Clocks and Calendars

At 3 o'clock, the hands are 90 degrees apart. For the first time after 3, the minute hand must gain distance. The formula for the angle between hands is |30H - 5.5M|. For 90 degrees, |30(3) - 5.5M| = 90. 90 - 5.5M = -90 (not possible) or 5.5M - 90 = 90. 5.5M = 180, M = 180 / 5.5 = 360/11 = 32 8/11.

At 4:20, the hour hand is at 4 + 20/60 = 4.33 hours. The angle from 12 is 4.33 * 30 = 130 degrees. The minute hand is at 20 minutes, which is 20 * 6 = 120 degrees. The angle between them is |130 - 120| = 10 degrees. The reflex angle is 360 - 10 = 350 degrees.

Time from Monday 8 AM to Tuesday 10 PM = 24 + 14 = 38 hours. 38 hours = 2280 minutes. The watch gains 5 seconds every 3 minutes. Total gain = (2280 / 3) * 5 = 760 * 5 = 3800 seconds. 3800 seconds = 63 minutes and 20 seconds. 10:00 PM + 1 hour 3 minutes 20 seconds = 11:03:20 PM.

For the hands to be opposite, the angle between them must be 180 degrees. At 7 o'clock, the minute hand is at 12 and the hour hand is at 7 (210 degrees). The minute hand must gain 5 minutes of space relative to the hour hand to be opposite. The formula is M = (2/11) * (30*H +/- 180). For H=7, M = (2/11) * (210 - 180) = (2/11) * 30 = 60/11 = 5 5/11 minutes past 7.

At 2:00, the hour hand is at 2 and the minute hand is at 12. Each hour mark represents 30 degrees (360/12). Thus, 2 * 30 = 60 degrees.

Total time from Monday 7 AM to next Monday 7 PM is 7 days + 12 hours = 180 hours. Total gain = 8 minutes + 7 minutes 36 seconds = 15.6 minutes = 936 seconds. Gain rate = 936 / 180 = 5.2 seconds per hour. To gain 8 minutes (480 seconds), time needed = 480 / 5.2 = 92.307 hours = 3 days and 20.3 hours. Monday 7 AM + 3 days 20 hours = Friday 3 AM.

In 25 hours there are 1,500 minutes. At 4 seconds gained every 20 minutes, the clock gains 75 x 4 = 300 seconds, which equals 5 minutes.

The angle between clock hands is given by |30H - 5.5M|. For 11:30, this is |30(11) - 5.5(30)| = |330 - 165| = 165 degrees.

At 10:30, the minute hand is at 6 (180 degrees). The hour hand moves 0.5 degrees per minute. At 10:00, it is at 300 degrees. In 30 minutes, it moves 15 degrees. Position = 300 + 15 = 315 degrees. Angle = 315 - 180 = 135 degrees.

The clock gains 15 minutes in 24 hours. In 16 hours (from 12 noon to 4 am), it gains (15/24) * 16 = 10 minutes. It will show 4:10 am.