This problem involves understanding sets and their relationships, specifically using the principle of inclusion-exclusion to determine the number of students who like romantic movies in a class. We will break down the information given and use a step-by-step approach to solve it.

Students and Their Movie Preferences

We are given information about a class of 100 students and their preferences for romantic and comedy movies. Let's define the key terms and the given data:

• Total students: The entire group we are considering is 100 students.
• Students who like romantic movies: Let's denote this set as \(R\).
• Students who like comedy movies: Let's denote this set as \(C\).
• Students who like both romantic and comedy movies: This is the intersection of the two sets, denoted as \(R \cap C\).
• Students who like neither romantic nor comedy movies: These are students outside both sets.

Let's list the numerical information provided:

We are also given a relationship between the number of students who like romantic movies and comedy movies:

• The number of students who like romantic movies is twice the number of students who like comedy movies. This can be written as \(N(R) = 2 \times N(C)\).

Calculating Movie Preferences

To find the number of students who like romantic movies, we first need to determine how many students like at least one type of movie (either romantic, comedy, or both). This is the union of the two sets, \(R \cup C\).

Students Liking At Least One Movie Type

We know the total number of students and the number of students who like neither type of movie. The students who like at least one type of movie are the total students minus those who like neither.

Total students \( = N(R \cup C) + \) Students liking neither

We have:

• Total students \( = 100 \)
• Students liking neither \( = 30 \)

So, the number of students who like at least one movie type is:

\( N(R \cup C) = \text{Total students} - \text{Students liking neither} \)

\( N(R \cup C) = 100 - 30 = 70 \)

Therefore, 70 students like either romantic movies, comedy movies, or both.

Applying the Inclusion-Exclusion Principle

The principle of inclusion-exclusion for two sets states:

\( N(R \cup C) = N(R) + N(C) - N(R \cap C) \)

We already know:

• \( N(R \cup C) = 70 \) (calculated above)
• \( N(R \cap C) = 20 \) (given in the problem)

Substitute these values into the formula:

\( 70 = N(R) + N(C) - 20 \)

Now, we can find the sum of students who like romantic movies and comedy movies by adding 20 to both sides:

\( N(R) + N(C) = 70 + 20 \)

\( N(R) + N(C) = 90 \)

Solving for Romantic Movies

We have two pieces of information now:

1. \( N(R) + N(C) = 90 \)
2. \( N(R) = 2 \times N(C) \) (given in the problem)

We can substitute the second equation into the first equation to solve for \(N(C)\):

Substitute \( N(R) = 2N(C) \) into \( N(R) + N(C) = 90 \):

\( 2N(C) + N(C) = 90 \)

\( 3N(C) = 90 \)

Divide by 3 to find \(N(C)\):

\( N(C) = \frac{90}{3} \)

\( N(C) = 30 \)

So, 30 students like comedy movies.

Now, we can find the number of students who like romantic movies using the relationship \( N(R) = 2 \times N(C) \):

\( N(R) = 2 \times 30 \)

\( N(R) = 60 \)

Therefore, 60 students in the class like romantic movies.

Summary of Findings

Let's summarize all the calculated and given values for clarity:

The calculations confirm that 60 students like romantic movies.