Question

In a village there are 60 families. Out of these 28 families speak only Tamil and 20 families speak only Urdu. How many families speak both Tamil and Urdu?

Answer 1

Hint: Here, in this given question, we are going to use Set Theory to solve this question. We will use the formula for the number of elements in the universal set that is equal to the sum of all the elements only in the first set, all the elements only in the second set and the elements which are in both the sets. That is $ \left| U \right|=\left| A-B \right|+\left| B-A \right|+\left| A\cap B \right| $ . We can take the set of family speaking Tamil as A and speaking Urdu as B and proceed.

Complete step-by-step answer:
In this given question we are given that in a village there are 60 families. Out of these 28 families speak only Tamil and 20 families speak only Urdu.
We are asked to find how many families speak both Tamil and Urdu.
In this question, we are going to use Set theory in order to solve it.
We will use the formula for the number of elements in the universal set that is equal to the sum of all the elements only in the first set, all the elements only in the second set and the elements which are in both the sets. That is $ \left| U \right|=\left| A-B \right|+\left| B-A \right|+\left| A\cap B \right| $ …………..(1.1)
Let the set of all the families speaking only Tamil be A and the set of all the families speaking Urdu be B.
U is the universal set and $ A\cap B $ is the set of families which speak both Tamil and Urdu.
Now, according to the given information,
No. of families who speak only Tamil is \[\left| A-B \right|=28\] .
No. of families who speak only Urdu is $ \left| B-A \right|=20 $ .
No. of elements in the Universal set is $ \left| U \right|=60 $ .
Now, putting these values in equation 1.1, we get,
 $ \begin{align}
  & \left| U \right|=\left| A-B \right|+\left| B-A \right|+\left| A\cap B \right| \\
 & \Rightarrow 60=28+20+\left| A\cap B \right| \\
 & \Rightarrow \left| A\cap B \right|=60-48 \\
 & \Rightarrow \left| A\cap B \right|=12................(1.2) \\
\end{align} $
From equation 1.2, we get that the set of families which speak both Tamil and Urdu contains 12 elements.
Therefore, we conclude that there are 12 families in the village which speak both Tamil and Urdu.

Note: Here, we have taken the set of families speaking only Tamil as A-B because those families which speak only Tamil belong only to set A and not to Set B. Similarly, using the same fact we have obtained the set of families speaking Urdu as B-A.