Question

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

Answer 1

Hint: To solve this question to find out how many people speak both Hindi and English, we should know about the basic concepts of sets. It states that for any two events, $A$ and $B$, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$, where $n\left( {} \right)$ represents the number of elements in that set. For example $n\left( A \right)$ represents the number of elements in $A$, $n\left( B \right)$ represents the number of elements in $B$, $n\left( A\cap B \right)$ represents the number of elements present in both and $n\left( A\cup B \right)$ represents the total number of elements taking part in the event. By using these relations, we can solve the question.

Complete step-by-step answer:

In this question, we have been asked to find the number of people who can speak both Hindi and English. We have been given that out of 950 persons, 750 can speak Hindi and 460 can speak English. Let us consider speaking Hindi as an event $A$, so the number of people speaking Hindi $=n\left( A \right)=750$.
And, consider speaking English as an event $B$, then the number of English speaking people $=n\left( B \right)=460$.
Now, we know that there were a total of 950 people in the group, so they are $=n\left( A\cup B \right)=950$.
We know that in sets, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$. So, we will substitute the values of $n\left( A\cup B \right)$, $n\left( A \right)$ and $n\left( B \right)$ in this equation and on doing so, we get,
$\begin{align}
  & 950=750+460-n\left( A\cap B \right) \\
 & \Rightarrow 950=1210-n\left( A\cap B \right) \\
\end{align}$
We will take the unknown units to the left hand side and all the numerical values on the right hand side. By doing so, we get,
$\begin{align}
  & n\left( A\cap B \right)=1210-950 \\
 & \Rightarrow n\left( A\cap B \right)=260 \\
\end{align}$
Therefore, we get the value of the number of people who speak both Hindi and English are 260.

Note: The possible mistakes that one can make while solving this question are the calculation mistakes by not keeping the signs of the numerical values in mind. Also, one can make a mistake by writing $-n\left( A\cap B \right)$ as $+n\left( A\cap B \right)$, which will give us the wrong answer.