Sets, Relations & Functions
Application of sets (word problems)
Grade 11
Question:
<p>In statistical survey of 1003 families of Kolkata, it was found that 63 families has neither a radio nor a TV, 794 families has a radio and 187 has TV. The number of families in that group having both a radio and a TV is</p>
<p>36</p>
<p>41</p>
<p>32</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle: |A ∪ B| = |A| + |B| - |A ∩ B|, where the total families equals those with at least one device plus those with neither.
<p><strong>Step 1:</strong> Identify the given information.</p><p>Total families = 1003</p><p>Families with neither radio nor TV = 63</p><p>Families with radio, |R| = 794</p><p>Families with TV, |T| = 187</p><p><strong>Step 2:</strong> Find families with at least one device.</p><p>Families with at least one device = Total - Neither = 1003 - 63 = 940</p><p><strong>Step 3:</strong> Apply inclusion-exclusion principle.</p><p>|R ∪ T| = |R| + |T| - |R ∩ T|</p><p>940 = 794 + 187 - |R ∩ T|</p><p>940 = 981 - |R ∩ T|</p><p>|R ∩ T| = 981 - 940 = 41</p><p><strong>Step 4:</strong> Verify: 794 + 187 - 41 = 940 ✓ and 940 + 63 = 1003 ✓</p><p>∴ Answer: <strong>41 families</strong> (Option B)</p>
Correct Answer: B