Estonian Mathematical Olympiad

Question

In the classroom there is a clock, whose minute and hour hands move with a constant angular velocity. The minute hand works correctly, but the hour hand moves at half the angular velocity of the minute hand. At 10:00, the clock shows the correct time. When will the clock show the correct time for the next time?

Accepted Answer

Imagine a correctly working clock next to the broken clock. The hour hand of the working clock moves 12 times slower than its minute hand, which means that its hour hand moves 6 times slower than the hour hand of the broken clock. As the minute hand of the broken clock is always at the correct position, the broken clock next shows the correct time when its hour hand has moved one more full circle compared to the hour hand of the correct clock, meaning that their indicated times differ by 12 hours.

Denote the number of hours passed by this point by

x

. Based on the previous information, we have the equation

6 x = x + 12

or

5 x = 12

, which yields

x = 2.4

. Thus the broken clock shows the correct time after 2 hours and

0.4 \cdot 60 = 24

minutes or at 12:24.