Question

In a group of 1000 people, there are 750 who can speak Hindi and 400 can speak Bengali. How many can speak Hindi only? How many can speak Bengali only? How many can speak both Bengali and Hindi?

Answer 1

Hint:To solve this question, we need to know the basic formula of two sets of an event, that is, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$, where $n\left( A \right)$ represents the number of elements in set A and $n\left( B \right)$ represents the number of elements in set B, $n\left( A\cup B \right)$ represents the number of events taking part in the event and $n\left( A\cap B \right)$ represents the number of elements present in both A and B. We will consider A as the people who can speak Hindi and B as the people who can speak Bengali.

Complete step-by-step answer:

In this question, we have been given that out of 1000 people, 750 can speak Hindi and 400 can speak Bengali. And we have to find the number of people who can speak only Hindi, the number of people who can speak only Bengali and the number of people who can speak Hindi and Bengali. To find the same, we will use the formula for two sets of an event, that is, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$. So, we will consider A as the group of people speaking Hindi and B as the people speaking Bengali. So, we can write, $n\left( A\cup B \right)$ as the total number of people in the group and $n\left( A\cap B \right)$ as the number of people who speak both Hindi and Bengali. So, we get,

$n\left( A \right)=750$

$n\left( B \right)=400$

$n\left( A\cup B \right)=1000$

Now, we will put these values in the formula. So, we get,

$750+400 -n\left( A\cap B \right)=1000$

$\Rightarrow 1150-n\left( A\cap B \right)=1000$

Now, simplifying further, we get,

$n\left( A\cap B \right)=1150-1000$

$n\left( A\cap B \right)=150$

Hence, we can say that 150 people speak both Hindi and Bengali.

Now, we have to find the number of people who speak only Hindi. So, we will find the difference between $n\left( A \right)$ and $n\left( A\cap B \right)$, because if we subtract the number of people who speak both Hindi and Bengali from the number of people who speak Hindi, we will get the number of people who speak only Hindi. So, we get,

$\Rightarrow n\left( \text{only Hindi speakers} \right)$= n(A)- n(A$\cap$ B)

$\Rightarrow n\left( \text{only Hindi speakers} \right)=750-150$

$\Rightarrow n\left( \text{only Hindi speakers} \right)=600$

Hence, we can say that there are 600 people who speak only Hindi.

Now we have to find the number of people who speak only Bengali. So, for that we will subtract the number of people who can speak both the languages from the number of people who can speak Bengali. So, we get,

$\Rightarrow n\left( \text{only Bengali speakers} \right)$ = n(B)- n(A$\cap$ B)

$\Rightarrow n\left( \text{only Bengali speakers} \right)=400-150$

$\Rightarrow n\left( \text{only Bengali speakers} \right)=250$

Hence, we can say that there are 250 people who speak only Bengali.

Note: We can also calculate the number of only Hindi speakers by subtracting the total number of Bengali speakers from the total number of people in the group. Similarly, we can find the number of Bengali speakers by subtracting the number of Hindi speakers from the total number of people. We can consider the Venn diagram as shown below

Here we have the number of A only as 600 and B only as 250 and A and B as 150.