Question: In a survey, 3/14 children liked only Mathematics and 70 didn't like Mathematics at all. Also, 9/14 children like Science but 20 like none of the subjects. Then

(i) Show the given information in a venn diagram.
(ii) How many children like both of the subjects?

Solution:
Let M and S represent the sets of students who like Mathematics and Science subjects, respectively.

Let the total number of students surveyed be x students.

According to the question,
$n(U) = x$ students
$n_o(M) = \frac{3}{14} \;of\;x$ students
$n(S) = \frac{9}{14} \; of \; x$ students
$n( \overline{M} ) = 70$ students
$n(\overline{M \cup S}) = 20$ students

To find: $n(M \cap S)$ = ?
(i) Showing the given information in a venn-digram:



(ii) We have,
$n(U) = n_o(M) + n(S) + n(\overline{M \cup S})$

$or, x = \dfrac{3x}{14} + \dfrac{9x}{12} + 20$

$or, x = \dfrac{12x}{14} + 20$

$or, x - \dfrac{12x}{14} = 20$

$or, \dfrac{14x -12x}{14} = 20$

$or, 2x = 20 × 14$
$or, 2x = 280$
$\therefore x = 140$
Also,
$n(\overline{M}) = n(S) - n(M \cap S) + n(\overline{M \cup S})$

$or, 70 = \dfrac{9 × 140}{14} - n(M \cap S) + 2$

$or, 70 - 20 = 90 - n(M \cap S) $

$or,  n(M \cap S)  = 90 - 50$

$\therefore n(M \cap S)  = 40$

Hence, the required number of students who like both the subjects are 40.

Explanation:

Here, the union has to be let x as the number of students of Maths and Science have been written in fractional form.

Here are the notes:


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