Question

In a group of 60 students, 36 read English newspapers, 22 read Hindi newspapers, and 12 read neither of the two. How many read both English and Hindi newspapers?

Answer 1

Hint:
Here, we will find how many students read both of the two newspapers. We will use sets, intersections and union to solve this problem. We will use the formula of union of two sets to find the number of students who read both newspapers.
Formula Used: The union of two sets can be found using the formula \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\].

Complete step by step solution:
A set is a well-defined collection of objects. For example: A set of odd numbers is a collection of all odd numbers.
Let \[E\] represent the set of students that read English newspapers.
Let \[H\] represent the set of students that read Hindi newspapers.
Now, we know that the number of elements in a set \[A\] is denoted by \[n\left( A \right)\].
We will write the number of elements in the sets \[E\] and \[H\].
In the group of 60 students, 36 students read English newspapers and 22 read Hindi newspapers.
Therefore, the number of elements in the set \[E\] is 36 and the number of elements in the set \[H\] is 22.
Thus, we get
\[n\left( E \right) = 36\] and \[n\left( H \right) = 22\]
Now, we will find the intersection and union of the two sets.
We know that the intersection of two sets is the set of elements that are in both the two sets.
Therefore, the intersection of the sets \[E\] and \[H\] represents the number of students who read both English and Hindi newspapers.
Therefore, the number of elements in the intersection of \[E\] and \[H\] is given by \[n\left( {E \cap H} \right)\].
We know that the union of two sets is the set of all the elements that are in the two sets.
Therefore, the union of the sets \[E\] and \[H\] represents the number of students who read either English or Hindi newspapers or both.
Therefore, the number of elements in the union of \[E\] and \[H\] is given by \[n\left( {E \cup H} \right)\].
Now, it is given that in the group of 60 students, 12 students read neither of the two newspapers.
Therefore, the number of students that read either of the two newspapers is the difference in the total number of students and the number of students who read neither of the two newspapers.
Thus, we get
\[\begin{array}{c}n\left( {E \cup H} \right) = 60 - 12\\ = 48\end{array}\]
Now, we know that the union of two sets can also be found using the formula \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\].
Therefore, using the formula, we get \[n\left( {E \cup H} \right) = n\left( E \right) + n\left( H \right) - n\left( {E \cap H} \right)\].
Substituting \[n\left( {E \cup H} \right) = 48\], \[n\left( E \right) = 36\], and \[n\left( H \right) = 22\] in the formula, we get
\[ \Rightarrow 48 = 36 + 22 - n\left( {E \cap H} \right)\]
Rewriting the equation, we get
\[ \Rightarrow n\left( {E \cap H} \right) = 36 + 22 - 48\]
Adding and subtracting the terms, we get
\[n\left( {E \cap H} \right) = 10\]

Therefore, the number of students who read both English and Hindi newspapers is 10.

Note:
For better understanding of set theory questions it’s always recommended to draw Vann diagrams. However it’s not needed in question in which we can directly apply formula. In the formula \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\] we have 4 terms. they’ll give any 3 values and ask us to find the remaining. This is how they can make many questions using a single formula.