you've got an analog watch with a second hand. how many times a day do all three of the watch's hands overlap?

• in 12 hrs, hour hand and minute hand overlap 11 times.
  12:00:00
  1:05:27 3/11
  2:10:54 6/11
  3 :16:21 9/11
  4:21:49 1/11
  5:27:14 4/11
  6:32:43 7/11
  7: 38:10 10/11
  8:43:38 2/11
  9:49:05 5/11
  10:54:32 8/11
  
  But all three ( hr min and sec hand) overlap only at 0000 hrs and 1200 hrs.
• 11 years agoHelpfull: Yes(25) No(0)
• At 12 midnight and 12 noon, all the three hands overlap.
  Reason: The hour hand moves 30 degrees in an hour, the minute hand moves 6 degrees in hour and the second hand moves 0.1 degrees. Finding the HCF of 30,6 and 0.1 and finding the next times when the hands meet, we get that at only those two times all the three hands gets overlapped.
• 11 years agoHelpfull: Yes(6) No(5)
• this is a concept of relative circular motion of 3 objects..
  here in one minute, seond hand wl cover 360 degree, minute hand 6 degree, hour hand 1/2 degree.
  step1:
  relative speed of second to minute hand= (360-6)=354 degree
  so they will meet after every (360/354)minute
  step2:
  relative speed of second to hour hand= (360-1/2)=719/2 degree
  so they will meet after every (360*2/719)= 720/719 minutes.
  step 3:
  now for all three hands will meet after every LHS(360/354,720/719) minutes..
  LHS=720/1 minutes
  so in a day they wl meet 24*60/720 times
  ans: 3 times..
  thank you
  
• 11 years agoHelpfull: Yes(6) No(2)
• 2 times...because at any other time minute and hour hand may overlap (11 times basically) but the second hand will differ from fraction of seconds
• 11 years agoHelpfull: Yes(2) No(1)
• 1 Day=24 hours= 2 times (12 hours)
  
  12 am 00:00:00
  1 am 01:01:01
  2 am 02:02:02
  .
  .
  .
  11 am 11:11:11
  12 pm 00:00:00
  .
  .
  .
  11 pm 11:11:11
  
  
  Net Total =24 times
• 11 years agoHelpfull: Yes(1) No(20)
• @Aman
  
  Good answer.
  Another thing is if the battery of the watch changed for once then 3 hands never will overlap. (i.e battery of the watch may not end exactly at 12.00)
• 11 years agoHelpfull: Yes(0) No(8)
• 2 times.. at 12.00 A.m and 12.00 P.m
• 10 years agoHelpfull: Yes(0) No(0)
• 22 times they will overlap
• 10 years agoHelpfull: Yes(0) No(3)
• In T hours, the minute hand completes T laps. In the same amount of time, the hour hand completes T/12 laps.
  
  The first time the minute and hour hands overlap, the minute hand would have completed 1 lap more than the hour hand. So we have T = T/12 + 1. This implies that the first overlap happens after T = 12/11 hours (~1:05 am). Similarly, the second time they overlap, the minute hand would have completed two more laps than the hour hand. So for N overlaps, we have T = T/12 + N.
  
  Since we have 24 hours in a day, we can solve the above equation for N
  
  24 = 24/12 + N
  24 = 2 + N
  N = 22
  
  Thus, the hands of a clock overlap 22 times a day. Thus the hands of the clock overlap at 12:00, ~1:05, ~2:10, ~3:15, ~4:20, ~5:25, ~6:30, ~7:35, ~8:40, ~9:45, ~10:50. Note that there is no ~11:55. This becomes 12:00.
• 9 years agoHelpfull: Yes(0) No(0)
• Well, according to Aman answer, the hours and minutes hand overlap at times which are not integer number of seconds (except the initial overlap at noon/midnight).
  And since the seconds hand is supposed to show integer number (of seconds) only, the overlap of all three hands will never occur (again, except the noon/midnight case)...
• 8 years agoHelpfull: Yes(0) No(0)
• analog watch overlaps only two times in 24 hours because when 0 degree angle is there and 360 degree angle is there.
• 6 years agoHelpfull: Yes(0) No(0)