Q:A clock is observed. The hour hand is exactly at the minute mark, and the minute hand is six minutes ahead of it. Later, the clock is observed again. This time, the hour hand is exactly on a different minute mark, and the minute hand is seven minutes ahead of it. How much time elapsed between the first and second observations?
A: The hour hand moves as the minute hand moves. For example, at 2:30, the hour hand is halfway between 2 and 3. The hour hand is exactly at the minute mark five times an hour on the 12, 24, 36, 48 and on the hour.
So 1:12 would make the hour hand pointing at 6 minutes and the minute hand pointing at 12 minutes.
24 minutes after on the minute hand would mean that the hour hand would have to be pointing at the 18 minute mark which would be more than halfway between 3 and 4.
36 minutes after the hour would mean that the hour hand would point exactly at the 6 and the hour hand only points exactly at the 6 when it is exactly at 6 o'clock.
48 minutes after the hour would mean that the hour hand would point at the 42 which would be a bit after 8 and not closer to nine.
So for the first observation the clock must be at 1:12.
The second observation cannot be 1:12 (since that it a 6minute difference) so we will now check the 24 minute after -- this would put the hour hand at the 17 which would work perfect with 3 as at 3:24, the hour hand has moved 2/5 between the 2 and 3.
Therefore the time elapsed between the first observation, 1:12, and the second observation, 3:24 is 2 hours and 12 minutes.
