Question

In a group of 500 people, 310 can speak Hindi, 260 can speak English. How many people can speak both English and Hindi?

Answer 1

Now here we have 310 people speak Hindi and 260 can speak English. Let us call set A and B as the set of people who speak English and people who speak Hindi respectively. Now $A\cup B$ represents the total number of people while $A\cap B$ represents the people speaking both the languages. Now we know that $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$ . Hence we will use this result to find the number of people who speak both the languages.

Complete step by step answer:
Now we know that the total number of people is 500.
It is given that the number of people that speak Hindi is 310.
Let A be the set of people that speak Hindi. Hence we get n(A) = 310, Where n(A) means number of elements in set A.
Now also the number of people speaking English is 260.
Let B be the set whose elements are the people who speak English.
Hence we get n(B) = 260, where n(B) means number of elements in set B.
Now in the group of 500, people are either Hindi speaking or English speaking or both.
Now what does set $A\cup B$mean?
$A\cup B$ consists of elements which are either in A or in B.
Hence the set $A\cup B$ represents the set of all people who speak Hindi or English.
Now $n\left( A\cup B \right)=500$ . Where $n\left( A\cup B \right)$ means the number of elements in the set $A\cup B$ .
Now let us understand what does $A\cap B$ means,
$A\cap B$ means the set of all the elements which are in A and B.
Hence $A\cap B$ represents the set of people speaking English and Hindi.
Hence $n\left( A\cap B \right)$ shows the number of people speaking English and Hindi.
Now we know that
$n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$
Now substituting the values in the equation we get,

& 500=310+260-n\left( A\cap B \right) \\\ & \Rightarrow 500=570-n\left( A\cap B \right) \\\ & \Rightarrow n\left( A\cap B \right)=570-500 \\\ & \Rightarrow n\left( A\cap B \right)=70 \\\ \end{aligned}$$ **Hence the number of people speaking both the languages English and Hindi are 70.** **Note:** Now note that here in the group of 500 people either people speak English or Hindi. Hence we could say that $\left( A\cup B \right)$ represents the total number of people. Also remember that the number of elements in $\left( A\cup B \right)$ is given by $$n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$$ and not $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)$ We can also understand this by drawing a venn diagram. Venn diagrams are nothing but representations of sets as closed figures. For example If A and B are two sets then we can represent them by  Note that the common part in both the figures represents the intersection of two sets.