Clock A gains $2$ minutes per hour over clock B; when do they next show the same time? suppose  we have two clock A and B.  each hour A  clock outruns  by  2 minute  then B  which  always shows correct time.in 1 January   both   clock  was corrected  and  it  was showed 16:00    question is what  what will be first time when both o'clock  will  show again the same time?there is list of  answers from which correct one is  16 January  16:00  please can anybody explain me why?
 A: If the two clock show the same time of day (in a 24-hour sense, since you've said 16:00), then clock A has gained some multiple of 24 hours on clock B.  Since clock A gains 2 minutes per hour, gaining 24 hours = 1440 minutes takes 720 hours, which is 30 days, exactly...
Okay, backing up, let's suppose we're talking 12-hour clocks, so they match up when A has gained 12 hours = 720 minutes, which takes 360 hours, which is exactly 15 days.  So, 15 days from 16:00 Jan 1 is 16:00 Jan 16, at which time clock B will correctly show 16:00 Jan 16 and clock A will show 04:00 Jan 17, so both will show "4 o'clock."
A: So let's set it up this way:
Jan 1
Hour 1:
Clock A: 16:00, Clock B: 16:02
Hour 2:
Clock A: 17:00, Clock B: 17:04
Hour 3:
Clock A: 18:00, Clock B: 18:06
60 minutes in clock A corresponds to 62 minutes in clock B.
We can see that in 30 hours Clock B will be a full hour ahead (30 * 2minutes per hour = 60 minutes extra)
Hour 30
Jan 3
Clock A: 0:00 Clock B: 1:00
Now in another 30 hours it will be 2 hours ahead
Jan 5
Clock A: 6:00 Clock B: 8:00
And we see the pattern that's the key to the solution: Every 30 hours the clock goes by one extra hour. We know that we need 24 extra hours to overlap by one day. 24 sets of 30 in Military time or 12 sets of 30 in regular (American) time.
The time at which the overlap will occur is 12*30 (24*30) hours past Jan 1 16:00 (4:00)
This is 15 or 30 days past Jan 1, at 4:00 (16:00)
