Question

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find :

(i) how many drink tea and coffee both

(ii) how many drink coffee but not tea.

Answer 1

(i) Let,

n(P) denote the total number of persons,

n(T) denote the num,ber of persons who drink tea and

n(C) denote the number of persons who drink coffee.

Then,

n (P) = 50, n(T- C) = 14, n(T) = 30

To find : $n(T\cap C)$

Clearly T is the disjoint union T - C and $T\cap C$

$\therefore T=(T-C)\cup (T\cap C)$

$\therefore n\left(T\right)=n(T-C)+n(T\cap C)$

$\Rightarrow 30=14+n(T\cap C)$

$\Rightarrow n(T\cap C)=30-14$

=16

Hence, 16 persons drink tea and coffee both.

(ii) To find : C - T

We know $n\left(P\right)=n\left(C\right)+n\left(T\right)-n(T\cap C)$

$\Rightarrow 50=n\left(C\right)+30-16$

$\Rightarrow 50=n\left(C\right)+14$

$\Rightarrow n\left(C\right)=50-14$

= 36

Now C is the disjoint nion of C- T and $T\cap C$

$\therefore C=(C-T)\cup (C\cap T)$

$\Rightarrow n\left(c\right)=n(C-T)+n(C\cap T)$

$\Rightarrow 36=n(C-T)+16$ [$\because n(T\cap C)=n(C\cap T)=16$]

$\Rightarrow n(C-T)=36-12$

= 20

Hence, 20 persons drink coffee but not tea.