Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at
- (a) 6 : 15 PM
- (b) 6 : 18 PM
- (c) 6 : 36 PM ✓ UPSC's answer
- (d) 6 : 48 PM
Why the answer is (c)
• At 6:00 the hour hand stands at the 6, which is 180° round the dial from the 12, and the minute hand stands at the 12.
• The minute hand moves 6° a minute and the hour hand 0.5° a minute, so the minute hand gains on it at 5.5° a minute.
• 'Ahead by 3 minutes' means three minute-spaces on the dial, and one minute-space is 6°, so the gap to be opened up is 18°.
• At m minutes past 6 the minute hand is at 6m degrees and the hour hand at 180 + 0.5m, so 6m − (180 + 0.5m) = 18 gives 5.5m = 198 and m = 36.
• The time is therefore 6:36 PM, which is option (c).
Why the other options are wrong
- (a) 6 : 15 PM
- At 6:15 the minute hand is at 90° and the hour hand at 187.5°, so the minute hand is still 97.5° — more than 16 minute-spaces — behind the hour hand, not ahead of it.
- (b) 6 : 18 PM
- At 6:18 the minute hand is at 108° and the hour hand at 189°, leaving it 81° behind; it has not yet caught the hour hand, let alone passed it by 3 minute-spaces.
- (d) 6 : 48 PM
- At 6:48 the minute hand is at 288° and the hour hand at 204°, so it is ahead by 84°, which is 14 minute-spaces rather than 3.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.