How many times in a 24-hour period are the hands of a clock in a straight line, diametrically opposite to each other?

To determine how many times in a day the hands of a clock are in a straight line, diametrically opposite to each other, we need to find how many times the angle between the hour hand and the minute hand is 180 degrees. Step 1: Understand the relative speed of the clock hands. The minute hand moves 360 degrees in 60 minutes, which is 6 degrees per minute. The hour hand moves 360 degrees in 12 hours (720 minutes), which is 0.5 degrees per minute. The relative speed of the minute hand with respect to the hour hand is 6 - 0.5 = 5.5 degrees per minute. Step 2: Calculate occurrences in a 12-hour period. For the hands to be diametrically opposite, they must be 180 degrees apart. In a 12-hour period, the minute hand overtakes the hour hand 11 times. Each time it overtakes, it also passes through the 180-degree position relative to the hour hand. This occurs once every hour, except for the interval between 5 o'clock and 7 o'clock, where the hands are diametrically opposite only once (exactly at 6:00). So, in a 12-hour period (e.g., from 12:00 PM to 12:00 AM, or 12:00 AM to 12:00 PM), the hands are diametrically opposite 11 times. The approximate times are: 6:00, 7:05, 8:10, 9:15, 10:20, 11:25, 12:30, 1:35, 2:40, 3:45, 4:50. Step 3: Calculate occurrences in a 24-hour day. A full day consists of two 12-hour periods. Therefore, the number of times the hands are diametrically opposite in a 24-hour day is 11 × 2 = 22 times. The final answer is B) 22. Last free one today — make it count tomorrow, or type /upgrade for unlimited.