Q. 145.0( 2 Votes )

In a town of 10,000 families, it was found that 40% of the families buy newspaper A, 20% buy newspaper B, 10% buy newspaper C, 5% buy A and B; 3% buy B and C, and 4% buy A and C. IF 2% buy all the three newspapers, find the number of families which buy

(i) A only,

(ii) B only,

(iii) none of A, B, and C.

Answer :

Given:


Total number of families = 10000


Percentage of families that buy newspaper A = 40


Percentage of families that buy newspaper B = 20


Percentage of families that buy newspaper C = 10


Percentage of families that buy newspaper A and B = 5


Percentage of families that buy newspaper B and C = 3


Percentage of families that buy newspaper A and C = 4


Percentage of families that buy all three newspapers = 2


To find:


(i) Number of families that buy newspaper A only


Consider the Venn Diagram below:



Number of families that buy newspaper A = n(A) = 40% of 10000


= 4000


Number of families that buy newspaper B = n(B) = 20% of 10000


= 2000


Number of families that buy newspaper C = n(C) = 10% of 10000


= 1000


Number of families that buy newspaper A and B = n(A B)


= 5% of 10000


= 500


Number of families that buy newspaper B and C = n(B C)


= 3% of 10000


= 300


Number of families that buy newspaper A and C = n(A C)


= 4% of 10000


= 400


Number of families that buys all three newspapers = n(A B C)=v


= 2% of 10000


= 200


We have,


n(A B) = v + t


500 = 200 + t


t = 500 – 200 = 300


n(B C) = v + s


300 = 200 + s


s = 300 – 200 = 100


n(A C) = v + u


400 = 200 + u


u = 400 – 200 = 200


p = Number of families that buy newspaper A only


We have,


A = p + t + v + u


4000 = p + 300 + 200 + 200


p = 4000 – 700


p = 3300


Therefore,


Number of families that buy newspaper A only = 3300


(ii) Number of families that buy newspaper B only


q = Number of families that buy newspaper B only


B = q + s + v + t


2000 = q + 100 + 200 + 300


q = 2000 – 600 =1400


Therefore,


Number of families that buy newspaper B only = 1400


(iii) Number of families that buys none of the newspaper


Number of families that buy none of the newspaper =


10000 – {n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)}


= 10000 – (4000 + 2000 + 1000 – 500 – 300 – 400 + 200)


= 10000 – 6000


= 4000


Therefore,


Number of families that buy none of the newspaper = 4000


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RS Aggarwal - Mathematics