Question

In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?

Answer 1

Let A & B denote the sets of the persons who can speak Hindi & Bengali, respectively.
$$\begin{gathered} \mathrm{Given}: \\ n\left(A\cup B\right)=1000 \\ n\left(A\right)=750 \\ n\left(B\right)=400 \\ n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right) \\ \Rightarrow 1000=750+400-n\left(A\cap B\right) \\ \Rightarrow n\left(A\cap B\right)=150 \\ \mathrm{Number}\mathrm{of}\mathrm{persons}\mathrm{who}\mathrm{can}\mathrm{speak}\mathrm{both}\mathrm{Hindi}\mathrm{and}\mathrm{Bengali}=n\left(A\cap B\right)=150 \\ \mathrm{Number}\mathrm{of}\mathrm{persons}\mathrm{who}\mathrm{can}\mathrm{speak}\mathrm{only}\mathrm{Hindi}=n\left(A-B\right)=n\left(A\right)-n\left(A\cap B\right) \\ =750-150 \\ =600 \\ \mathrm{Number}\mathrm{of}\mathrm{persons}\mathrm{who}\mathrm{can}\mathrm{speak}\mathrm{only}\mathrm{Bengali}=n\left(B-A\right)=n\left(B\right)-n\left(A\cap B\right) \\ =400-150 \\ =250 \end{gathered}$$