In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all 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
Solution:
Let H be the set of people who read the newspaper H.
T be the set of people who read the newspaper T.
I will be the set of people who read the newspaper I.
It is given that
n (H) = 25, n (T) = 26, n (I) = 26, n (H ∩ T) = 11, n (H ∩ I) = 9, n (T ∩ I) = 8, n (H ∩ T ∩ I) = 3
(i) The number of people who read at least one of the newspapers.
We know that
n (H υ T υ I) = n (H) + n (T) + n (I) - n (H ∩ T) - n (H ∩ I) - n (T ∩ I) + n (H ∩ T ∩ I)
= 25 + 26 + 26 - 11 - 9 - 8 + 3
= 80 - 28
= 52
Hence, 52 people read at least one of the newspapers.
(ii) Let x people read newspapers H and T only, y people read newspapers T and I only,
and z people read newspapers H and I
Now, draw the Venn diagram for the given problem.
We can see that,
n ( H ∩ T ∩ I) = a = 3
n ( H ∩ T) = x + a = 11 n (H ∩ I) = z + a = 9 n (T ∩ I) = y + a = 8
Now,
(x + a) + (y + a) + (z + a) = 11 + 8 + 9
x + 3 + y + 3 + z + 3 = 28
x + y + z = 28 - 9
x + y + z = 19
Number of people who read exactly two newspapers = x + y + z = 19
Number of people who read two or more newspapers = 19 + 3 = 22
Therefore,
Number of people who read 3 exactly one newspaper = 52 - 22 = 30
Hence, 30 people read exactly one newspaper
NCERT Solutions Class 11 Maths Chapter 1 Exercise ME Question 15
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all 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.
Summary:
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, 3 read all three newspapers is given. We have found that 30 people read exactly one newspaper