Question

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find :

(i) How many students were studying Hindi ?

(ii) How many students were studying English and Hindi ?

Answer 1

(i) Let n(P) denote total number of students

n(E) denote number of students studying English language

n(H) denote number of students studying Hindi language

n(S) denote number of students studying Sanskrit language

Then , n(P) = 100, n(E - H)= 23, $n(E\cap S)=8,n\left(E\right)=26,n\left(S\right)=48$

$n(S\cap H)=8$ , n {$(E\cup H\cup S{)}^{\prime }$} = 24

Number of students studying English only = 18

We have,

n {$(E\cup H\cup S{)}^{\prime }$} = 24

$\Rightarrow n\left(P\right)-n(E\cup H\cup S)=24$

$\Rightarrow 100-24=n(E\cup H\cup S)$

$n(E\cup H\cup S)$ = 76

We have $n(E\cup H\cup S)=n\left(E\right)+n\left(H\right)+n\left(S\right)$

-$n(E\cap H)-n(H\cap S)-(E\cap S)+n(E\cap H\cap S)$

$\Rightarrow 76=26+n\left(H\right)+48-3-8-8+3$

$\Rightarrow 76=26+n\left(H\right)+48-16$

$\Rightarrow 76=26+32+n\left(H\right)$

$\Rightarrow n\left(H\right)=76-58=18$

$\therefore$ 18 students were studying Hindi.

(ii) From (i) we have $n(E\cap H)$ = 3

$\therefore$ 3 students were studying both English and Hindi.