Clocks and Calendars

The angle between hands is |30H - 5.5M|. For 2:30, |30(2) - 5.5(30)| = |60 - 165| = 105 degrees.

At 12:00, the minute hand points East. This means the 12 o'clock position is East. Moving clockwise, 3 is South, 6 is West, 9 is North. At 4:30, the hour hand is halfway between 4 and 5. Since 3 is South and 6 is West, 4 is South-West and 5 is West-South-West. Halfway between 4 and 5 is South-West.

From 8 AM to 2 PM is 6 hours. The hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. 6 hours * 30 degrees/hour = 180 degrees.

At 8 o'clock, the hands are 40 minute spaces apart. For them to be in a straight line (180 degrees), they must be 30 minutes apart. The minute hand must gain 40 - 30 = 10 minute spaces. Time = 10 * (60/55) = 120/11 = 10 10/11 minutes past 8.

When clock hands change places, the time T (in minutes past 5) satisfies the condition that the hour hand is at 5 + T/60 and the minute hand is at T. After swapping, the new positions are T/12 (for the hour hand) and 5*5 + T/12 (for the minute hand). Solving the system leads to T = 300/13 + 300/13, resulting in 32 4/13 minutes past 5.

From Sunday 6:10 am to Wednesday 2:50 pm is 3 days and 8 hours and 40 minutes (80 hours and 40 minutes). The watch gains 12 minutes in 24 hours (1440 minutes), so it gains 1 minute every 120 minutes. In 4840 minutes, it gains 4840 / 120 = 40.33 minutes. Subtracting the gain from the watch time gives the correct time.

The watch gains 3 minutes in 60 hours (from Tuesday 1 pm to Friday 1 am). To show the correct time, it must gain 1 minute. 1 minute is gained in 20 hours. Adding 20 hours to Tuesday 1 pm results in Wednesday 9 am.

At 4 o'clock, the hands are 20 minute spaces apart. For a 60 degree angle, they must be 10 minute spaces apart. The minute hand must gain 10 minute spaces. Time = 10 * (60/55) = 120/11 = 10 and 10/11 minutes past 4.

Angle = |30H - 5.5M|. H=10, M=25. Angle = |300 - 137.5| = 162.5. Reflex angle = 360 - 162.5 = 197.5 degrees.

At 5:30, the angle is 15 degrees. To be 90 degrees apart, the minute hand must gain 75 degrees on the hour hand. The relative speed is 5.5 degrees per minute. Time = 75 / 5.5 = 150/11 = 13 and 7/11 minutes past 5:30. Total time = 30 + 13 and 7/11 = 43 and 7/11 minutes past 5.

At 2:25, the minute hand is at 25 minutes (150 degrees). The hour hand is at 2*30 + (25/60)*30 = 60 + 12.5 = 72.5 degrees. The angle is |150 - 72.5| = 77.5 degrees.

Standard coincidence time is 65 5/11 minutes. The watch coincides every 65 3/11 minutes, meaning it is faster. Gain = (65 5/11 - 65 3/11) / (65 3/11) * 24 * 60 = (2/11) / (718/11) * 1440 = (2/718) * 1440 = 2880 / 718 = 4.01 minutes.