2003 AMC 12B Problem 11

In $60$ real minutes the watch advances only $57$ minutes $36$ seconds $= 57.6$ minutes. So when the watch shows $t$ minutes past noon, the real elapsed time is $\dfrac{60}{57.6}t = \dfrac{25}{24}t$ minutes. The watch reads $10{:}00$ PM after $600$ recorded minutes, so the real elapsed time is $$\frac{25}{24}(600) = 625$$ minutes $= 10$ hours $25$ minutes past noon. The actual time is $10{:}25$ PM. Thus, the correct answer is C.