(i)there are 30 students who neither like romantic movies nor comedy movies,
(ii)the number of students who like romantic movies is twice the number of students who like comedy movies, and
(iii)the number of students who like both romantic movies and comedy movies is 20.
How many students in the class like romantic movies?
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:
- $ |R| + |C| = 90 $
- $ |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.
Important Questions from Numerical Reasoning
- Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
