Question

In a class consisting of 100 students, 20 know English and 20 do not know Hindi and 10 know neither English nor Hindi. The number of students knowing both Hindi and English is
A. 5
B. 10
C. 15
D. 20

Answer 1

Hint: We draw Venn diagrams for the given situation and use set theory to find the required number of students. We use the total given number of students as a union and calculate the number of students knowing Hindi using formula of compliment and number of students knowing either English or Hindi using formula of compliment. Use the formula of set theory to calculate the number of students knowing both English and Hindi.
If \[n(A),n(B)\] denotes number of elements in each set A and B respectively, then their union i.e. \[n(A \cup B)\]and their intersection i.e. \[n(A \cap B)\]gives us the formula \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]

Complete step by step answer:
We are given that
Total number of students \[ = 100\]
\[U = 100\]
Let us denote English by A and Hindi by B
Number of students who know English\[ = 20\]
\[n(A) = 20\]
Number of students who don’t know Hindi\[ = 20\]
\[n(B)' = 20\]
We know the number of students not knowing Hindi is complement to the number of students knowing Hindi. We subtract number of students not knowing Hindi from total number of students to calculate number of students knowing Hindi.
\[ \Rightarrow n(B) = U - n(B)'\]
\[ \Rightarrow n(B) = 100 - 20\]
\[ \Rightarrow n(B) = 80\]
Number of students who know neither English nor Hindi\[ = 10\]
\[n(A \cup B)' = 10\]
Number of students who know either English or Hindi is given by subtracting the number of students knowing either language from the total number of students.
\[ \Rightarrow n(A \cup B) = U - n(A \cup B)'\]
\[ \Rightarrow n(A \cup B) = 100 - 10\]
\[ \Rightarrow n(A \cup B) = 90\]
Then the number of students who know both the languages is given by\[n(A \cap B)\]
Then we can draw a Venn diagram depicting the number of students who know A, B and both A and B. Here U is the universal set which depicts the total number of students.

We use the formula\[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]to find the value of intersection of A and B.
Substitute the value of\[n(A) = 20\],\[n(B) = 80\]and\[n(A \cup B) = 90\]in the formula
\[ \Rightarrow 90 = 20 + 80 - n(A \cap B)\]
Add the terms in RHS
\[ \Rightarrow 90 = 100 - n(A \cap B)\]
Shift all constant terms to one side of the equation
\[ \Rightarrow n(A \cap B) = 100 - 90\]
Calculate the difference
\[ \Rightarrow n(A \cap B) = 10\]
\[\therefore \] Number of students who know both English and Hindi is equal to 10

\[\therefore \]Option B is correct.

Note: Students might make the mistake of assuming the union of the two elements as the total number of elements. Keep in mind union denoted the elements that are in A, B or in both A and B but we have elements that are neither in A, nor in B also. So, the universal set denotes the total number of elements available.