Key Concepts and Formulas

• Principle of Inclusion-Exclusion for Two Sets: For any two sets $A$ and $B$, the percentage of elements in their union is given by $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Subset Property: The percentage of elements in an intersection of sets ($P(A \cap B)$) cannot exceed the percentage of elements in any of the individual sets ($P(A)$ or $P(B)$).
• Total Percentage: The percentage of any union of sets cannot exceed $100\%$.

Step-by-Step Solution

Step 1: Define Variables and Apply Inclusion-Exclusion Let $A$ be the event that a person reads newspaper A, and $B$ be the event that a person reads newspaper B. We are given:

• $P(A) = 63\%$
• $P(B) = 76\%$
• $P(A \cap B) = x\%$ (the percentage of people reading both newspapers)

Using the Principle of Inclusion-Exclusion, the percentage of people reading at least one newspaper is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substituting the given values: $P(A \cup B) = 63\% + 76\% - x\%$ $P(A \cup B) = (139 - x)\%$ Explanation: This formula helps us calculate the total percentage of people engaged in reading at least one newspaper by summing the individual percentages and subtracting the overlap (those reading both) to avoid double-counting.

Step 2: Determine the Upper Bound for $x$ The percentage of people reading both newspapers ($x\%$) must be a subset of those reading newspaper A and a subset of those reading newspaper B. Therefore, $x$ cannot be greater than the percentage of people reading either newspaper individually. $x \le P(A) \quad \Rightarrow \quad x \le 63$ $x \le P(B) \quad \Rightarrow \quad x \le 76$ To satisfy both conditions, $x$ must be less than or equal to the minimum of the two percentages: $x \le \min(63, 76)$ $x \le 63$ Explanation: If $x$ were, for example, 65%, it would imply that 65% of people read both newspapers. However, only 63% read newspaper A in total, making it impossible for 65% to read both. This establishes a logical upper limit for $x$.

Step 3: Determine the Lower Bound for $x$ The total percentage of people in the city is $100\%$. The percentage of people reading at least one newspaper ($P(A \cup B)$) cannot exceed this total. $P(A \cup B) \le 100\%$ From Step 1, we know $P(A \cup B) = (139 - x)\%$. So, we can write the inequality: $139 - x \le 100$ Rearranging the inequality to solve for $x$: $139 - 100 \le x$ $39 \le x$ Explanation: If $x$ were too small, say 30%, then $P(A \cup B)$ would be $139 - 30 = 109\%$, which is impossible as it exceeds the total population of 100%. This inequality establishes a necessary minimum value for $x$.

Step 4: Combine Bounds and Identify Possible Values for $x$ From Step 2, we have the upper bound: $x \le 63$. From Step 3, we have the lower bound: $x \ge 39$. Combining these, the possible range for $x$ is: $39 \le x \le 63$ Explanation: This range represents all logically consistent values for the percentage of people reading both newspapers, given the initial survey data.

Step 5: Evaluate the Given Options We need to find which of the given options falls within the range $[39, 63]$. The options are: (A) 37 (B) 65 (C) 29 (D) 55

Let's check each option:

• (A) 37: $37 < 39$. This value is outside the range.
• (B) 65: $65 > 63$. This value is outside the range.
• (C) 29: $29 < 39$. This value is outside the range.
• (D) 55: $39 \le 55 \le 63$. This value is within the range.

Since the question asks for a possible value of $x$, and 55 falls within our derived valid range, it is a possible value.

Common Mistakes & Tips

• Confusing "at least one" with "exactly one": Ensure you are using the Inclusion-Exclusion principle correctly to represent the union ($P(A \cup B)$), not just the sum of individual probabilities.
• Forgetting the $100\%$ constraint: Always remember that the total percentage of people cannot exceed $100\%$, which is crucial for establishing the lower bound of $x$.
• Ignoring the subset property: The intersection ($x\%$) must be less than or equal to each individual set's percentage, providing the upper bound for $x$.

Summary

The problem requires us to find a possible value for the percentage of people reading both newspapers ($x\%$) given the percentages of people reading newspaper A ($63\%$) and newspaper B ($76\%$). We used the Principle of Inclusion-Exclusion to establish an expression for the percentage of people reading at least one newspaper ($P(A \cup B) = 139 - x\%$). We then applied two key constraints: the percentage reading both newspapers cannot exceed the percentage reading either newspaper individually ($x \le 63$), and the total percentage of people reading at least one newspaper cannot exceed $100\%$ ($139 - x \le 100 \Rightarrow x \ge 39$). Combining these, we found the valid range for $x$ to be $39 \le x \le 63$. By checking the given options against this range, we identified 55 as the only possible value.

The final answer is \boxed{55} which corresponds to option (D).