A watch that gains uniformly is 2 minutes slow at 12:00 noon on Tuesday and 4 minutes 48 seconds fast at 2:00 p.m. on the following Tuesday. At what exact time did this watch show the correct time?

Introduction / Context:
In this problem, we deal with a watch that gains time at a uniform rate. We are given two readings: initially the watch is slow, and later it is fast. The task is to determine when the watch must have been exactly correct. Such questions involve understanding uniform gain (or loss) in time and applying concepts of linear variation over a given interval, which are very common in aptitude tests dealing with faulty clocks and watches.

Given Data / Assumptions:

• At 12:00 noon on Tuesday, the watch is 2 minutes slow (that is, it shows 2 minutes less than the correct time).

• At 2:00 p.m. on the following Tuesday, the same watch is 4 minutes 48 seconds fast.

• One week plus 2 hours elapses between these two instants.

• The watch gains time uniformly, meaning its error changes at a constant rate.

• We must find the exact time between these two instants when the error is zero (the watch is correct).

Concept / Approach:
Since the gain is uniform, the time error changes linearly from -2 minutes to +4 minutes 48 seconds over a known time span. First, convert all errors to minutes and find the total change in error. Then compute the rate of change of error per hour. Next, determine how long it takes to move from -2 minutes to 0 minutes at that constant rate. Adding this duration to the initial time gives the moment when the watch showed the correct time.

Step-by-Step Solution:
Step 1: Convert the final error to minutes. 4 minutes 48 seconds = 4 + 48/60 = 4.8 minutes fast. Step 2: Initial error at noon on Tuesday = -2 minutes (slow). Step 3: Final error at 2:00 p.m. on the following Tuesday = +4.8 minutes (fast). Step 4: Total change in error = 4.8 - (−2) = 6.8 minutes. Step 5: Time between noon Tuesday and 2:00 p.m. the next Tuesday = 7 days + 2 hours. Step 6: In hours, this is 7 * 24 + 2 = 168 + 2 = 170 hours. Step 7: Rate of change of error per hour = 6.8 minutes / 170 hours = 0.04 minutes per hour. Step 8: To go from −2 minutes to 0 error, the change needed is +2 minutes. Step 9: Time required to correct this −2 minute error = 2 / 0.04 = 50 hours after noon on Tuesday. Step 10: 48 hours after Tuesday noon is Thursday noon, and 2 more hours gives Thursday 2:00 p.m.

Verification / Alternative check:
We can verify by checking the fraction of total time. The total error change is 6.8 minutes over 170 hours. The fraction needed to move from −2 minutes to 0 is 2 / 6.8 = 1 / 3.4. Multiplying this fraction by the total time: 170 * (1 / 3.4) = 50 hours. This matches our earlier calculation, confirming that the watch is correct exactly 50 hours after noon on Tuesday, which is 2:00 p.m. on Thursday.

Why Other Options Are Wrong:
• 12:00 noon on Wednesday: This is only 24 hours after the start, which is too early to correct the full −2 minute error at the given rate.
• 3:00 p.m. on Thursday: This is 51 hours after noon Tuesday and would mean the watch has gained slightly more than needed to reach zero error, making it fast instead of correct at that time.
• 2:00 p.m. on Wednesday: This is 26 hours after the start, still not enough time to eliminate the entire −2 minute error at the calculated rate of 0.04 minutes per hour.

Common Pitfalls:
Students often miscalculate the total time interval between the two observations, forgetting that “the following Tuesday” is a full week later. Others may forget to convert seconds into minutes correctly or may accidentally treat the gain as per day rather than per hour in the calculation. Carefully converting all quantities into consistent units (hours and minutes) and recognizing the linear nature of the error change helps avoid these mistakes.

Final Answer:
The watch showed the correct time at 2:00 p.m. on Thursday.