John's clock is broken. The minute hand rotates around the clock correctly, but the hour hand is stuck in the three o'clock position. Suppose John first looks at the clock when the hands are aligned and it shows 3:15. He looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring - 135°. How many degrees could the minute hand have rotated to reach its current position?

It moved 135º + k*360º, with k a non negative integer.

Step-by-step explanation:

In order for the hands to for an angle of - 135º = 5/4 π, the minute hand should be at 37.5 minutes, because at 0 minutes they form an angle of π/2 and for each 7.5 minutes we move the minute hand to the left the angle increases in π/4.

In order for the minute hand to be at 37.5 minutes, it should have passed 37.5-15 = 22.5 minutes, or 22 minutes and 30 seconds, plus any multiple of an hour (because if exactly one hour passes, then both hands will be at the same place).

Since the minute hand moves throught the right, it should have moved 360-135-90 = 135º to the right (90º was its initial position and - 135 the end one), plus any multiple of 360º.

So, the answer is, it moved 135º + k * 360º, with k a non negative integer (k is the number of hours passed).