Coincidence of Clock Hands with Overlap Time Examples
Coincidence of Clock Hands means the hour and minute hands overlap at the same position. Their relative speed determines the time between two successive coincidences. This page explains the coincidence of clock hands formula, the number of coincidences in a given period, exact meeting times and solved aptitude examples.
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What Is Coincidence of Clock Hands?
The minute hand moves at 6° per minute, while the hour hand moves at 0.5° per minute. Therefore, the minute hand gains on the hour hand at 5.5° per minute. Since one complete gain is 360°, the time between two coincidences is 360 ÷ 5.5 = 720/11 minutes. The hands coincide 11 times in 12 hours and 22 times in 24 hours.
Coincidence of Clock Hands Video Lessons
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Clock Coincidence, Right Angle and Opposite Times
Learn how to determine clock times when the hour and minute hands coincide, form a right angle, or lie in opposite positions.
Coincidence of Clock Hands Formula & Tricks
Important Formulas
The minute hand gains 5.5 degrees on the hour hand every minute.
The minute hand must gain 360 degrees to overlap the hour hand again.
Here, H is the hour number from 0 to 11 and m is the number of minutes after H o'clock. The formula gives the next coincidence in that hour interval.
The initial coincidence at 12:00 is included in the usual count over a closed 12-hour cycle.
Quick Tricks
For consecutive coincidence questions, add 720/11 minutes to the known coincidence time.
The hands coincide 11 times in 12 hours, not 12 times, because the minute hand gains only 11 complete rounds on the hour hand in that period.
To find the coincidence between H and H + 1 o'clock, substitute H in the formula and simplify the minutes.
Coincidence of Clock Hands Concepts
Relative Motion of the Clock Hands
The minute hand covers 360° in 60 minutes, so its speed is 6° per minute. The hour hand covers 30° in 60 minutes, so its speed is 0.5° per minute. Their relative speed is 6 − 0.5 = 5.5° per minute.
Time Between Two Coincidences
After one overlap, the minute hand must gain another 360° on the hour hand. Thus, time = 360 ÷ 5.5 = 720/11 minutes. The decimal form is approximately 65.45 minutes.
Finding the Exact Coincidence Time
At H o'clock, the hour hand is H × 30° ahead of the 12 mark. The minute hand covers this angular distance at the relative rate of 5.5° per minute. Therefore, m = 30H ÷ 5.5 = 60H/11.
Counting Coincidences in a Day
The coincidence at 12:00 starts a 12-hour cycle, and the hands meet at 11 distinct positions before returning to the same starting position. Hence, the count is 11 per 12 hours and 22 per 24 hours.
Coincidence of Clock Hands Visual Guide
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Clock hands coinciding at 12:00
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Coincidence of Clock Hands: Revision Points
Use relative angular speed to calculate overlap times and intervals.
- Minute-hand speed = 6° per minute.
- Hour-hand speed = 0.5° per minute.
- Relative speed = 5.5° per minute.
- Time between coincidences = 720/11 minutes = 65 5/11 minutes.
- Coincidence after H o'clock occurs at H:60H/11 minutes.
- There are 11 coincidences in 12 hours and 22 in 24 hours.
- The exact minute may be fractional, such as 5 5/11 or 38 2/11 minutes.
Coincidence of Clock Hands FAQs
What is the coincidence of clock hands formula?
The time between two successive coincidences is 720/11 minutes, or 65 5/11 minutes. The coincidence after H o'clock occurs at m = 60H/11 minutes.
At what time do the hands coincide between 3 and 4 o'clock?
Using m = 60H/11, m = 60 × 3/11 = 180/11 = 16 4/11 minutes. Therefore, the hands coincide at 3:16 4/11.
How many times do clock hands coincide in 12 hours?
The hands coincide 11 times in 12 hours. They coincide 22 times in 24 hours.
Why do the hands not coincide 12 times in 12 hours?
The hands return to the same position at the end of the 12-hour cycle, so that endpoint is the starting coincidence repeated. There are 11 distinct overlaps during one 12-hour cycle.
When do the hands coincide between 9 and 10 o'clock?
m = 60 × 9/11 = 540/11 = 49 1/11 minutes. The time is 9:49 1/11.
What is the time gap between two consecutive coincidences?
The gap is 720/11 minutes, which equals 65 5/11 minutes or approximately 1 hour 5 minutes 27 3/11 seconds.