Question

(a) A florist has certain number of garlands. 110 of them have champak, 50 have jasmine flowers and 30 garlands have both the flowers. Find the total number of garlands with him.

(b) In a class of 60 students everyone should select either Mathematics or Science or both. If 45 students select Mathematics, 30 students select Science, how many students do select both the subjects?

(c) In a survey of 1000 families of T.V. viewers in a city, it was found that 750 families view Kannada programmes, 400 families view Hindi programmes and 300 families view both the programme. Find how many families do view (i) Kannada programmes only (ii) Hindi programmes only (iii) neither of the programmes.

Answer 1

(a) Let A denote the set of garlands having champak flowers and B denote the set of garlands having jasmine flowers.

Then A ∩ B denotes the set of garlands that have both champak and jasmine flowers.

∴ n(A) = 110, n(B) = 50 and n(A ∩ B) = 30

The total number of garlands is given by n(A∪B).

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

∴ n(A ∪ B) = 110 + 50 − 30

⇒ n(A ∪ B) = 130

Therefore, the total number of garlands is 130.

(b) Let M denote the set of students who select Mathematics and S denote the set of students who select Science.

∴ n(M ∪ S) = 60, n(M) = 45 and n(S) = 30

Number of students who select both Mathematics and Science = n(M ∩ S)

n(M ∪ S) = n(M) + n(S) − n(M ∩ S)

∴ 60 = 45 + 30 − n(M ∩ S)

⇒ 60 = 75 − n(M ∩ S)

⇒ n(M ∩ S) = 75 − 60 = 15

Thus, 15 students select both Mathematics and Science.

(c) Let U be the set of all T.V. viewers who took part in the survey.

Let K denote the set of families that view Kannada programmes and H denote the set of families that view Hindi programmes.

∴ n(U) = 1000, n(K) = 750, n(H) = 400 and n(K ∩ H) = 300

K ∪ H is the set of families who view either Kannada or Hindi programme.

(K ∪ H)′ is the set of families who view neither Kannada nor Hindi programme.

n(K ∪ H) = n(K) + n(H) − n(K ∩ H)

⇒ n(K ∪ H) = 750 + 400 − 300

⇒ n(K ∪ H) = 850

(i) Number of families that view only Kannada programmes = n(K) − n(K ∩ H) = 750 − 300 = 450

(ii) Number of families that view only Hindi programmes = n(H) − n(K ∩ H) = 400 − 300 = 100

(iii) Number of families that view neither programme = n(K ∪ H)′ = n(U) − n(K ∪ H) = 1000 − 850 = 150