75 students in a class like tea or coffee or both. Out of them 10 like both the drinks. The ratio of the number of students who like tea to those who like coffee is 2:3.

1. Find the number of students who like tea.
2. Find the number of students who like coffee only.
3. Represent the above information in a Venn-diagram.

Solution:

Let U be the set of total number of students.

Let T and C be the sets of students who like tea and coffee, respectively.

According to the question,

$n(U) = 75$

$n(T \cap C) = 10$

$n(T \cup C)^c = 0$

$n(T) : n(C) = 2:3$

Let the number in the ratio be x, we get,

$n(T):n(C) = 2x:3x$

$n(T) = 2x$

$n(C) = 3x$

Using formula,

$n(U) = n(T) + n(C) - n(T \cap C)$

$or, 75 = 2x + 3x - 10$

$or, 75 + 10 = 5x$

$or, 85 = 5x$

$or, 5 × 17 = 5 × x$

$\therefore x = 17$

Now,

$n(T) = 2× 17 = 34$

And,

$n_o(C) = n(C) - n(T \cap C) $

$or, n_o(C) = 3×17 - 10$

$\therefore n_o(C) = 41$

(i) The required number of students who liked tea is 34.

(ii) The required number of students who liked coffee only is 41.

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