For the next Four (21-24) questions that follow:
In a city, three daily newspapers A, B, C are published, 42% read A, 51% read B, 68% read C, 30% read A and B, 28% read Band C 36% read A and C, 8% do not read any of the three newspapers.
What is the percentage of persons who read all the three papers?

Options

A. 0.2
B. 0.25
C. 0.3
D. 0.4

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and Venn diagrams. Let's denote the percentage of people who read all three newspapers (A, B, and C) as \( X \).

### Step 1: Define the given percentages
- Let \( P(A) = 42\% \) (percentage of people who read A)
- Let \( P(B) = 51\% \) (percentage of people who read B)
- Let \( P(C) = 68\% \) (percentage of people who read C)
- Let \( P(A \cap B) = 30\% \) (percentage of people who read both A and B)
- Let \( P(B \cap C) = 28\% \) (percentage of people who read both B and C)
- Let \( P(A \cap C) = 36\% \) (percentage of people who read both A and C)
- Let \( P(\text{none}) = 8\% \) (percentage of people who do not read any newspapers)

### Step 2: Set up the equations
Using the Venn diagram, we can express the percentages of people reading only one or two newspapers in terms of \( X \):
- People who read only A: \( P(A) - P(A \cap B) - P(A \cap C) + X = 42 - (30 - X) - (36 - X) + X = 42 - 30 + X - 36 + X + X = 42 - 30 - 36 + 3X = -24 + 3X \)
- People who read only B: \( P(B) - P(A \cap B) - P(B \cap C) + X = 51 - (30 - X) - (28 - X) + X = 51 - 30 + X - 28 + X + X = 51 - 30 - 28 + 3X = -7 + 3X \)
- People who read only C: \( P(C) - P(A \cap C) - P(B \cap C) + X = 68 - (36 - X) - (28 - X) + X = 68 - 36 + X - 28 + X + X = 68 - 36 - 28 + 3X = 4 + 3X \)

### Step 3: Total percentage of readers
The total percentage of people who read at least one newspaper is \( 100\% - 8\% = 92\% \). Thus, we can write:
\[ \text{Only A} + \text{Only B} + \text{Only C} + P(A \cap B) + P(B \cap C) + P(A \cap C) - 2X + X = 92\% \]
Substituting the values we calculated:
\[ (-24 + 3X) + (-7 + 3X) + (4 + 3X) + (30) + (28) + (36) - 2X = 92 \]
Simplifying this:
\[ (-24 - 7 + 4 + 30 + 28 + 36) + (3X + 3X + 3X - 2X) = 92 \]
\[ 67 + 7X = 92 \]

### Step 4: Solve for \( X \)
Now, we can solve for \( X \):
\[ 7X = 92 - 67 \]
\[ 7X = 25 \]
\[ X = \frac{25}{7} \approx 3.57\% \]

### Conclusion
The percentage of persons who read all three papers is approximately \( 3.57\% \).