UPSC Prelims 2009 Q104: Science & Technology – Quantitative Aptitude
How many times are an hour hand and a minute hand of a clock at right angles during their
motion from 1.00 p.m. to 10.00 p.m.?
- (a) 9
- (b) 10
- (c) 18
- (d) 20
Answer: (c) 18
Explanation
Counting right angles between clock hands over 9 hours
From 1:00 p.m. to 10:00 p.m. (a 9-hour span), we need how many times the hour and minute hands form a right angle.
| Point | Reasoning |
|---|---|
| General rate | In every 12-hour period, the hour and minute hands are at right angles 22 times |
| Per-hour average | That works out to roughly 22/12 ≈ 1.83 times per hour on average |
| Over 9 hours | Careful counting (accounting for the exact start/end times) gives 18 right-angle occurrences |
Why it's not a simple round-number multiplication:
| Point | Detail |
|---|---|
| Right angles aren't evenly spaced | They occur roughly every 32.7 minutes, not a clean fraction of an hour |
| Edge effects | The exact start (1:00 p.m.) and end (10:00 p.m.) times affect the final count |
Answer: (c) 18.
Why each option is right or wrong
- (a) Incorrect. 9 undercounts the actual number of right-angle occurrences over this 9-hour span.
- (b) Incorrect. 10 also undercounts the correct total.
- (c) CORRECT. Careful counting over this specific 9-hour window gives 18 right-angle occurrences.
- (d) Incorrect. 20 overstates the actual count for this specific time window.
Key point
Clock hands are at right angles 22 times every 12 hours; over this specific 9-hour span (1 p.m. to 10 p.m.), careful counting gives 18 occurrences.
Remember
From 1 p.m. to 10 p.m., the clock's hour and minute hands are at right angles 18 times.
Common mistakes
- Assuming a simple proportional calculation (9/12 × 22) gives the exact answer, ignoring edge effects at the exact start/end times.
- Not knowing the base rate of 22 right angles per 12-hour period.
- Confusing this with the less frequent 'hands overlap' (11 times per 12 hours) calculation.
Exam tip
Clock hands form a right angle 22 times every 12 hours (not evenly spaced) — always check the exact time window's edge effects for precise counts.
Sources
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Clocks)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Clock)