In a survey of a community, it was found that 65% of people liked folk songs, 55% liked modern songs and 10% of people did not like both type of songs

(i) Illustrate the above information in a Venn-diagram.

(ii) If 360 people liked both types of songs, how many people were surveyed? Ans: 1200

Solution:

Let U be the set of total people surveyed in the community.

Let F and M be the sets of people who like folk songs and modern songs, respectively.

Given,

$n(F) = 65%$

$n(M) = 55%$

$n(\overline{F \cup M} ) = 10%$

$n(F \cap M) = 360$

To find: (ii) n(U) = ?

Using formula,

$n(U) = n(F) + n(M) - n(F \cap M) + n(\overline{F \cup M})$

$or, 100% = 65% + 55% - n(F \cap M) + 10%$

$or, 100% = 130% - n(F \cap M)$

$or, n(F \cap M) = 130% - 100%$

$\therefore n(F \cap M)= 30%$

Also,

$n(F \cap M) = 360$

$or, 30% of n(U) = 360$

$or, \dfrac{30}{100} × n(U) = 360$

$or, n(U) = \dfrac{360×100}{30}$

$\therefore n(U) = 1200$

Hence, the required total number of people who were surveyed in the community is 1200.

Explanation:

In a set, the universal set always consists of all the required elements. Hence, the universal set is always 100%.

Other sets are dependent upon the universal set. So, the percentage of elements contained by any set is derived from the Universal Set (Super Set).

#SciPiPupil