Sets, Relations & Functions
Application of sets (word problems)
Grade 11
Question:
<p>A survey shows that 63% of the people watch a news channel whereas, 76% watch an entertainment channel at a particular time. If \(x\%\) of the people watch both types of channels, then</p>
<p>\(x = 35\)</p>
<p>\(x \geq 63\)</p>
<p>\(39 \leq x \leq 63\)</p>
<p>\(x = 39\)</p>
Step-by-Step Solution
Key Concept: Use the principle of inclusion-exclusion: the sum of individual percentages minus overlap cannot exceed 100%, and the overlap must be at least the difference between their sum and 100%. This gives bounds on x: max(0, 63+76-100) ≤ x ≤ min(63, 76).
<p><strong>Step 1:</strong> Apply inclusion-exclusion principle. If A = people watching news (63%) and B = people watching entertainment (76%), then:</p><p>n(A ∪ B) = n(A) + n(B) − n(A ∩ B)</p><p>n(A ∪ B) = 63 + 76 − x = 139 − x</p><p><strong>Step 2:</strong> Since the union cannot exceed 100% of total population:</p><p>139 − x ≤ 100</p><p>x ≥ 39</p><p><strong>Step 3:</strong> Also, the intersection cannot exceed the smaller set:</p><p>x ≤ min(63, 76) = 63</p><p><strong>Step 4:</strong> Additionally, intersection must be non-negative and the complement constraint gives x ≥ 63 + 76 − 100 = 39</p><p><strong>Conclusion:</strong> The valid range is <strong>39 ≤ x ≤ 63</strong></p><p>∴ Answer: C (requires identifying that x must satisfy 39 ≤ x ≤ 63)</p>
Correct Answer: C