Question

In a group of 500 students, there are 475 students who can speak Hindi and 200 can speak Bengali. What is the number of students who can speak Hindi only?
A. 275
B. 300
C. 325
D. 350

Answer 1

Hint: We will apply the relation of $n\left( {A \cup B} \right)$, $n\left( A \right)$, $n\left( B \right)$, and $n\left( {A \cap B} \right)$ to find the number of people who speak both languages. Then we will subtract the number of people who speak both languages from the total number of people who speak Hindi to get the number of people who speak Hindi only.

Formula Used:
$n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$

Complete step by step solution:
Given that, the total number of students is 500 and 475 students can speak Hindi, and 200 can speak Bengali.
Let $n\left( H \right)$ represent the number of students who speak Hindi.
Let $n\left( B \right)$ represent the number of students who speak Bengali.
So, $n\left( {H \cup B} \right) = 500$
Since 475 students can speak Hindi. So $n\left( H \right) = 475$.
Since 200 can speak Bengali. So $n\left( B \right) = 200$.
Now we will apply the formula $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$ to calculate the number of students who can speak both languages.
$n\left( {H \cup B} \right) = n\left( H \right) + n\left( B \right) - n\left( {H \cap B} \right)$
Substitute the value of $n\left( {H \cup B} \right)$, $n\left( H \right)$, and $n\left( B \right)$
$ \Rightarrow 500 = 475 + 200 - n\left( {H \cap B} \right)$
$ \Rightarrow 500 = 675 - n\left( {H \cap B} \right)$
$ \Rightarrow 500 - 675 = - n\left( {H \cap B} \right)$
$ \Rightarrow - 175 = - n\left( {H \cap B} \right)$
$ \Rightarrow n\left( {H \cap B} \right) = 175$
The number of students who speak both is 175.
The number of students who speak Hindi only is
$ = {\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{students}}\,{\rm{who}}\,{\rm{speak}}\,{\rm{Hindi}} - {\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{students}}\,{\rm{who}}\,{\rm{speak}}\,{\rm{both}}\,{\rm{languages}}$
$ = 475 - 175$
$ = 300$

Option ‘B’ is correct

Note: In the question you have to find the number of students who can speak only Hindi. It means the number of students who can speak Hindi but are unable to speak Bengali.
To this type of question, you have to consider the number of students who can speak both languages. Then subtract the number of students who can speak both languages from the number of students who can speak Hindi.