If both the hour hand and the minute hand start at the same position at 12 o'clock, when is the first time, correct to a fraction of a minute, that the two hands will be together again?

Answer 1

This will happen a little after 1 o clock.

Let's do this in terms of degrees, where the entire clock is 360°.

Every minute, the minute hand moves through (1/60) of the clock, which is 6°

Every minute, the hour hand moves through (1/60) of the gap between two hours (the gap itself is 30°), which is 0.5°

At 1:05, the minute hand will be 30° away from 12 o clock, and the hour hand will be ( 30 + (0.5)*5 ) = 32.5° away from 12 o clock, hence they will be 2.5° apart.

Now,

2.5° plus the distance traveled by hour hand = distance traveled by minute hand

=> 2.5° + (t)*(0.5°/min) = (t)*(6°/min)
Which gives you t = 0.4545 min
Which gives you t = 27.27 seconds

Hence, they will meet at t = 1:05:27.27

Answer 2

maybe one hour and five minutes like five past one?

Answer 3

Do it the way Arora did. I did not notice the fraction of a minute part.