Question

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

Answer 1

Hint: For solving this problem we assume that the total number of persons as set \['\mu '\] and persons who speak Hindi as \['A'\] and the persons who speak English as \['B'\]. The number of persons who are speaking Hindi only is calculated by \[A-\left( A\cap B \right)\]. We know the value of \['A'\] but for finding the value of \[\left( A\cap B \right)\] we use the relation \[\mu =A+B-\left( A\cap B \right)\].

Complete step by step answer:
Let us assume that total number of persons present as
\[\mu =950\]
Let us assume that the persons who speak Hindi as \['A'\], so we can take
\[A=750\]
Let us assume that the persons who speak English as \['B'\], so we can take
\[B=460\]
We know that the relation of \['\mu ,A,B'\] is
\[\mu =A+B-\left( A\cap B \right)\]
Now, by substituting the values of \['\mu ,A,B'\] in above equation we get
\[\begin{align}
  & \Rightarrow 950=750+460-\left( A\cap B \right) \\
 & \Rightarrow \left( A\cap B \right)=1210-950 \\
 & \Rightarrow \left( A\cap B \right)=260 \\
\end{align}\]
Now, let us assume that the number of persons who speak Hindi only as \['x'\].
We know that the number of persons who speak Hindi is given as
\[x=A-\left( A\cap B \right)\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow x=750-260 \\
 & \Rightarrow x=490 \\
\end{align}\]

Therefore, we can say that there are 490 people who speak Hindi only.

Note: This problem can be solved in another method.
Let us assume that total number of persons present as
\[\mu =950\]
Let us assume that the persons who speak Hindi as \['A'\], so we can take
\[A=750\]
Let us assume that the persons who speak English as \['B'\], so we can take
\[B=460\]
Now, let us assume that the number of persons who speak Hindi only as \['x'\].
We know that the number of persons who speak Hindi is given as
\[x=A-\left( A\cap B \right)\]
We know that the relation of \['\mu ,A,B'\] is
\[\mu =A+B-\left( A\cap B \right)\]
The above equation is modified as
\[\Rightarrow A-\left( A\cap B \right)=\mu -B\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow x=950-460 \\
 & \Rightarrow x=490 \\
\end{align}\]
Therefore, we can say that there are 490 people who speak Hindi only.