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 one paper?

Options

A. 0.38
B. 0.48
C. 0.51
D. None of these

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and some basic set theory concepts.

### Step 1: Define the Variables
Let:
- \( N(A) \) = Percentage of people reading newspaper A = 42%
- \( N(B) \) = Percentage of people reading newspaper B = 51%
- \( N(C) \) = Percentage of people reading newspaper C = 68%
- \( N(A \cap B) \) = Percentage of people reading both A and B = 30%
- \( N(B \cap C) \) = Percentage of people reading both B and C = 28%
- \( N(A \cap C) \) = Percentage of people reading both A and C = 36%
- \( N(A \cup B \cup C)' \) = Percentage of people not reading any newspaper = 8%

### Step 2: Calculate the Total Percentage Reading at Least One Newspaper
The percentage of people reading at least one newspaper is:
\[ N(A \cup B \cup C) = 100\% - N(A \cup B \cup C)' = 100\% - 8\% = 92\% \]

### Step 3: Apply the Inclusion-Exclusion Principle
Using the inclusion-exclusion principle for three sets, we have:
\[ N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(B \cap C) - N(A \cap C) + N(A \cap B \cap C) \]
Substituting the known values:
\[ 92 = 42 + 51 + 68 - 30 - 28 - 36 + N(A \cap B \cap C) \]

### Step 4: Simplify the Equation
Calculating the right-hand side:
\[ 92 = 42 + 51 + 68 - 30 - 28 - 36 + N(A \cap B \cap C) \]
\[ 92 = 161 - 94 + N(A \cap B \cap C) \]
\[ 92 = 67 + N(A \cap B \cap C) \]
Thus,
\[ N(A \cap B \cap C) = 92 - 67 = 25 \]

### Step 5: Calculate the Percentage of People Reading Only One Newspaper
To find the percentage of people reading only one newspaper, we use the formula:
\[ N(\text{only A}) = N(A) - N(A \cap B) - N(A \cap C) + N(A \cap B \cap C) \]
\[ N(\text{only A}) = 42 - 30 - 36 + 25 = 1 \]
\[ N(\text{only B}) = N(B) - N(A \cap B) - N(B \cap C) + N(A \cap B \cap C) \]
\[ N(\text{only B}) = 51 - 30 - 28 + 25 = 18 \]
\[ N(\text{only C}) = N(C) - N(A \cap C) - N(B \cap C) + N(A \cap B \cap C) \]
\[ N(\text{only C}) = 68 - 36 - 28 + 25 = 29 \]

### Step 6: Total Percentage of People Reading Only One Newspaper
Now, we sum the percentages of people reading only one newspaper:
\[ N(\text{only A}) + N(\text{only B}) + N(\text{only C}) = 1 + 18 + 29 = 48 \]

### Step 7: Final Percentage Calculation
The total percentage of people reading only one newspaper is:
\[ \text{Percentage} = 48\% \]

### Conclusion
The percentage of persons who read only one paper is **48%**.