Question

There are $600$ students in a school. If $400$ of them can speak Telugu, and $300$ can speak Hindi, then the number of students who can speak both Telugu and Hindi are
A. $100$
B. $200$
C. $300$
D. $400$

Answer 1

Hint: In this question, we are to find the number of students who can speak both languages. For this, we use the formula and by substituting the given values into it, we get the required value. Here the total students represent the union and the students who can speak both represents the intersection.

Formula Used: Set: A collection of objects in which it is possible to decide whether a given object belongs to the collection or not is said to be a set. Those objects are nothing but the elements in the set.
Sets are represented in two ways: The roaster method and the set builder form
Roaster method:
The set of vowels – \[\left\{ a,\text{ }e,\text{ }i,\text{ }o,\text{ }u \right\}\]
The set of natural numbers – \[\left\{ 1,\text{ }2,\text{ }3,\text{ }\ldots . \right\}\]
Set builder form:
The set of vowels – \[\left\{ x:x\text{ }is\text{ }a\text{ }vowel\text{ }in\text{ }English\text{ }alphabet \right\}\]
The set of natural numbers – \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number \right\}\]
Some of the important mathematical sets:
$N$ - the natural number’s set - $N=\{1,2,3...\}$
$Z$- the set of integers - $Z=\{0,\pm 1,\pm 2,\pm 3,...\}$
If two sets $A,B$ where the set $A$ is said to be the subset of $B$ i.e., $A\subseteq B$ then every element of $A$ is in the set $B$ and the set $A$ is said to be the proper subset of $B$ i.e., \[A\subset B\] then $A\subseteq B$ and $A\ne B$.
Some of the important set operations:
$\begin{align}
 & n(A\cup B)=n(A)+n(B)-n(A\cap B) \\
 & n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C) \\
\end{align}

Complete step by step solution: Given that, there are $600$ students in a school. Here $400$ of them can speak Telugu and $300$ can speak Hindi.
Consider the set of students who speak Telugu as $T$ and the set of students who speak Hindi as $H$.
Then, the number of students who speak Telugu is $n(T)=400$, the number of students who speak Hindi is $n(H)=300$, and the total number of students in the school is $n(T\cup H)=600$.
So, the number of students who can speak both Telugu and Hindi is represented by $n(T\cap H)$
From the theorem we have
 $n(A\cup B)=n(A)+n(B)-n(A\cap B)$
Then,
$\begin{align}
  & n(T\cup H)=n(T)+n(H)-n(T\cap H) \\
 & \Rightarrow n(T\cap H)=n(T)+n(H)-n(T\cup H) \\
\end{align}$
On substituting the given values, we get
$n(T\cap H)=400+300-600=100$
Therefore, the number of students who can speak both Telugu and Hindi is $100$ students.

Option ‘A’ is correct

Note: Here, we have given all the required values for calculating the number of students who can speak both Telugu and Hindi. So, we can easily calculate by substituting them in the predefined formula.