intersection of sets M and D, denoted as $|M \cap D|$.

The principle of inclusion-exclusion for two sets states:

$|M \cup D| = |M| + |D| - |M \cap D|$

We know:

• $|M \cup D|$ = Number of students liking at least one activity = 55
• $|M|$ = Number of students who like music = 45
• $|D|$ = Number of students who like dancing = 50

We want to find $|M \cap D|$, which is the number of students who like both music and dancing. Let's substitute the known values into the formula:

$\text{55} = \text{45} + \text{50} - |M \cap D|$

$\text{55} = \text{95} - |M \cap D|$

Now, we can rearrange the equation to solve for $|M \cap D|$:

$|M \cap D| = \text{95} - \text{55}$

$|M \cap D| = \text{40}$

Therefore, the number of students in the class who like both music and dancing is 40.

Summary Table

Conclusion

By using the information about the total students, those who like music, those who like dancing, and those who like neither, we calculated that 55 students like at least one activity. Applying the inclusion-exclusion principle, we found that 40 students like both music and dancing.

Revision Table: Students and Activities

Here’s a quick look back at the key numbers and what they represent in this problem about students liking music and dancing.

Additional Information: Set Theory Basics for Overlapping Groups

Problems like this, dealing with groups of people or items that have overlapping characteristics, are commonly solved using set theory concepts. The total collection is often called the universal set (U). The subgroups are represented as sets (like M for Music, D for Dancing).

Key concepts used here:

• Union ($A \cup B$): Represents elements that are in set A, or in set B, or in both. In this problem, students who like music OR dancing OR both.
• Intersection ($A \cap B$): Represents elements that are in both set A and set B. In this problem, students who like music AND dancing.
• Complement ($A'$): Represents elements in the universal set that are NOT in set A. In this problem, students who do not like music.
• Neither ($A' \cap B'$): Represents elements that are neither in set A nor in set B. This is the complement of the union: $(A \cup B)'$. So, $|A' \cap B'| = |U| - |A \cup B|$.
• Inclusion-Exclusion Principle: For two sets A and B, $|A \cup B| = |A| + |B| - |A \cap B|$. This formula accounts for the overlap ($|A \cap B|$) being counted twice when you add $|A|$ and $|B|$. By subtracting the intersection once, you get the correct count for the union.

Understanding these basic set theory principles is very helpful for solving problems involving surveys or data with overlapping categories.