UPSC Prelims 2001 Q144: Science & Technology – Quantitative Aptitude
In a survey, it was found that 80% of those surveyed owned a car while 60% of those
surveyed owned a mobile phone. If 55% owned both a car and a mobile phone, what per cent of
those surveyed owned a car or a mobile phone or both?
- (a) 65%
- (b) 80%
- (c) 85%
- (d) 97.5%
Answer: (c) 85%
Explanation
Per cent of those surveyed who owned a car or a mobile phone or both
| Set | Per cent |
|---|---|
| Car (C) | 80% |
| Mobile phone (M) | 60% |
| Both (C ∩ M) | 55% |
| Car or phone or both (C ∪ M) | 80 + 60 − 55 = 85% |
Given: 80% of those surveyed owned a car, 60% owned a mobile phone and 55% owned both. By the addition rule, n(C ∪ M) = n(C) + n(M) − n(C ∩ M) = 80 + 60 − 55 = 85%.
Answer: (c) 85%.
Why each option is right or wrong
- (a) Incorrect. 65% is too small; it is less than those owning a car.
- (b) Incorrect. 80% is just the car owners alone.
- (c) CORRECT. 80 + 60 − 55 = 85% own a car or phone.
- (d) Incorrect. 97.5% does not follow from the data.
Key point
15% owned neither a car nor a mobile phone.
Remember
80 + 60 − 55 = 85% own a car or phone or both.
Common mistakes
- Adding 80 + 60 and forgetting to subtract the overlap.
- Taking only the larger group (80%).
- Subtracting the overlap of both sets twice.
Exam tip
Union = A + B − both; the answer must be at least the larger set.
Sources
- NCERT Class XI — Mathematics, chapter 'Sets' (union of sets)
- R. S. Aggarwal — Quantitative Aptitude, chapter 'Venn diagrams'