Question: In a group of \[1000\] people, each can speak either Hindi or English. There are \[750\] people who ...

Question

In a group of $1000$ people, each can speak either Hindi or English. There are $750$ people who can speak Hindi and $400$ can speak English. The number of people who can speak Hindi only is
A) $300$
B) $400$
C) $600$
D) $450$

Answer 1

Solution

We can solve this problem by using a general substitution method and also using the Venn diagram method. Given in the problem is the information about a number of people speaking two languages in a given number of groups of people. We have to find a number of people for the required result by using probability relations. Then, using the formula and given information we can find the number of students who can speak both and then the number of people who can speak only Hindi.

Formula used: We will apply the formula of $n(H \cap E) = n(H) + n(E) - n(H \cup E)$ .
Here.
$H$ means number of students who can speak Hindi and
$E$ means the number of students who can speak English.

Complete step-by-step answer:
It is given that; total number of people is $1000$ .
Number of people who can speak Hindi is $750$ .
Number of people who can speak English is $400$ .
We have to find the number of people who can speak Hindi only.
So, as per the given information
$n(H \cup E) = 1000$
$n(H) = 750$
$n(E) = 400$
Let us consider the number of students who can speak Hindi and English is $x$ that is $n(H \cap E) = x$ .
First, we have to find the value of $n(H \cap E)$ .
We know that,
$n(H \cap E) = n(H) + n(E) - n(H \cup E)$
Substitute the values in the above formula we get,
$\Rightarrow$$$n(H \cap E) = 750 + 400 - 1000$$ Simplifying we get,$ \Rightarrow$$$n(H \cap E) = 150 $So, the number of people who can speak Hindi only is$ n(H) - n(H \cap E) $. Substitute the values we get, The number of people who can speak Hindi only is$ 750 - 150 = 600 $. Hence, the number of people who can speak Hindi only is$ 600$$

$\therefore$ The correct answer is option C.

Note: We can solve the sum by using a Venn diagram.

Here, the red shaded part indicates the number of people who can speak Hindi is $750$ .
The blue shaded part indicates the number of people who can speak English is $400$ .
The green shaded part indicates the number of people who can speak both Hindi and English.
The total number of people is $1000$ .
We have to find the value of the green shaded part.
So, the value of green shaded part is
$750 + 400 - 1000 = 150$
The number of people who can speak Hindi only is $750 - 150 = 600$ .