Key Concepts and Formulas

This problem involves applying the principles of set theory to a population distribution problem. The key concepts are:

• Set Operations: Understanding how to find the proportion of elements in sets that are unique to one set, common to both, or not in either. Specifically, we'll use the concept of the complement of an intersection within a set.
• Mutually Exclusive Events: The problem requires breaking down the population into disjoint (mutually exclusive) groups based on their newspaper readership to avoid double-counting when calculating advertisement viewership.
• Conditional Percentages: Applying a percentage to a specific subgroup of the population.

The relevant formulas are:

• Percentage of people reading only A: $P(A \text{ only}) = P(A) - P(A \cap B)$
• Percentage of people reading only B: $P(B \text{ only}) = P(B) - P(A \cap B)$
• Percentage of a subgroup looking into advertisements: $P(\text{Ad} | \text{Subgroup}) \times P(\text{Subgroup})$

Step-by-Step Solution

Let $A$ be the set of people who read newspaper A, and $B$ be the set of people who read newspaper B. We are given the following percentages of the total city population:

• $P(A) = 25\%$
• $P(B) = 20\%$
• $P(A \cap B) = 8\%$ (percentage who read both A and B)

Step 1: Decompose the population into disjoint groups based on newspaper readership. To accurately calculate the percentage of people looking into advertisements, we need to consider three mutually exclusive groups:

1. Those who read only newspaper A ($A \setminus B$).
2. Those who read only newspaper B ($B \setminus A$).
3. Those who read both newspapers A and B ($A \cap B$).

Step 2: Calculate the percentage of people in each disjoint group.

• Percentage reading only A: This group consists of people who read A but do not read B. $P(A \text{ only}) = P(A) - P(A \cap B)$ $P(A \text{ only}) = 25\% - 8\% = 17\%$

• Percentage reading only B: This group consists of people who read B but do not read A. $P(B \text{ only}) = P(B) - P(A \cap B)$ $P(B \text{ only}) = 20\% - 8\% = 12\%$

• Percentage reading both A and B: This is directly given. $P(A \cap B) = 8\%$

Step 3: Calculate the percentage of people looking into advertisements from each disjoint group. We are given the following conditional percentages of people looking into advertisements within these groups:

• $30\%$ of those who read A but not B look into advertisements.
• $40\%$ of those who read B but not A look into advertisements.
• $50\%$ of those who read both A and B look into advertisements.

Now, we calculate the contribution of each group to the total percentage of advertisement lookers:

• Advertisement lookers from the "only A" group: $P(\text{Ad} \cap (A \setminus B)) = P(\text{Ad} | A \setminus B) \times P(A \setminus B)$ $P(\text{Ad} \cap (A \setminus B)) = 30\% \times 17\%$ $P(\text{Ad} \cap (A \setminus B)) = \frac{30}{100} \times 17 = 0.30 \times 17 = 5.1\%$

• Advertisement lookers from the "only B" group: $P(\text{Ad} \cap (B \setminus A)) = P(\text{Ad} | B \setminus A) \times P(B \setminus A)$ $P(\text{Ad} \cap (B \setminus A)) = 40\% \times 12\%$ $P(\text{Ad} \cap (B \setminus A)) = \frac{40}{100} \times 12 = 0.40 \times 12 = 4.8\%$

• Advertisement lookers from the "both A and B" group: $P(\text{Ad} \cap (A \cap B)) = P(\text{Ad} | A \cap B) \times P(A \cap B)$ $P(\text{Ad} \cap (A \cap B)) = 50\% \times 8\%$ $P(\text{Ad} \cap (A \cap B)) = \frac{50}{100} \times 8 = 0.50 \times 8 = 4.0\%$

Step 4: Sum the contributions from all disjoint groups to find the total percentage of advertisement lookers. Since the groups ($A \setminus B$), ($B \setminus A$), and ($A \cap B$) are mutually exclusive, the total percentage of the population looking into advertisements is the sum of the percentages calculated in Step 3.

$P(\text{Ad}) = P(\text{Ad} \cap (A \setminus B)) + P(\text{Ad} \cap (B \setminus A)) + P(\text{Ad} \cap (A \cap B))$ $P(\text{Ad}) = 5.1\% + 4.8\% + 4.0\%$ $P(\text{Ad}) = 13.9\%$

The percentage of the population who look into advertisements is $13.9\%$.

Common Mistakes & Tips

• Double Counting: A common mistake is to apply the advertisement percentages directly to $P(A)$ and $P(B)$ without first isolating the "only A" and "only B" groups. This would lead to overcounting the people who read both newspapers. Always break down the population into disjoint sets first.
• Venn Diagram Visualization: Drawing a Venn diagram is highly recommended. Place $8\%$ in the intersection. Then, calculate $25\% - 8\% = 17\%$ for "A only" and $20\% - 8\% = 12\%$ for "B only". This visually confirms the disjoint groups.
• Percentage Calculations: Be careful when calculating percentages of percentages. For example, $30\%$ of $17\%$ is $(30/100) \times (17/100)$, but in this context, since we are working with percentages of the total population, $30\%$ of $17\%$ of the population means $0.30 \times 17\%$, which is $5.1\%$ of the total population.

Summary

The problem was solved by partitioning the city's population into three mutually exclusive groups based on their newspaper reading habits: those who read only A, those who read only B, and those who read both. The percentage of people looking into advertisements was then calculated for each of these disjoint groups by applying the given conditional percentages. Finally, these individual contributions were summed to obtain the total percentage of the population who look into advertisements.

The final answer is $\boxed{13.9\%}$ which corresponds to option (D).