Question

In a group of $800$ people, $550$ can speak Hindi and $450$ can speak English. How many people can speak both Hindi and English $?$
(A) $100$
(B) $150$
(C) $200$
(D) None of these

Answer 1

Hint: We know that the total number of people is equal to the number of people who speak Hindi plus the number of people who speak English minus number of people who speak both Hindi and English. Therefore, with the help of this formula of \[A\] union $B$ we can easily find out \[A\] intersection $B$ .

Formula used: We know that $A \cup B = A + B - \left( {A \cap B} \right)$ .

Complete step by step solution: The total number of people in a group is $800$ . The number of people who can speak Hindi is $550$ and the total number of people who can speak English is $450$ .
Let’s $H$ denote the set of people who speak Hindi. Therefore, $n\left( H \right) = 550$ . Let’s $E$ denote the set of people who speak English. Therefore, $n\left( E \right) = 450$ .
Therefore, we can write the total number of people as $n\left( {H \cup E} \right) = 800$ .
Now we know the formula $A \cup B = A + B - \left( {A \cap B} \right)$ . From this formula we can write $n\left( {H \cup E} \right) = n\left( H \right) + n\left( E \right) - n\left( {H \cap E} \right)$ where $n\left( {H \cap E} \right)$ is the number of people who speaks both Hindi and English.
Now, put the value in the formula $n\left( {H \cup E} \right) = n\left( H \right) + n\left( E \right) - n\left( {H \cap E} \right)$
$ \Rightarrow n\left( {H \cup E} \right) = n\left( H \right) + n\left( E \right) - n\left( {H \cap E} \right)$
$ \Rightarrow 800 = 550 + 450 - n\left( {H \cap E} \right) $
Now, we have to find $n\left( {H \cap E} \right)$
$ \Rightarrow n\left( {H \cap U} \right) = 550 + 450 - 800$
$\Rightarrow n\left( {H \cap U} \right) = 1000 - 800 = 200$
Therefore, the total number of people who speak both Hindi and English are $200$ .

Hence, the correct option is (C).

Note: Students should be able to apply the formula of \[A\] union $B$ in this question and they should be able to identify what value we should put in the formula. So, students should be careful when putting the values in the formula. This question can also be solved by using a Venn diagram.