# In a Group of 50 Persons, 14 Drink Tea but Not Coffee and 30 Drink Tea. Find: (I) How May Drink Tea and Coffee Both; (Ii) How Many Drink Coffee but Not Tea. - Mathematics

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea.

#### Solution

Let A & B denote the sets of the persons who drink tea & coffee, respectively .

$\text{ Given }:$
$n\left( A \cup B \right) = 50$
$n\left( A \right) = 30$
$n\left( A - B \right) = 14$
$(i) n\left( A - B \right) = n\left( A \right) - n\left( A \cap B \right)$
$\Rightarrow 14 = 30 - n\left( A \cap B \right)$
$\Rightarrow n\left( A \cap B \right) = 16$
$\text{ Thus, 16 persons drink tea and coffee both } .$
$(ii) n\left( A \cup B \right) = n\left( A \right) + n\left( B \right) - n\left( A \cap B \right)$
$\Rightarrow 50 = 30 + n\left( B \right) - 16$
$\Rightarrow n\left( B \right) = 36$
$\text{ We have to find } n\left( B - A \right)$
$\Rightarrow n\left( B - A \right) = n\left( B \right) - n\left( A \cap B \right)$
$\Rightarrow n\left( B - A \right) = 36 - 16 = 20$
$\text{ 20Thus, 20 persons drink coffee but not tea } .$

Concept: The Empty Set
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.8 | Q 8 | Page 47