In one full day of 24 hours, how many times are the hands of a clock exactly at right angles to each other?

Introduction / Context:
This question asks for the number of times in a day that the hour hand and the minute hand of a clock are at a right angle, that is, separated by 90 degrees. Such counting questions are common in clock problems and require understanding how often certain angular relationships occur in a 12 hour cycle and then extending that to 24 hours.

Given Data / Assumptions:

• A day has 24 hours.
• We consider an ordinary analog clock with hour and minute hands.
• We define a right angle as a 90 degree separation between the two hands.
• The motions of the hands are uniform.

Concept / Approach:
In 12 hours, the hour and minute hands are at right angles a certain number of times. Because 24 hours is simply two cycles of 12 hours, we can count the number of right angle occurrences in one 12 hour period and then double that count. The main task is to know or derive how many right angle positions exist in 12 hours. For each hour interval, there are usually two moments when the hands form a right angle, though in some cases one of these can fall outside the interval.

Step-by-Step Solution:
Step 1: Over 12 hours, the minute hand makes 12 full revolutions, while the hour hand makes one. Step 2: It is a known result from clock theory that in 12 hours, the hands are at right angles 22 times. Step 3: One way to see this is that, in each hour, the relative motion of the hands leads to approximately two right angle positions, so across 11 closer intervals there are 22 such occurrences. Step 4: Because 24 hours equals two blocks of 12 hours, the pattern of right angle occurrences repeats in the second half of the day. Step 5: Therefore, in 24 hours, the total number of times the hands are at right angles is 22 * 2 = 44. Step 6: Hence, the answer is 44 times in one day.

Verification / Alternative check:
For a more detailed reasoning, note that the relative angular speed of the hands is 5.5 degrees per minute. The angle between them increases or decreases at this rate. The condition for a right angle is that the difference in their positions is 90 degrees or 270 degrees. Solving these equations for all times in a 12 hour range leads to 22 distinct solutions. Doubling this count for 24 hours again gives 44, confirming the standard result.

Why Other Options Are Wrong:
54: This would imply 27 right angles in 12 hours, which is more than the known maximum for the relative motion.

64: This would mean more than 2.5 right angle occurrences per hour on average, which is impossible given the uniform movement.

22: This is the number for 12 hours only, not for the entire day.

36: This count does not correspond to any standard cycle result and is neither 1.5 nor 2 times the known 12 hour value.

Common Pitfalls:
Many learners confuse the counts for coincidences and right angles. The hands coincide 22 times in 24 hours, but they are at right angles twice as often, that is, 44 times. Another mistake is assuming exactly two right angle positions in every single hour without considering the transitions near 11 and 12, which can slightly alter the straightforward count. Remembering the benchmark values 22 coincidences and 44 right angle positions per day is very helpful.

Final Answer:
The hands of a clock are at right angles 44 times in one day.