# Solution.

From the initial and final positions of the hour hand, it is clear that the student started writing before 9:30 and finished after 1:30 PM. Therefore, the exact time $x$, spent on the work, satisfies the inequalities $4 < x < 5$. If after completing the work, the minute hand travels the distance the hour hand had traveled from the initial position of the minute hand on the clock face, this will add $x/12$ hours to $x$, since the minute hand moves 12 times faster than the hour hand. As a result, from the start of the exam, $x + x / 12$ hours will pass, and this number must be an integer, as the minute hand will complete a whole number of revolutions, and it must be greater than $x$ and the closest to $x$. Therefore, $x + x / 12 = 5$, $x = 60 / 13$ hours.