Problem Analysis

We are given a total of 100 students and information about their preferences for romantic and comedy movies.

• Total students: $ |U| = 100 $
• Students liking neither romantic nor comedy movies: $ |(R \cup C)'| = 30 $
• Students liking romantic movies is twice those liking comedy movies: $ |R| = 2 \times |C| $
• Students liking both romantic and comedy movies: $ |R \cap C| = 20 $
• We need to find the number of students who like romantic movies: $ |R| $.

Calculating Students Liking at Least One Genre

The number of students who like at least one of the two genres (romantic or comedy) can be found by subtracting those who like neither from the total number of students.

Number of students liking at least one genre: $ |R \cup C| = |U| - |(R \cup C)'| $

$ |R \cup C| = 100 - 30 = 70 $

Setting Up Equations

Using the principle of inclusion-exclusion for two sets:

$ |R \cup C| = |R| + |C| - |R \cap C| $

Substitute the known values:

$ 70 = |R| + |C| - 20 $

Rearranging the equation gives:

$ |R| + |C| = 70 + 20 $

$ |R| + |C| = 90 $

Solving for Romantic Movie Fans

We have two equations:

1. $ |R| + |C| = 90 $
2. $ |R| = 2 \times |C| $

Substitute the second equation into the first:

$ (2 \times |C|) + |C| = 90 $

Combine the terms:

$ 3 \times |C| = 90 $

Solve for the number of students who like comedy movies:

$ |C| = \frac{90}{3} = 30 $

Now, use the relationship $ |R| = 2 \times |C| $ to find the number of students who like romantic movies:

$ |R| = 2 \times 30 = 60 $

Therefore, 60 students like romantic movies.