Sets, Relations & Functions
Venn Diagrams and Conditional Counting
Grade 11
Question:
<p>Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then, the percentage of the population who look into advertisements is</p>
<p>(a) 13.5</p>
<p>(b) 13</p>
<p>(c) 12.8</p>
<p>(d) 13.9</p>
Step-by-Step Solution
Key Concept: Partition the population into disjoint subsets (A only, B only, both) and apply the given percentages to each subset, then sum the results.
<p><strong>Step 1:</strong> Let the population of city be 100.</p><p><strong>Step 2:</strong> Then, $n(A) = 25$, $n(B) = 20$ and $n(A \cap B) = 8$</p><p><strong>Step 3:</strong> From the Venn diagram:</p><p>$n(A \cap B^c) = n(A) - n(A \cap B) = 25 - 8 = 17$</p><p>$n(A^c \cap B) = n(B) - n(A \cap B) = 20 - 8 = 12$</p><p><strong>Step 4:</strong> Percentage of population who look into advertisements is:</p><p>$$= \frac{30}{100} \times n(A \cap B^c) + \frac{40}{100} \times n(A^c \cap B) + \frac{50}{100} \times n(A \cap B)$$</p><p>$$= \frac{30}{100} \times 17 + \frac{40}{100} \times 12 + \frac{50}{100} \times 8$$</p><p>$$= 5.1 + 4.8 + 4 = 13.9$$</p><p>∴ Answer is (d) 13.9</p>
Correct Answer: D