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? How many can speak both Hindi and Bengali?

Answer 1

$Let$

Total number of People $n\left(P\right)=1000$

People who speak Hindi $n\left(H\right)=750$

People who speak Bengali $n\left(B\right)=400$

We have

$P=n(H\cup B)$

$\Rightarrow n\left(P\right)=n\left(H\right)+n\left(B\right)–n(H\cap B)$

$\Rightarrow 1000=750+400–n(H\cap B)$

$n(H\cap B)=150$

∴ $150Peoplecanspeakbothlanguages.$

$H=(H–B)\cup (H\cap B)$ (union is disjoint)

∴ $n\left(H\right)=n(H–B)+n(H\cap B)$

$750=n(H–B)+150$

$n(H–B)=600$

∴ $600peoplespeakHindi.$

$B=(B–H)\cup (H\cap B)$ (union is disjoint)

∴ $n\left(B\right)=n(B–H)+n(H\cap B)$

$400=n(B–H)+150$

$n(B–H)=250$

∴ $250peoplespeakBengali.$