Question

In a town three daily news papers X, Y, Z are published. 52% of the people read paper X, 61 % read Y and 78% read Z. 40% read X and Y; 38% read Y and Z, 46% read X and Z; 18% do not read any of the three news papers. Find the percentage of persons who read all the three papers.

Answer 1

Let the number of person is the town be 100.

Now it is given n(X) = 52, n(Y) = 61, n(Z) = 78, n(X ∩ Y) = 40, n(Y ∩ Z) = 38

n(X ∩ Z) = 46, n(X ∪ Y ∪ Z) = 100 – 18 = 82

We have

n(X ∪ Y ∪ Z) = n(X) + n(Y) + n(Z) – n(X ∩ Y) – n(Y ∩ Z) – n(X ∩ Z) + n(X ∩ Y ∩ Z).

⇒ 82 = 52 + 61 + 78 – 40 – 38 – 46 + n(X ∩ Y ∩ Z)

= 191 – 124 + n(X ∩ Y ∩ Z) = 67 + (X ∩ Y ∩ Z)

∴ n(X ∩ Y ∩ Z) = 82 – 67 = 15

Here 15% of the people read all the three papers.