Clocks and Calendars

The total time from Monday noon to the following Monday 2 p.m. is 7 days and 2 hours = 170 hours. The watch gained 3 minutes + 3 minutes 48 seconds = 6.8 minutes. To show the correct time, it must gain 3 minutes. The time taken is (3 / 6.8) * 170 hours = 75 hours. 75 hours is 3 days and 3 hours. Monday noon + 3 days 3 hours = Thursday 3 p.m.

At 3:30, the hour hand is halfway between 3 and 4 (15 degrees past 3). The minute hand, due to the defect, is at 6 (180 degrees from 12). The angle between 3 and 6 is 90 degrees. Subtracting the 15 degrees the hour hand moved, we get 75 degrees.

The total time from Sunday noon to Wednesday 3 PM is 75 hours. The first clock gains 2 minutes every 24 hours, so in 75 hours it gains (2/24)*75 = 6.25 minutes. Thus, when the clock shows 3:00 PM, the true time is 3:00 PM minus 6 minutes and 15 seconds, which is 2:53:45 PM, closest to 2:54 PM.

The clock loses 3% of 168 hours (1 week) = 5.04 hours. It gains 2% of 168 hours = 3.36 hours. The net loss is 1.68 hours (1 hour 40 minutes 48 seconds). Subtracting this from 12:00:00 results in 10:19:12.

At 9 o'clock, the hands are 45 minutes apart. To be opposite, the minute hand must gain 15 minutes on the hour hand. Time = (15 * 60) / 55 = 900 / 55 = 180 / 11 = 16 and 4/11 minutes past 9.

The angle between hands is calculated as |30H - 5.5M|. For 3:40, H=3 and M=40. Angle = |30*3 - 5.5*40| = |90 - 220| = 130 degrees.

At 4:20, the minute hand is exactly on the 4 mark, which corresponds to 120 degrees from the 12 o'clock position. The hour hand, which moves at 0.5 degrees per minute, has moved 10 degrees past the 4 mark in the 20 minutes since 4:00. Therefore, the angle between the two hands is 10 degrees.

At 4:15, the hour hand is at 4 + 15/60 = 4.25 hours. The position in degrees from 12 is 4.25 * 30 = 127.5 degrees. The minute hand is at 15 minutes, which is 15 * 6 = 90 degrees. The difference is 127.5 - 90 = 37.5 degrees.

The hands of a clock form a 90-degree angle twice every hour, except for the periods between 2-4 and 8-10, where it happens only three times instead of four. This results in 22 occurrences in 12 hours, and 44 in a full day.

The watch gains 5 seconds every 3 minutes, which is 100 seconds per hour. From 6 a.m. to 3:15 p.m. is 9.25 hours. Total gain = 9.25 * 100 = 925 seconds = 15 minutes and 25 seconds. Subtracting this from 3:15 p.m. gives approximately 3:00 p.m.

A clock face is 360 degrees. Each hour mark represents 360/12 = 30 degrees. At 2 o'clock, the angle between the 12 and 2 is 2 * 30 = 60 degrees.

At 4:30, the minute hand is at 6 (180 degrees). The hour hand moves 0.5 degrees per minute; at 30 minutes past 4, it is at 120 + (30 * 0.5) = 135 degrees. The difference is 180 - 135 = 45 degrees.

A leap year has 366 days, which consists of 52 complete weeks and 2 extra days. These 2 extra days can be any of the 7 consecutive pairs of days of the week. Out of these 7 possible pairs, exactly 2 pairs contain a Monday, giving a probability of 2/7.