Q1391 Jack looked at the clock next to his front door as he left home one afternoon to visit Jill and watch a TV programme. Arriving exactly as the programme started, he set out for home again when it finished one hour later. As he did so he looked at her clock and noticed that it showed the same time as his had done when he left home. Puzzling over how Jill's clock could be so wrong, Jack travelled home at half the speed of his earlier journey. When he arrived home he saw from his clock that the whole expedition had taken two hours and fifteen minutes. He still hadn't worked out about Jill's clock and so he called her up on the phone. Jill explained that her clock was actually correct (as was Jack's), but it was an ``anticlockwise clock'' on which the hands travel in the opposite direction from usual. Jack had been in such a hurry to leave that he hadn't noticed the numbers on the clock face going the ``wrong'' way around the dial. At what time did Jack leave home?
SOLUTION Jack's travel time home was twice that of his outward journey; the total travel time, plus the $60$ minutes' visit, adds up to $2$ hours $15$ minutes. So Jack's outward travel time was $25$ minutes, and the time when he looked at Jill's clock was $1$ hour $25$ minutes after looking at his own. Since it then appeared to show the time at which Jack left, double this time plus the $1$ hour $25$ minute stay must add up to $12$ hours (if you have trouble seeing why, draw pictures of the two clocks). So the time at which Jack left was 5:17:30.
Comment. The times involved could have added up to $24$ hours instead of $12$, in which case Jack's departure time would have been 11:17:30. However this does not fit in with the information that he ``left home one afternoon''.