In a town three daily news papers X, Y, Z are published. 52% of the pe

1.	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» 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.

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»

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.