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?

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

maybe one hour and five minutes like five past one?

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

To find the first time when the hour and minute hand of a clock are together again after starting at the same position at 12 o'clock, we need to determine the time it takes for the minute hand to overtake the hour hand.

First, let's calculate the relative speed at which the minute hand overtakes the hour hand:

The minute hand moves 360 degrees in 60 minutes, so its speed is 360/60 = 6 degrees per minute.
The hour hand moves 360 degrees in 12 hours, which is 12 * 60 = 720 minutes. Therefore, its speed is 360/720 = 0.5 degrees per minute.

Since the minute hand moves at a faster speed, it will eventually overtake the hour hand. The time it takes for this to happen can be found by dividing the initial angular separation between the two hands by the relative speed.

At 12 o'clock, the hour hand and the minute hand are on top of each other, so the initial angular separation is 0 degrees.

Dividing this by the relative speed of 6 - 0.5 = 5.5 degrees per minute, we get:

0 / 5.5 = 0 minutes.

Therefore, the first time the hour and minute hand will coincide again after starting at the same position at 12 o'clock is exactly 0 minutes past 12 o'clock.