This is tricky. Fasten your seat belt. It's going to be a boompy ride.
If it's a 12-hour clock (doesn't show AM or PM), then it has to gain
12 hours in order to appear correct again.
How many times must it gain 3 minutes in order to add up to 12 hours ?
(12 hours) x (60 minutes/hour) / (3 minutes) = 240 times
It has to gain 3 minutes 240 times, in order for the hands to be in the correct positions again. Each of those times takes 1 hour. So the job will be complete in 240 hours = 10 days .
Check:
In 10 days, there are 240 hours.
The clock gains 3 minutes every hour ==> 720 minutes in 240 hours.
In 720 minutes, there are 720/60 = 12 hours yay !
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If you are on a military base and your clocks have 24-hour faces,
then at the same rate of gaining, one of them would take 20 days
to appear to be correct again.
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Note:
It doesn't have to be an analog clock. Cheap digital clocks can
gain or lose time too (if they run on a battery and don't reference
their rate to the 60 Hz power that they're plugged into).
