In a period of 24 hours (one full day), how many times are the hour hand and the minute hand of a clock in a straight line but pointing in exactly opposite directions (forming an angle of 180 degrees)?

Introduction / Context:
This question focuses on how often the hour and minute hands of a clock are directly opposite each other in a day. When the hands are in a straight line but opposite in direction, they form a 180 degree angle. Understanding how frequently this condition occurs requires knowledge of the relative speeds of the hands and the standard results for their various alignments. This concept forms an important part of clock-based aptitude questions.

Given Data / Assumptions:

• We consider a 12-hour analog clock with hour and minute hands.
• The minute hand completes 1 full revolution (360 degrees) in 60 minutes.
• The hour hand completes 1 full revolution (360 degrees) in 12 hours (720 minutes).
• We are interested in how many times per day (24 hours) the hands are in a straight line but opposite in direction, i.e., separated by 180 degrees.

Concept / Approach:
The minute hand gains on the hour hand because it moves faster. The relative speed of the minute hand compared to the hour hand determines how frequently certain alignments occur. It is known that in 12 hours, the hands coincide 11 times and are opposite each other 11 times. Since a full day has 24 hours, which is two such 12-hour cycles, we simply double the number of occurrences for the 12-hour period to obtain the result for 24 hours.

Step-by-Step Solution:
Step 1: In 12 hours, the minute hand completes 12 full revolutions, while the hour hand completes 1 revolution. Step 2: The difference in the number of revolutions between the minute and hour hands in 12 hours is 11. Step 3: Between consecutive times when the hands are opposite (180 degrees apart), the relative position changes by one-half revolution. Step 4: Over each 12-hour period, the hands are opposite each other exactly 11 times (a standard clock result). Step 5: A full day is 24 hours, which consists of two 12-hour cycles. Step 6: Therefore, in 24 hours, the number of times the hands are opposite = 2 * 11 = 22 times.

Verification / Alternative check:
We can reason based on symmetry: the behavior of the clock from 12:00 to 12:00 occurs identically in each 12-hour block (for example, from 12:00 midnight to 12:00 noon and then from 12:00 noon to 12:00 midnight). Since we know there are 11 times the hands are opposite in each 12-hour interval, adding the two intervals gives 22. This symmetry-based reasoning confirms the calculation without needing to list every occurrence.

Why Other Options Are Wrong:
• 11 times: This is the count for 12 hours, not for the full 24-hour day.
• 33 times: This would imply an additional half-cycle of occurrences, which does not fit the regular pattern of the clock's motion.
• 44 times: This would incorrectly double the already doubled result, and there is no mathematical basis for such a number in a 24-hour period.

Common Pitfalls:
A typical error is to confuse the number of times the hands coincide with the number of times they are opposite, or to misremember the standard results. Another common mistake is to think that in 12 hours they are opposite 12 times, which is incorrect. Remembering that the number of times the minute hand gains a full revolution on the hour hand in 12 hours is 11 helps in deriving the right counts for both coincidences and opposite alignments.

Final Answer:
In one full day of 24 hours, the hands of a clock are in a straight line but opposite in direction 22 times.