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Albert Einstein was standing on the station platform thinking about relativity when he noticed that he could see two station clocks. Each clock was digital, showing only hours and minutes.

He observed that the display on one clock changed to the next minute $10$ seconds before the correct time, whereas the display on the other clock changed to the next minute $10$ seconds after the correct time.

For what fraction of the day did the clocks show the same time?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.  

This problem is taken from the UKMT Mathematical Challenges.