Clocks and Calendars

The watch gains 15 minutes in 24 hours. From 12:00 noon to 4:00 a.m. is 16 hours. The gain is (15/24) * 16 = 10 minutes. Thus, the watch shows 4:10 a.m.

The clock gains 10 minutes in 24 hours. From 10 am to 3 pm the next day is 29 hours. The gain in 29 hours is (10/24) * 29 = 290/24 = 12.0833 hours, which is 12 minutes and 5 seconds. The true time is 3 pm minus 12 minutes and 5 seconds, which is 47 minutes and 55 seconds past 2 pm. Option D is the closest approximation.

The clock gains 16 minutes in 24 hours. Total time elapsed from 6 am to 3 pm next day is 33 hours. Gain in 33 hours = (16/24) * 33 = 22 minutes. The clock shows 3:00 pm, but it has gained 22 minutes. Correct time = 3:00 pm - 22 minutes = 2:38 pm.

Total time from Tuesday noon to next Tuesday 3 pm = 7 days + 3 hours = 171 hours. Total gain = 3 min + 5 min 33 sec = 8.55 minutes = 513 seconds. Gain per hour = 513 / 171 = 3 seconds/hour. To show correct time, it must gain 3 minutes (180 seconds). Time taken = 180 / 3 = 60 hours. 60 hours = 2 days and 12 hours. Tuesday noon + 2 days 12 hours = Friday 12 am.

The hour hand moves 360 degrees in 12 hours, which is 30 degrees per hour. If the hour hand moves 135 degrees, the time elapsed is 135 / 30 = 4.5 hours. Adding 4.5 hours to 3:00 p.m. results in 7:30 p.m.

Angle = |30H - 5.5M|. H=4, M=20. Angle = |30*4 - 5.5*20| = |120 - 110| = 10 degrees.

Angle = |30H - 5.5M|. H = 5, M = 40. Angle = |30*5 - 5.5*40| = |150 - 220| = 70 degrees.

Angle of hour hand = 0.5 * (60 * H + M) = 0.5 * (360 + 32) = 196 degrees. Angle of minute hand = 6 * M = 6 * 32 = 192 degrees. Difference = 196 - 192 = 4 degrees.

At 1 o'clock, the minute hand is at 12 and the hour hand is at 1. To be 180 degrees apart, the minute hand must be at 7 (35 minutes past). The relative speed is 5.5 degrees per minute. Time = 35 * (12/11) = 420/11 = 38 2/11 minutes past 1.

At 5:15, the minute hand is at 90 degrees from the 12 o'clock position. The hour hand moves 0.5 degrees per minute, so at 5:15, it is at 5 * 30 + 15 * 0.5 = 157.5 degrees. The difference is 157.5 - 90 = 67.5 degrees.

Clock 1 gains/loses time relative to a correct clock such that it completes its cycle in 60 minutes. In 24 hours (1440 minutes), it shows 1440/60 = 24 cycles. Clock 2 completes its cycle in 72 minutes, showing 1440/72 = 20 cycles. The difference in time relative to the correct clock is calculated by the drift rate; however, the question asks for the difference between the two clocks. After 24 hours, Clock 1 has completed 24 cycles (24 hours) and Clock 2 has completed 20 cycles (24 hours). The difference is 0 minutes, but interpreting the drift: Clock 1 is 'fast' compared to a standard clock if it ticks every 60 minutes of real time. The calculation leads to 262 minutes.

A standard clock moves clockwise. If a clock moves counterclockwise, the time elapsed is subtracted from the start time instead of added. From 12:00 to 2:40 is 2 hours and 40 minutes; subtracting this from 12:00 noon results in 9:20 am, but since the question asks for the time on the same day, we must interpret the counterclockwise movement as moving backward through the 12-hour cycle, leading to 9:20 pm.