What is the time?

When do the hour and the minute hands of the clock lie on top of each other, and also between the numbers 1 and 2?

One such time is aroud 1:05 - at 1 o'clock and 5.454... minutes. This can be easily found by solving the linear equation x=12(x-5) => x=60/11.

Another time you possibly didn't think about is 12:00 o'clock. At that time, the hour and minute hands are on top of each other and just between the two digits of the number 12.