Puzzle ZDJB 

Workings

Hint
The answer occurs on the hour.

Answers
Midnight, 10 days later. They will both show 4 o'clock.
Midnight, 30 days later. They will both show 12 o'clock.
Reasoning #1
If the first clock were to show 2 o'clock, it would have gained 2 hours. In order for the second clock to also show 2 o'clock, it would have had to have lost 10 hours. This is a total of 12 hours gained and lost.

If the first clock were to show 6 o'clock, it would have gained 6 hours. In order for the second clock to also show 2 o'clock, it would have had to have lost 6 hours. This is also total of 12 hours gained and lost.

It doesn't matter what time they both show, in order for the clocks to show the same time, the total time gained and lost is always 12 hours.

We know that for every hour that has passed, the total time gained and lost is 1 + 2 = 3 minutes.

The total time gained and lost will equal 12 hours when 12 x 60 ÷ 3 = 240 hours have passed.

The first clock will have gained 240 x 1 minutes = 240 minutes = 4 hours.

The second clock will have lost 240 x 2 minutes = 480 minutes = 8 hours.

So, they will both show 4 o'clock, 10 days later.

Reasoning #2
In the first answer, we can see that 10 days later, the clocks both show 4 o'clock.

If we move forward another 10 days, both clocks would show 8 o'clock.

If we move forward another 10 days, both clocks would show 12 o'clock.

This will be the correct time, as we are moving a whole number of days each time, and we started at 12 o'clock.

So, they will both show 12 o'clock, 30 days later.