Solving the Newspaper Reading Population Problem Using Inclusion-Exclusion Principle

In a small town, three newspapers—A, B, and C—are published. The challenge is to find the number of people in the town who read all three newspapers based on the given data. Let's break down the solution step-by-step using the principle of inclusion-exclusion.

Given Data

42% of the population read newspaper A (nA) 68% of the population read newspaper B (nB) 51% of the population read newspaper C (nC) 30% of the population read both A and B (nA ∩ B) 28% of the population read both B and C (nB ∩ C) 36% of the population read both A and C (nA ∩ C) 18% of the population do not read any of the newspapers

Solution

First, we need to find the percentage of people who read at least one of the three newspapers. Since 18% do not read any newspaper, the percentage of people who read at least one newspaper is:

100 - 18 82%

According to the principle of inclusion-exclusion for three sets:

n(A ∪ B ∪ C) nA nB nC - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) n(A ∩ B ∩ C)

Let x be the percentage of people who read all three newspapers (nA ∩ B ∩ C).

Substituting the known values into the equation:

82 42 68 51 - 30 - 28 - 36 x

Simplifying the equation:

82 161 - 94 x

82 67 x

Solving for x:

x 82 - 67 15

Hence, 15% of the population reads all three newspapers.

Alternative Calculation Method

Alternatively, we can use the raw counts for the calculation:

nA 42 nB 51 nC 68 nA ∩ B 30 nB ∩ C 28 nC ∩ A 36 nA ∪ B ∪ C' 8 (people who do not read at least one paper)

So nA ∪ B ∪ C 100 - 8 92

Using the inclusion-exclusion principle:

nA ∪ B ∪ C nA nB nC - nA ∩ B - nB ∩ C - nC ∩ A nA ∩ B ∩ C

92 42 51 68 - 30 - 28 - 36 nA ∩ B ∩ C

92 161 - 94 nA ∩ B ∩ C

92 67 nA ∩ B ∩ C

nA ∩ B ∩ C 92 - 67 25

Thus, 25 people read all three newspapers.

Venn Diagram Visualization

If you want to visualize this problem, a Venn diagram would be helpful. The total population is 100%, and 18% do not read any newspaper. This leaves 82% reading at least one newspaper. In the Venn diagram, each overlap would represent the percentage of people reading the respective combinations of newspapers. By solving the equations, we can determine the exact number of people in the intersection where all three newspapers are read.

Thank you for reading this explanation. If you have any questions or need further assistance, please feel free to ask.