In one full day of 24 hours, how many times do the hour hand and the minute hand of a clock exactly coincide?

Introduction / Context:
This question asks for the number of times in a full day that the hour and minute hands of a clock exactly coincide, that is, occupy the same position. Counting such events is a standard result in clock problems and reinforces understanding of relative motion between the hands over a 12 hour cycle and a 24 hour day.

Given Data / Assumptions:

• A day has 24 hours.
• We consider a normal analog clock with hour and minute hands.
• Two hands coincide when the angle between them is zero degrees.
• The motion of the hands is uniform and continuous.

Concept / Approach:
In 12 hours, the minute hand laps or overtakes the hour hand a certain number of times. Each overtaking corresponds to one coincidence. Because 24 hours is simply two such 12 hour cycles back to back, we can determine the number of coincidences in 12 hours and then double this value to cover the whole day. The key is to remember that the hands do not coincide 12 times in 12 hours but slightly fewer times due to their relative speeds.

Step-by-Step Solution:
Step 1: Consider the relative speed of the hands. The minute hand moves at 6 degrees per minute, and the hour hand at 0.5 degrees per minute, so the relative speed is 5.5 degrees per minute. Step 2: To coincide, the minute hand must gain a full 360 degrees over the hour hand. Step 3: The time between two consecutive coincidences is: interval = 360 / 5.5 minutes = 65 5/11 minutes. Step 4: Now look at a 12 hour period. In 12 hours, the minute hand laps the hour hand exactly 11 times, resulting in 11 coincidences. Step 5: Because 24 hours contains two such 12 hour periods, the total number of coincidences in a full day is 11 * 2 = 22. Step 6: Therefore, the hands coincide 22 times in one day.

Verification / Alternative check:
We can convince ourselves by considering the positions of the hour hand at each coincidence. In 12 hours, there are 11 coincidences, each roughly one hour and five minutes apart. If we start at 12 o clock at noon and track through the next 12 hours, we see that the last coincidence before 12 midnight occurs slightly before the hour, so we do not get a twelfth coincidence in that range. The pattern repeats in the next 12 hours, giving another 11 coincidences, consistent with the total of 22 per day.

Why Other Options Are Wrong:
20: This would imply only 10 coincidences in 12 hours, which is fewer than the known result.

21: Also inconsistent with the 11 coincidences in a single 12 hour period.

23: This would mean one of the 12 hour periods has 12 coincidences, which is not possible.

24: A common incorrect guess based on one coincidence per hour, but the hands do not coincide every hour exactly.

Common Pitfalls:
Students often assume that since there are 12 hours, the hands coincide 12 times in 12 hours, and hence 24 times in a day. This ignores the exact timing and relative speed of the hands. Another mistake is confusing the number of right angle positions (44 times per day) with the number of coincidences (22 times per day). Keeping these two counts separate helps avoid confusion.

Final Answer:
The hour and minute hands coincide 22 times in one day.