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 only A but neither B nor C?

Options

A. A) 4%
B. B) 3%
C. C) 1%
D. D) None of these

Answer

To solve the problem of finding the percentage of persons who read only newspaper A but neither B nor C, we will follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = percentage of people who read all three newspapers A, B, and C.
- We know the following percentages:
- \( |A| = 42\% \) (people who read A)
- \( |B| = 51\% \) (people who read B)
- \( |C| = 68\% \) (people who read C)
- \( |A \cap B| = 30\% \) (people who read both A and B)
- \( |B \cap C| = 28\% \) (people who read both B and C)
- \( |A \cap C| = 36\% \) (people who read both A and C)
- \( |A \cup B \cup C|' = 8\% \) (people who do not read any of the three newspapers)

### Step 2: Calculate Total Readers
Since 8% do not read any newspapers, the percentage of people who read at least one newspaper is:
\[ |A \cup B \cup C| = 100\% - 8\% = 92\% \]

### Step 3: Set Up the Equation
Using the principle of inclusion-exclusion for three sets, we can express the total percentage of readers as:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C| \]
Substituting the known values:
\[ 92\% = 42\% + 51\% + 68\% - 30\% - 28\% - 36\% + x \]

### Step 4: Simplify the Equation
Now, simplify the equation:
\[ 92\% = 42 + 51 + 68 - 30 - 28 - 36 + x \]
Calculating the left side:
\[ 92 = 42 + 51 + 68 - 30 - 28 - 36 + x \]
\[ 92 = 42 + 51 + 68 - 94 + x \]
\[ 92 = 61 + x \]
Thus, we can isolate \( x \):
\[ x = 92 - 61 = 31 \]

### Step 5: Calculate Only A
Now we need to find the percentage of people who read only A. The formula for only A is:
\[ |A \text{ only}| = |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) \]
Substituting the known values:
\[ |A \text{ only}| = 42\% - (30\% + 36\% - 31\%) \]
Calculating:
\[ |A \text{ only}| = 42\% - (30 + 36 - 31) \]
\[ |A \text{ only}| = 42\% - 35\% \]
\[ |A \text{ only}| = 7\% \]

### Final Answer
The percentage of persons who read only newspaper A but neither B nor C is **7%**.