In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

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In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Solution:

Let’s consider the set of people who speak Hindi as H, and
the set of people who speak English as E.

It is known that

$n(H~\cup ~E)\text{ }=\text{ }400$

$n\left( H \right)\text{ }=\text{ }250$

$n\left( E \right)\text{ }=\text{ }200$

This can be written as

$n(H~\cup ~E)\text{ }=~n\left( H \right)\text{ }+~n\left( E \right)\text{ }~-n(H~\cap ~E)$

Now, substitute the values

$400\text{ }=\text{ }250\text{ }+\text{ }200\text{ -}~n(H~\cap ~E)$

Calculating further

$400\text{ }=\text{ }450\text{ -}~n(H~\cap ~E)$

As a result, we get

$n(H~\cap ~E)\text{ }=\text{ }450-400$

$n(H~\cap ~E)\text{ }=\text{ }50$

As a result, 50 people can speak both Hindi and English.

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