Clocks and Calendars

One right angle is 3 hours on a clock face. Three right angles equal 9 hours. Starting at 10, moving 9 hours clockwise (10 + 9 = 19) corresponds to 19 - 12 = 7.

At 10:40, the hour hand is at 10 + 40/60 = 10.666 hours. The angle from 12 o'clock is 10.666 * 30 = 320 degrees. The minute hand is at 40 minutes, which is 40 * 6 = 240 degrees. The difference is 320 - 240 = 80 degrees.

At 4:00, the hour hand is at 4 and the minute hand is at 12. Each hour mark represents 30 degrees (360/12). The angle is 4 * 30 = 120 degrees.

A clock face is 360 degrees. Each hour mark is 30 degrees (360/12). At 3 o'clock, the hour hand is at 3 and the minute hand is at 12. The angle is 3 * 30 = 90 degrees.

Angle = |30H - 5.5M|. H=3, M=40. Angle = |30*3 - 5.5*40| = |90 - 220| = 130 degrees.

Angle = |30H - 5.5M|. At 3:25, H=3, M=25. Angle = |30(3) - 5.5(25)| = |90 - 137.5| = 47.5 degrees, which is 95/2 degrees.

Angle = |30H - 5.5M|. H = 10, M = 30. Angle = |300 - 165| = 135 degrees.

At 6:45, the hour hand is at 6 + 45/60 = 6.75 hours. Each hour is 30 degrees, so 6.75 * 30 = 202.5 degrees. The minute hand is at 45 minutes, which is 45 * 6 = 270 degrees. The difference is 270 - 202.5 = 67.5 degrees.

From 8 a.m. to 8 p.m. is 12 hours. The watch loses 30 seconds per hour. Total loss = 12 * 30 = 360 seconds = 6 minutes. Actual time is 8 p.m., so watch time is 7:54 p.m.

At 6 o'clock, the hands are 30 minutes apart. The minute hand gains 5.5 degrees per minute relative to the hour hand. To overlap, the minute hand must cover 30 minutes distance. Time = 30 / (11/60) = 1800/11 = 163.63 minutes? No, the formula is T = (2/11) * (30*H + M). For H=6, M=0, T = (2/11) * 180 = 360/11 = 32 and 8/11 minutes past 6.