Question

In a city, there are 25 Hindi medium schools, 18 English medium schools and 7 schools have both the medium. Find (i) How many schools are there in all in the city. (ii) How many schools have a Hindi medium only? (iii) How many schools have English medium only?

Answer 1

Hint: In the above question, we will use the set theory to solve the problem using its properties. First of all, we will take a variable to the number of Hindi and English medium school and then, we will use the formula of set theory as below,
\[n\left( A\cap B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cup B \right)\]
Where, n(A) and n(B) means number of element in the respective set.
\[n\left( A\cap B \right)\] Means number of elements common in set n(A) and n(B).
\[n\left( A\cup B \right)\] Means number of elements in set n(A) or set n(B).

Complete step-by-step answer:

We have been given that, there are 25 Hindi medium schools, 18 English medium schools and 7 schools have both the medium.
Let the number of Hindi medium to be n(A) and English medium to be n(B).
Then, we have,
\[\begin{align}
  & \text{n}\left( \text{A} \right)=\text{25} \\
 & \text{n}\left( \text{B} \right)=\text{18} \\
\end{align}\]
Number of English or Hindi medium school \[n\left( A\cup B \right).\]

Number of English and Hindi medium school\[\Rightarrow n\left( A\cap B \right)=7\]
We know that, according to set theory,
\[\begin{align}
  & n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right) \\
 & \Rightarrow n\left( A\cup B \right)=25+18-7 \\
 & \Rightarrow n\left( A\cup B \right)=36 \\
\end{align}\]
(i) Hence, there are 36 schools in the city.
(ii) In order to find the number of schools having Hindi medium only, we will have to subtract the common part of n(A), n(B) i.e. \[n\left( A\cap B \right)\] from n(A).
\[\begin{align}
  & \text{Only Hindi medium school=n}\left( A \right)-n\left( A\cap B \right) \\
 & \Rightarrow 25-7 \\
 & \Rightarrow 18 \\
\end{align}\]
(iii) In order to find the number of schools having English medium only, we will have to subtract the common part of n(A), n(B) i.e \[n\left( A\cap B \right)\] from n(B).
\[\begin{align}
  & \text{Only English medium school=n}\left( B \right)-n\left( A\cap B \right) \\
 & \Rightarrow 18-7 \\
 & \Rightarrow 11 \\
\end{align}\]
Therefore, the total number of schools in the city is 36, only Hindi medium school is 18 and only English medium school is 11.

Note: We can use the Venn diagram to solve these kinds of questions as shown below,

Here, n(A) is a number of Hindi medium schools, n(B) is the number of English medium schools and \[n\left( A\cap B \right)\] represent both Hindi and English medium schools.
Now, to find the total number of schools, we have to add numbers inside both circles and then deduct the common portion, so we get it as 25+18-7=36.
Now, to find the total number of only Hindi medium schools, we have to consider circle n(A) alone. Then, we have to deduct the shaded region from the circle, so we get it as 25-7=18.
Now, to find the total number of only English medium schools, we have to consider circle n(B) alone. Then, we have to deduct the shaded region from the circle, so we get it as 18-7=11.
Also, remember the meaning of \[\cup \] i.e. union and \[\cap \] i.e. intersection if the two statements are joined by 'and' then we use intersection and if two statements are joined by 'or' then, we use union.