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?

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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."

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thanks very much yes i was thinking about this problem by this way but wanted to be sure and also know exactly way how to solve problems like this one because it is one task from national examination for gaining grant from state thanks ones again –  dato datuashvili Jun 26 '11 at 7:35
@user3196: I've written a number of contest problems based around the idea of clocks running at different rates and it's usually a matter of thinking about the rate at which the difference between the two clocks changes (or the rate at which one fast or slow clock deviates from the actual time), then figuring out how to use that rate to answer the actual question. –  Isaac Jun 26 '11 at 7:43
yes you are right. in general i have found myself that in this case A o'clock outruns by 1 hour in 30 hour just wanted exactly way how solve it because it is lack of time in exam as ussualy so ask –  dato datuashvili Jun 26 '11 at 7:48

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)

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great explanation @aengle thanks u very much –  dato datuashvili Jun 26 '11 at 7:39
one question more please so if it would be not 2 but by n minute n<60 then it's solution will be what? –  dato datuashvili Jun 26 '11 at 7:43
@user3196 Now that you know how to do the problem for $n=2$ why not figure it out yourself for any $n<60$? –  barf Jun 26 '11 at 12:06
JTL is right. Life is about learning, and mathematics is about generalizing. Half of the fun of solving the problem in a particular case $n=2$ is realizing you've figured out how to solve it for $n<60$! –  mathmath8128 Jun 26 '11 at 16:33