Question

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Answer 1

We will use the formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ and put in the values by assuming A and B to be the event as given in the question and get the required answer.

Complete step-by-step answer:
Let A be the set of people who can speak English.
So, n (A) = 250, where n is the number of elements in the set.
Let B be the set of people who can speak Hindi.
So, n (B) = 200, where n is the number of elements in the set.
Now, we need to find a number of people who can speak both, which will be in the set $n(A \cap B)$.
Now, we know that we have formula involving union and intersection of two sets which is given by the following expression:-
$\Rightarrow n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Now, we will put in the values since total group is of 400, so $n(A \cup B) = 400$
Putting this and the values of individual sets, we will then get:-
$\Rightarrow 400 = 250 + 200 - n(A \cap B)$
Simplifying the calculations on the right hand side by adding 250 and 200 to get the following:-
$\Rightarrow 400 = 450 - n(A \cap B)$
Taking 400 from addition in the left hand side to subtraction in the right hand side and taking the term which needs to be found on the right hand side in addition, to get the following expression:-
$\Rightarrow n(A \cap B) = 450 - 400$
Simplifying the calculation on the right hand side in the above equation, we will then obtain the following expression:-
$\Rightarrow n(A \cap B) = 50$

Hence, there are 50 people who can speak both Hindi and English.

Note:
The students must commit to memory the following formulas:-
$\Rightarrow n(A \cup B) = n(A) + n(B) - n(A \cap B)$
The students must also note that they can easily think and imagine this formula as well. If we have 200 people who can either drink milk and coca – cola, now there will definitely be some people who drink both. Now, if 150 drink milk, 100 drink coca – cola, then we will have both, they will also contain people who drink both, since there are only 200 people in all, hence, we subtract and get the answer.
The students may verify the same using the Venn diagram as well.