When Do A Clock’s Hands Overlap?
It has been nearly 2 months since I last posted. I thought it was time for my next post. So, I decided to write about “time”
Right from a young age, I have had a fascination for clocks (especially analog clocks with the minute and hour hands). For me, the beauty of a clock lies in its hands
Given this, many times I have wondered how often do the hands of a clock meet. Obviously, they meet at 12 o clock. But after that, how many times do they meet? A cursory glance at the clock tell that they meet at approximately 1:05, 2:10, 3:15 etc. But that did not satisfy me. Is it possible to determine exactly at what time they meet?
Few years back, while preparing for the CAT exam, I encountered a lot of such clock-related questions while preparing for the Quantitative Ability section. That was when, I came up with a formula to determine at what time exactly the hands of a clock meet
The Basic Theory Behind Clocks
No. I am not going into the theory of how clocks/watches are manufactured or how they run. There is a separate field of study for that called horology. Here, I will be explaining the mathematics behind the movement of the hours and minutes hand of an analog clock
Consider a clock as a circle. As you know, a circle has 360 degrees. In a 1-hour rotation, the minute hand covers one entire circle around the clock.
In 1 hour (60 minutes), it covers 360 degrees.
Hence, in 1 minute, the minute hand will cover 360/60 = 6 degrees. This is the first important thing to remember
Now, in the meanwhile, the hour hand is not sitting idle. By the time the minute hand has completed one full rotation (1 hour), the hour hand would have moved ahead by 1 hour. In terms of degrees, how much is this?
A clock is numbered from 1 to 12. Total of 360 degrees. So, the distance between 2 numbers in a clock is 360/12 = 30 degrees. In a 1-hour time frame, the hour hand moves 30 degrees. This is the next point you should keep in mind
In a 60 minute rotation of the minute hand, the hour hand moves 30 degrees
Hence, in 1 minute, the hour hand moves 30/60 = 0.5 degrees
The Actual Calculation
Assume the time is 1o clock. The minute hand is at the number 12 (or 0). The hour hand is at the 1 clock mark i.e 30 degrees ahead of the minute hand
Let the time at which both hands meet be y minutes from now
In one minute, the minute hand covers 6 degrees. Thus, y minutes from now, the minute hand will be at (0 + 6y) = 6y degrees
Similarly, y minutes from now, the hour hand would have moved 0.5*y degrees
But it was already 30 degrees ahead of the minute hand (at 1o clock). So, after y minutes, the hour hand will be at (30 + 0.5y) degrees
Equating these two, we get 0 + 6y = 30 + 0.5y
Re-arranging, we get 5.5y = 30 => y = 5.45 minutes
Converting this to seconds we get 5 mins 27 secs
This implies that both hands meet at 5 mins 27 secs past 1 i.e at 1:05:27
Extending this further
This is for 1o clock, what about 2o clock?
Simple. At 2o clock, the hour hand will be 30*2 = 60 degrees ahead of the minute hand
Hence the equation becomes, 0+6y = 30*2 + 0.5y = 60 + 0.5y =>y = 60/5.5 = 10.90 = 10 mins 54 secs
The hands meet at 2:10:54
Generalizing, we get 0+6y = 30*x + 0.5y
=> y = 30*x/5.5 where x is the hour
If I put x = 6, I get y = 32.73 = 32 mins 44 secs implying that the hands meet at 06:32:44
Substituting 1, 2 … 11 in the above formula, we get the entire list of times at which the hands intersect. At x=11, you get y = 60 implying they meet at 60 mins after 11 i.e at 12 clock
With this, the clock has completed one full rotation.
Hence, we conclude that the hour and minute hand totally meet 11 times in a 12-hour duration
One way to confirm our calculations is that, since the hands meet 11 times in a 12-hour duration, they should meet every 12/11 hours = 1.09 hrs i.e every 1 hr 5mins 27 secs which matches with the timings we have obtained above
Thanks for reading!!
Originally published at http://infinitesimallysmallcom.wordpress.com on February 27, 2022.
