Find Hindi Only, Bengali Only and Both Speakers
Question:
In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali only? How many can speak both Hindi and Bengali?
Solution
\[ n(H\cup B)=1000 \] \[ n(H)=750,\quad n(B)=400 \] \[ n(H\cup B)=n(H)+n(B)-n(H\cap B) \] \[ 1000=750+400-n(H\cap B) \] \[ 1000=1150-n(H\cap B) \] \[ n(H\cap B)=150 \]Hindi only
\[ n(H)-n(H\cap B) \] \[ =750-150 \] \[ =600 \]Bengali only
\[ n(B)-n(H\cap B) \] \[ =400-150 \] \[ =250 \]Both Hindi and Bengali
\[ =150 \]Hence,
\[ \boxed{\text{Hindi only }=600} \] \[ \boxed{\text{Bengali only }=250} \] \[ \boxed{\text{Both Hindi and Bengali }=150} \]