Question

65% of the students in a class like cartoon movies, 70% like horror movies and 75% like war movies. What is the smallest percentage of students liking all the three types of movies?
(a) 30%
(b) 20%
(c) 10%
(d) 5%
(e) None of these

Answer 1

Hint: We start solving the problem by assigning the variables for the events of liking cartoon movies, horror movies and war movies. We then find the total percentage of students who are not liking cartoon movies, horror movies and war movies by subtracting the given values from 100%. We then find the maximum number of students who don't all the three movies by adding the obtained values. We then subtract this value from 100% to get the required value of the smallest percentage of students liking all the three types of movies.

Complete step by step answer:
According to the problem, we are given that 65% of the students in a class like cartoon movies, 70% like horror movies and 75% like war movies. We need to find the smallest percentage of students liking all the three types of movies.
Let us assume ‘A’ be the event of liking cartoon movies, ‘B’ be the event of liking horror movies and ‘C’ be the event of liking war movies, then we get $\overline{A}$ as the event of not liking cartoon movies, $\overline{B}$ as the event of not liking horror movies and $\overline{C}$ as the event of not liking war movies.
So, we have $n\left( A \right)=65\%$, $n\left( B \right)=70\%$ and $n\left( C \right)=75\%$.
We know that $n\left( \overline{A} \right)=100\%-n\left( A \right)\Leftrightarrow n\left( \overline{A} \right)=35\%$.
$\Rightarrow n\left( \overline{B} \right)=100\%-n\left( B \right)\Leftrightarrow n\left( \overline{B} \right)=30\%$.
\[\Rightarrow n\left( \overline{C} \right)=100\%-n\left( C \right)\Leftrightarrow n\left( \overline{C} \right)=25\%\].
Now, let us find the maximum percentage of students not liking all the movies which is the sum of $n\left( \overline{A} \right)$, $n\left( \overline{B} \right)$, $n\left( \overline{C} \right)$.
So, we get the maximum percentage of students not liking all the movies as $35\%+30\%+25\%=90\%$.
We know that the remaining 10% of the people will like all three types of movies which will be the required minimum percentage of students.

So, the correct answer is “Option c”.

Note: We should know that the maximum number of students that were not liking all the movies will be possible only if the events $\overline{A}$, $\overline{B}$ and $\overline{C}$ are disjoint or distinct events. We can also solve this problem by taking a class of students with strength 100 and then considering all the given cases will lead us to the required answer. Similarly, we can expect the problems to find the maximum number of students liking all the three movies.