Question

In a group of $500$ people $200$ can speak Hindi alone while only $125$ can speak English alone. The number of people can speak both English and Hindi is:
A) $175$
B) $325$
C) $300$
D) $375$

Answer 1

Hint: We will add the people who can speak Hindi only and the people who can speak English only and then subtract the sum from the total number of people from the group to find the number of people who can speak both languages.

Complete step-by-step solution:
Given: It is given that there are $500$ people in a group. Out of these people $200$ people are present who can speak Hindi only and $125$ people are present who can speak English only.
To Find: We have to find the number of people who can speak both English and Hindi.
To find the people who can speak both the languages we will apply the formula here, which is:-
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) + P\left( {A \cap B} \right)$
Where,
$P\left( {A \cup B} \right)$ represents all the people of group,
$P\left( A \right)$ represents the people who can speak Hindi,
$P\left( B \right)$ represents the people who can speak English only,
$P\left( {A \cap B} \right)$ represents the people who can speak both Hindi and English.
And we are given the values of:-
$P\left( {A \cup B} \right) = 500$
$P\left( A \right) = 200$
$P\left( B \right) = 125$
And $P\left( {A \cap B} \right)$ is to be found.
Now, we will put all the values given in the formula.
$500 = 200 + 125 + P\left( {A \cap B} \right)$
Now, solve this for $P\left( {A \cap B} \right)$.
$
  500 - 200 - 125 = P\left( {A \cap B} \right) \\
  175 = P\left( {A \cap B} \right) \\
$
So, there are 175 people in the group out of 500 people who can speak both Hindi and English.
Therefore, option (A) is correct.
Note: the formula we have applied here can be applied in different way, that is $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$ but when $P\left( A \right)$ and $P\left( B \right)$ are given completely , now we have applied formula $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) + P\left( {A \cap B} \right)$ because we have mentioned properly that $P\left( A \right)$ is the people who can speak Hindi only and $P\left( B \right)$ is the people who can speak English and there was nothing common between $P\left( A \right)$ and $P\left( B \right)$.