Question

Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English ? How many speak only French ? How many of them speak English and French both ?

Answer 1

Let A be the set of persons speaking English and B be the set of persons speaking French.

So, n (A) = 72; n (B) = 43; $\mathrm{n}(\mathrm{A}\cup \mathrm{B})=100$

Now,

$n\left(A\right)+n\left(B\right)=n(A\cup B)+n(A\cap B)$

$$\begin{gathered} \Rightarrow \mathrm{n}(\mathrm{A}\cap \mathrm{B})=72+43-100 \\ \Rightarrow \mathrm{n}(\mathrm{A}\cap \mathrm{B})=15 \\ \mathrm{So},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{person}\mathrm{who}\mathrm{speak}\mathrm{French}\mathrm{and}\mathrm{English}\mathrm{both}\mathrm{is}15. \end{gathered}$$

Also,

$\mathrm{n}\left(\mathrm{A}\right)=\mathrm{n}(\mathrm{A}-\mathrm{B})+\mathrm{n}(\mathrm{A}\cap \mathrm{B})$

$$\begin{gathered} \Rightarrow \mathrm{n}(\mathrm{A}-\mathrm{B})=72-15 \\ \Rightarrow \mathrm{n}(\mathrm{A}-\mathrm{B})=57 \\ \mathrm{So},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{person}\mathrm{who}\mathrm{speak}\mathrm{only}\mathrm{English}\mathrm{is}57. \end{gathered}$$

And,

$$\begin{gathered} \mathrm{n}\left(\mathrm{B}\right)=\mathrm{n}(\mathrm{B}-\mathrm{A})+\mathrm{n}(\mathrm{A}\cap \mathrm{B}) \\ \Rightarrow \mathrm{n}(\mathrm{B}-\mathrm{A})=43-15 \\ \Rightarrow \mathrm{n}(\mathrm{B}-\mathrm{A})=28 \\ \mathrm{So},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{person}\mathrm{who}\mathrm{speak}\mathrm{only}\mathrm{French}\mathrm{is}28. \end{gathered}$$