For clock puzzles, how many times in one day are the hands of a clock in a straight line but pointing in opposite directions?

Introduction / Context:
This puzzle question belongs to the standard topic of clock problems in quantitative aptitude. It asks how many times in a full day the hour hand and the minute hand of a clock lie in a straight line but are pointing in opposite directions. Understanding the relative motion of clock hands and the patterns they follow over twelve hours and twenty four hours is essential for solving many related questions.

Given Data / Assumptions:

• The clock is a regular twelve hour analog clock with an hour hand and a minute hand.
• The phrase in a straight line but opposite in direction means the two hands are separated by exactly 180 degrees.
• The time period of interest is one full day of twenty four hours.
• We assume the clock runs uniformly with no mechanical errors.

Concept / Approach:
The basic concept is that the minute hand and hour hand move at different speeds. The minute hand moves 360 degrees in 60 minutes, that is 6 degrees per minute, while the hour hand moves 360 degrees in 12 hours, that is 0.5 degrees per minute. The relative speed between them is 5.5 degrees per minute. In clock problems, alignments such as hands together or hands in opposite directions occur at regular intervals based on this relative speed. Over a twelve hour period, the hands are in opposite directions 11 times, so over a full twenty four hour day, that count doubles.

Step-by-Step Solution:
Step 1: Note that the minute hand speed is 6 degrees per minute and the hour hand speed is 0.5 degrees per minute. Step 2: Compute the relative angular speed as 6 minus 0.5 which equals 5.5 degrees per minute. Step 3: For the hands to be in opposite directions, the angular separation between them must be 180 degrees. Step 4: The time between consecutive instances when the hands are opposite is given by 180 divided by 5.5 minutes. Step 5: The total time in twelve hours is 12 hours multiplied by 60 which equals 720 minutes. Step 6: The number of times the hands are opposite in twelve hours is 720 divided by the interval in minutes, which results in 11 occurrences. Step 7: In a full day of twenty four hours, the pattern repeats twice, so the total number of opposite alignments is 11 multiplied by 2 which equals 22.
Verification / Alternative check:
An alternative way to check is to remember the standard results for clock problems. In twelve hours, the hands coincide 11 times and are opposite 11 times. Therefore, in twenty four hours, the number of coincidences is 22 and the number of times they are opposite is also 22. This rule is widely used in aptitude books and confirms that 22 is the correct count without needing to calculate each time specifically.

Why Other Options Are Wrong:
20: This underestimates the actual count and does not follow from the twelve hour pattern of 11 occurrences.
24: This number mistakenly assumes one opposite alignment per hour, which is not accurate because the intervals are slightly more than one hour apart.
48: This value doubles an incorrect assumption of 24 per day and is much higher than the correct count.

Common Pitfalls:
A common error is to assume that the hands are opposite exactly every hour, leading to 24 as an answer. Another mistake is mixing up the count for hands in the same direction with the count for hands in opposite directions. To avoid these pitfalls, it helps to memorize that in twelve hours, there are 11 coincidences and 11 opposite alignments, giving 22 each over a full day. Using the relative speed method reinforces this pattern and prevents confusion.

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