UPSC Prelims 2009 Q102: Science & Technology – Quantitative Aptitude
In an examination, there are three subjects A, B and
C. A student has to pass in each subject. 20% students failed in A, 22% students failed in B and
16% failed in
C. The total number of students passing the whole examination lies between
- (a) 42% and 84%
- (b) 42% and 78%
- (c) 58% and 78%
- (d) 58% and 84%
Answer: (b) 42% and 78%
Explanation
Finding the range of students passing all three subjects
With failure rates of 20% (A), 22% (B), and 16% (C), we need the possible range for students passing all three.
| Bound | Reasoning |
|---|---|
| Minimum pass % | Occurs when failures don't overlap at all: total failed = 20+22+16 = 58%, so minimum passing = 100−58 = 42% |
| Maximum pass % | Occurs when failures overlap as much as possible (all failing students fail in multiple subjects, concentrated within the largest failure group): the largest single failure rate is 22% (subject B), so at most 22% could have failed overall, giving maximum passing = 100−22 = 78% |
Why this range makes sense:
| Point | Detail |
|---|---|
| Worst case (no overlap) | Failures add up directly, minimizing the pass percentage |
| Best case (maximum overlap) | Failures concentrate within the largest failing group, maximizing the pass percentage |
Answer: (b) 42% and 78%.
Why each option is right or wrong
- (a) Incorrect. 84% overstates the maximum possible passing percentage.
- (b) CORRECT. The passing percentage ranges between 42% (no overlap) and 78% (maximum overlap).
- (c) Incorrect. 58% as a lower bound doesn't match the correct no-overlap calculation.
- (d) Incorrect. Both bounds in this option are inconsistent with the correct overlap analysis.
Key point
The percentage of students passing all three subjects lies between 42% (if failures don't overlap at all) and 78% (if failures overlap as much as possible, concentrated within the largest single failure rate of 22%).
Remember
With failure rates 20%, 22%, 16%: passing-all-three percentage ranges between 42% and 78%.
Common mistakes
- Forgetting to consider both extremes — no overlap and maximum overlap — when establishing a range.
- Using the sum of failure rates as the upper bound instead of correctly identifying it as the lower bound for the PASS percentage.
- Not identifying the largest individual failure rate as the key figure for the maximum-overlap scenario.
Exam tip
For 'pass all subjects' range problems: minimum pass = 100% − (sum of failure rates); maximum pass = 100% − (largest single failure rate).
Sources
- NCERT Class XI Mathematics — Sets
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Percentages and Venn diagrams)