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 to them speak English and French both ?

Answer 1

Let $A\to$ Set of people who speak English.

$B\to$ Set of people who speak French.

$A-B\to$ Set of people who speak English and not French.

$B-A\to$ Set of people who speak French and not English.

$A\cap B\to$ Set of people who speak both English and French.

Given

$n\left(A\right)=72n\left(B\right)=43n(A\cup B)=100$

Now,

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

$=72+43-100$

$=15$

$\therefore$ Number of persons who speak both English and French are 15

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

$\Rightarrow n(A-B)=n\left(A\right)-n(A\cap B)$

$=72-15$

$=57$

And

$\Rightarrow n(B-A)=n\left(B\right)-n(A\cap B)$

$=43-15$

$=28$

$\therefore$ Number of people speaking English only are 57.

and Number of people speaking French only are 28.