A watch that gains uniformly is 2 minutes slow at Monday noon and 4 minutes 48 seconds fast at 2 p.m. the following Monday; at what time was the watch exactly correct?

Introduction / Context:
This is a classic clock gain and loss problem from quantitative aptitude. The watch in question gains time uniformly, meaning it runs consistently faster than real time. At one reading, the watch is slow; at a later reading, it is fast. The question asks you to determine the exact time when the watch showed the correct time. Such problems test your understanding of uniform rate of gain or loss and how to convert time differences into rate based calculations.

Given Data / Assumptions:

• At Monday noon, the watch is 2 minutes slow compared to the correct time.
• At 2 p.m. on the following Monday, the same watch is 4 minutes 48 seconds fast.
• The watch gains time uniformly between these two observations.
• We assume no resets or interruptions occurred during this period.

Concept / Approach:
The core idea is to treat the difference between the two states of the watch as the total gain over a known time interval. At the first observation, the watch is 2 minutes behind. At the second observation, it is 4 minutes 48 seconds ahead. The total gain over this interval is therefore 2 minutes plus 4 minutes 48 seconds. Once we know the total gain and the length of the interval, we can compute the rate of gain per hour. Finally, we use that rate to determine how long it takes the watch to gain 2 minutes, which is the amount needed to move from being 2 minutes slow to being exactly correct.

Step-by-Step Solution:
Step 1: Compute the time interval from Monday noon to 2 p.m. the following Monday. From Monday noon to the next Monday noon is 7 days, or 7 multiplied by 24 equals 168 hours. Adding 2 extra hours gives 170 hours in total. Step 2: Determine the total gain of the watch over this interval. Initially, it is 2 minutes slow. Finally, it is 4 minutes 48 seconds fast. The total gain equals 2 minutes plus 4 minutes 48 seconds. Step 3: Convert 4 minutes 48 seconds to minutes. 48 seconds is 48 divided by 60 which equals 0.8 minutes. So 4 minutes 48 seconds equals 4.8 minutes. Step 4: Add the two parts of the gain: 2 minutes plus 4.8 minutes equals 6.8 minutes of total gain over 170 hours. Step 5: Compute the rate of gain per hour. Rate equals 6.8 divided by 170 minutes per hour, which simplifies to 0.04 minutes per hour, that is 2.4 seconds per hour. Step 6: To find when the watch was correct, calculate how long it takes to gain 2 minutes, because starting from 2 minutes slow, gaining 2 minutes makes it exactly correct. Step 7: Time to gain 2 minutes equals 2 divided by 0.04 hours, which is 50 hours. Step 8: Add 50 hours to the starting point, Monday noon. Forty eight hours from Monday noon takes you to Wednesday noon. Adding 2 more hours gives Wednesday 2 p.m. Step 9: Therefore, the watch was exactly correct at 2 p.m. on Wednesday.
Verification / Alternative check:
As a check, note that from Monday noon to Wednesday 2 p.m. is exactly 50 hours. At this point, the watch has gained 2 minutes and moved from being 2 minutes slow to exactly correct. From Wednesday 2 p.m. to the following Monday 2 p.m. is the remaining 120 hours. At the gain rate of 0.04 minutes per hour, over 120 hours the watch gains 120 multiplied by 0.04 equals 4.8 minutes. Thus, by the next Monday at 2 p.m. it is 4.8 minutes fast, matching the condition given in the problem. This confirms that our calculation and answer are consistent.

Why Other Options Are Wrong:
2 p.m. on Tuesday: This corresponds to only 26 hours after Monday noon. At the gain rate, the watch would not yet have gained the full 2 minutes needed to become correct.
3 p.m. on Thursday: This is much later than the correct time and would yield a gain larger than 2 minutes, so the watch would be fast by then.
1 p.m. on Friday: This time is also far beyond the moment when the watch becomes correct and does not match the uniform gain pattern needed to satisfy both given conditions.

Common Pitfalls:
One common mistake is to forget to add the initial delay and final advance when calculating total gain, leading to using only 4 minutes 48 seconds instead of 2 plus 4 minutes 48 seconds. Another pitfall is mishandling the conversion of seconds into decimal minutes, which can distort the rate calculation. Finally, some students miscount hours between the two observations or forget that 7 full days are involved. Carefully tracking the time interval, converting all times to minutes, and methodically computing the gain per hour avoids these errors.

Final Answer:
The watch was exactly correct at 2 p.m. on Wednesday.