UPSC Prelims 2008 Q7: Science & Technology – Quantitative Aptitude
In an examination, 70% of the students passed in the Paper I, and 60% of the students passed
in the Paper II. 15% of the students failed in both the papers while 270 students passed in both
the papers. What is the total number of students?
- (a) 600
- (b) 580
- (c) 560
- (d) 540
Answer: (a) 600
Explanation
Finding total students from overlapping pass/fail percentages
With 70% passing Paper I, 60% passing Paper II, 15% failing both, and 270 students passing both, we need the total number of students.
| Step | Calculation |
|---|---|
| Pass at least one paper | 100% − 15% (fail both) = 85% |
| Pass both papers (inclusion-exclusion) | P(I) + P(II) − P(at least one) = 70 + 60 − 85 = 45% |
| 45% of total = 270 students | Total = 270 / 0.45 = 600 |
Why inclusion-exclusion applies here:
| Point | Detail |
|---|---|
| Formula | P(A∩B) = P(A) + P(B) − P(A∪B) |
| P(A∪B) | Equals 100% minus the percentage failing both |
Answer: (a) 600.
Why each option is right or wrong
- (a) CORRECT. 45% of the total equals 270, so total = 270/0.45 = 600.
- (b) Incorrect. 580 does not match the correct percentage-based calculation.
- (c) Incorrect. 560 also doesn't match.
- (d) Incorrect. 540 understates the correct total.
Key point
Using inclusion-exclusion: 45% of students passed both papers (70+60−85, where 85% is 'at least one pass'). Since 45% = 270 students, the total is 270/0.45 = 600.
Remember
70% pass I, 60% pass II, 15% fail both, 270 pass both: total students = 600.
Common mistakes
- Forgetting to first calculate 'at least one pass' (100% minus 'fail both') before applying inclusion-exclusion.
- Using the wrong formula, directly adding or subtracting percentages without the inclusion-exclusion correction.
- Making a division error in the final total-students calculation.
Exam tip
For overlapping pass/fail percentage problems: use P(A∩B) = P(A)+P(B)−P(A∪B), where P(A∪B) = 100% − (% failing both).
Sources
- NCERT Class XI Mathematics — Sets
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Percentage)