Question

In a group of $950$ persons, $750$ can speak Hindi and $460$ can speak English. Find how many can speak both Hindi and English.

Answer 1

Hint: This question is based on the concept of set theory. Try to recall all the formula of set theory which relate the union and intersection of the two sets. In this question, one set is of those persons who can speak Hindi and the other set is of those persons who can speak English.

Before proceeding with this question, we must know the formula that is required to solve this question. Let us consider two sets, set $A$ and set $B$. In set theory, we have a formula that relates the number of terms in the intersection and the union of the two sets $A$ and $B$. The formula is,
$n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)...............\left( 1 \right)$
For this question, let us denote set $A$ for those persons who can speak Hindi and let us denote set $B$ for those persons who can speak English.
It is given that there are a total $950$ persons in the group. Hence, we can say,
$n\left( A\cup B \right)=950..........\left( 2 \right)$
It is given that $750$ persons can speak Hindi. Hence, we can say,
$n\left( A \right)=750.............\left( 3 \right)$
Also, it is given that $460$ persons can speak English. Hence, we can say,
$n\left( B \right)=460.............\left( 4 \right)$
In the question, we have to find the number of persons that can speak both Hindi and English. This means, we have to calculate the number of elements in the intersection of the sets $A$ and $B$. Substituting $n\left( A\cup B \right)$ from equation $\left( 2 \right)$, $n\left( A \right)$ from equation $\left( 3 \right)$ and $n\left( B \right)$ from equation $\left( 4 \right)$ in equation $\left( 1 \right)$, we get,
$\begin{aligned} & 950=750+460-n\left( A\cap B \right) \\\ & \Rightarrow n\left( A\cap B \right)=750+460-950 \\\ & \Rightarrow n\left( A\cap B \right)=260 \\\ \end{aligned}$
Hence, the number of persons that can speak both Hindi and English is $260$.

Note: In this question, it may become easier for a student to analyze the data given in the question if he/she analyzes this question with the help of a venn diagram. So, one can also do this question with the help of a venn diagram to avoid any small mistakes.