Tuesday, May 02, 2006

Hands of a clock

How many times in a day do the hands of a clock overlap? Can you list out the times too? Explain your methodology and not just list the answer.

Level: Medium

posted by Karthik Nagaraj at 8:11 AM

7 Comments:

Anonymous Anonymous said...

Every hour the minute hand travels 360 degrees and the hour hand travels 30 degrees.This is got by dividing the clock dial in to 12 equal parts.

Every hour the hour hand and the minute hand meet atmost once.

Hence in a day both the hands meet 24 times.

Coming to the times at which they meet.

In a minute the hour hand travels 0.5 degrees and the minute hand travels 6 degrees.

At the end of each hour the hour hand has moved to 30 degrees, 60 degrees(30*2), 90 degrees(30*3) and so on.At the end of each hour the minute hand will be at 12(0 degrees).
To catch up with the hour hand the minute hand has to travel from 0 th position
30+x in the first hour,
60+2x in the second hour,
90+3x in the third hour and so on.



It would require the minute hand
(360+(30+x))/6 minutes, say t1, to meet with the hour hand for the first time.
Considering the (30+x) distance travelled by the hour hand at the same time, t1 is also equal to (30+x)/0.5, say t2.
equating t1 and t2
(360+(30+x))/6=(30+x)/0.5
x=30/11

so substituting for x in either t1 or t2
they meet after 720/11 minutes from 12:00 midnight.

Subsequently,
they meet after 2*(30+(30/11))/0.5 from 12:00 midnight.

3*(30+(30/11))/0.5 from 12:00 midnight and so on.

1:58 AM, May 04, 2006  
Blogger Karthik Nagaraj said...

Mayura,
Sorry the answer is not 24. Write out the times when it will overlap and then youwill see why it is not 24. Maybe are you counting something twice? Think about it !

8:20 AM, May 05, 2006  
Anonymous Anonymous said...

the minute hand goes around 24 times,
the hour hand goes around 2 times.
assume both start at 00 00 together,the answer is 24-2 = 22. it is simlar to a car race where 2 cars start together and the cars do 24 and2 laps resp.

7:22 AM, May 06, 2006  
Blogger Karthik Nagaraj said...

Vikram,

This has made me think - you have gotten the answer right (it is 22) but I'm wondering if your logic is completely correct. Let me think about this more. In the meantime, work out the actual times they overlap with each other.

Good thinking!

1:39 PM, May 06, 2006  
Blogger Karthik Nagaraj said...

I think I agree with you on this. Because according to your logic in a 12 hour period, the hour hand goes around once and the minute hand 12 times, so they overlap 12-1=11 times and times 2 for a 24 hour period. Good reasoning. Now the next part - what the times do they overlap?

10:40 AM, May 10, 2006  
Anonymous Anonymous said...

Bobach writes:
It takes 5.454545... minutes more each hour for the hands to cross. Therefore, they will cross at:
1:05 5/11
2:10 10/11
3:16 4/11
4:21 9/11
5:27 3/11
6:32 8/11
7:38 2/11
8:43 7/11
9:49 1/11
10:54 6/11
12:00
for both am and pm.

8:34 PM, May 12, 2006  
Blogger Karthik Nagaraj said...

Yes Bobach,
Thats correct. Just to express it mathemtically we have the hour hand complete one rotation of 360 degrees in 12 hours -> speed is 1/2 degree per minute. The minute hand completes one rotation in 60 minutes -> 6 degrees per minute. At 1:00 the hands make an angle of 30 degrees. Lets say they meet after x minutes. So the hour hand would have travelled x/2 degrees from its 30 degrees position and the minute hand would have travelled 6x degrees. So we get an equation as:

30+x/2 = 6x

x=5.454545 -> 60/11

So they meet at 1:05:27.27

Now at 2:00 the equation will give 120/11 -> 10.90909, 2:10:54.54

At 3:00 the equation will give
180/11 -> 3:11:21.81 and so on ad Bobach has already written it out. Credits go to Vikram and Bobach.

So, finally this puzzle is SOLVED!

8:43 AM, May 13, 2006  

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