A watch that gains uniformly, is two minutes slow at noon on Monday and is 4 minutes 48 seconds fast at 2 pm on the following Monday. When was the watch showing the correct time?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2 pm on Wednesday
1. Calculate the Total Elapsed Time
First, find the total time from Monday noon to the following Monday at 2 pm:
- From Monday noon to the next Monday noon = $7 \text{ days} = 7 \times 24 = 168 \text{ hours}$
- From next Monday noon to Monday 2 pm = $2 \text{ hours}$
- Total time elapsed = $168 + 2 = 170 \text{ hours}$
2. Calculate the Total Time Gained
The watch went from being 2 minutes slow to 4 minutes 48 seconds fast.
- Total gain = $2 \text{ minutes} + 4 \text{ minutes } 48 \text{ seconds}$
- Convert 48 seconds to minutes: $48/60 = 0.8 \text{ minutes}$
- Total gain = $2 + 4.8 = 6.8 \text{ minutes}$
3. Determine the Time to Gain 2 Minutes
The watch will show the correct time when it gains exactly the 2 minutes it was lagging at the start. Since it gains uniformly, we can set up a ratio:
- The watch gains $6.8 \text{ minutes}$ in $170 \text{ hours}$.
- Let $T$ be the time taken to gain $2 \text{ minutes}$.
$\frac{6.8 \text{ min}}{170 \text{ hours}} = \frac{2 \text{ min}}{T}$
$T = \frac{2 \times 170}{6.8}$
$T = \frac{340}{6.8} = 50 \text{ hours}$
4. Find the Exact Time and Day
Now, add 50 hours to the starting time (Monday noon):
- 48 hours after Monday noon = Wednesday noon.
- Remaining 2 hours after Wednesday noon = 2 pm on Wednesday.
Conclusion
The watch showed the correct time at 2 pm on Wednesday.