In a group of students, the ratio of the number of students who liked music and sports is 9:7. Out of which 25 liked both the activities, 20 liked music only and 15 liked none of the activities.

1. Represent the above information in a Venn-diagram.
2. Find the total number of students in the group.

Solution:

Let U be the set of total students.

Let M and S be the sets of students who like music and sports, respectively.

According to the question,

$n(M \cap S) = 25$

$n(M \cup S)^c = 15$

$n_o(M) = 20$

$n(M):n(S) = 9:7$

Let x be the number in the ratio.

$n(M):n(S) = 9x :7x$

$n(M) = 9x$

$n(S) = 7x$

We know,

$n(M) = n_o(M) + n(M \cap S)$

$or, 9x = 20 + 25$

$or, 9x = 45$

$or, 9×x = 9×5$

$\therefore x = 5$

So,

$n(M) = 9×5 = 45$

$n(S) = 7×5 = 35$

Using formula,

$n(U) = n(M) + n(S) - n(M \cap S) + n(M \cup S)^c$

$or, n(U) = 45 + 35 - 25 + 15$

$or, n(U) = 80 - 10$

$\therefore n(U) = 70$

Hence, the required number of students who were surveyed is 70.

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