To determine when the two faulty clocks will show the same time again, we need to find the time interval after which the time difference between them becomes zero.
Let's consider the time elapsed in hours as 't'. The first faulty clock gains 10 minutes every hour, which is equivalent to 10/60 = 1/6 hours. Therefore, the time shown by the first clock after 't' hours will be 10 AM + t/6 hours.
Similarly, the second faulty clock loses 5 minutes every hour, which is equivalent to -5/60 = -1/12 hours. Thus, the time shown by the second clock after 't' hours will be 10 AM - t/12 hours.
To find when the clocks will show the same time, we need to equate the expressions:
10 AM + t/6 = 10 AM - t/12
Now we can solve this equation:
t/6 + t/12 = 0
Multiplying through by 12, we get:
2t + t = 0
3t = 0
t = 0
The solution tells us that the time difference between the clocks becomes zero after 0 hours, which means they will show the same time again immediately. Therefore, the two faulty clocks will show the same time again right after they were set at 10 AM.