Q. 12

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read the newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, and 3 read all the three newspapers. Find

(i) The number of people who read at least one of the newspapers,

(ii) The number of people who read exactly one newspaper.

Answer :

Given:


- Total number of people = 60


- Number of people who read newspaper H = 25


- Number of people who read newspaper T = 26


- Number of people who read newspaper I = 26


- Number of people who read newspaper H and I both = 9


- Number of people who read newspaper H and T both = 11


- Number of people who read newspaper T and I both = 8


- Number of people who read all three newspapers = 3


To Find:


(i) The number of people who read at least one of the newspapers


Let us consider,


Number of people who read newspaper H = n(H)= 25


Number of people who read newspaper T = n(T) = 26


Number of people who read newspaper I = n(I) =26


Number of people who read newspaper H and I both =


n(H I ) = 9


Number of people who read newspaper H and T both =


n(H T ) = 11


Number of people who read newspaper T and I both =


n(T I ) = 8


Number of people who read all three newspapers =


n(H T I) = 3


Number of people who read at least one of the three newspapers= n(H Ս T Ս I)


Venn diagram:



We know that,


n(H Ս T Ս I) = n(H) + n(T) + n(I) – n(H I) – n(H T) –


n(T I) + n(H T I)


= 25 + 26 + 26 – 9 – 11 – 8 + 3


= 52


Therefore,


Number of people who read at least one of the three newspapers = 52


(ii) The number of people who read exactly one newspaper


Number of people who read exactly one newspaper =


n(H Ս T Ս I) – p – q – r – s


Where,


p = Number of people who read newspaper H and T but not I


q = Number of people who read newspaper H and I but not T


r = Number of people who read newspaper T and I but not H


s = Number of people who read all three newspapers = 3


p + s = n(H T) …(1)


q + s = n(H I) …(2)


r + s = n(T I) …(3)


Adding (1), (2) and (3)


p + s + q + s + r + s = n(H T) + n(H I) + n(T I)


p + q + r + 3s = 9 + 11 + 8


p + q + r + s + 2s = 28


p + q + r + s = 28 - 2×3


p + q + r + s = 22


Now,


n(H Ս T Ս I) – p – q – r – s = n(H Ս T Ս I) – (p + q + r + s)


= 52 -22


= 30


Hence, 30 people read exactly one newspaper.


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