Question

A watch which gains uniformly is 2 minutes slow at 10:00 AM on Monday and is \(2\frac 12\) minutes fast at 1:00 PM the following day. When did it show the correct time ?

• A. 10:30 PM Tuesday
• B. 10:30 PM Monday
• C. 10:45 PM Monday
• D. 10:00 PM Monday

Answer 1

The Correct Option is D

To solve this problem, we need to determine the time when the watch shows the correct time. The watch gains uniformly, meaning that the rate at which it gains is constant over time.

1. Initially, at 10:00 AM on Monday, the watch is 2 minutes slow. Therefore, the actual time is 10:02 AM on Monday.
2. By 1:00 PM on Tuesday, the watch is 2.5 minutes fast. Therefore, the actual time is 12:57:30 PM on Tuesday.
3. We need to find the time duration from 10:00 AM Monday to 1:00 PM Tuesday, which is 27 hours (from 10:00 AM Monday to 10:00 AM Tuesday is 24 hours, plus another 3 hours from 10:00 AM to 1:00 PM).
4. During this time span, the watch changes from being 2 minutes slow to being 2.5 minutes fast. That's a total change of 4.5 minutes (from -2 to +2.5).
5. We now need to find the rate at which the watch gains time. The gain rate can be calculated as:
   \(\frac{4.5 \, \text{minutes}}{27 \, \text{hours}} = \frac{1}{6} \, \text{minutes per hour}\)
6. To determine when the watch shows the correct time, it should have neither gained nor lost time relative to the actual time. Since it was 2 minutes slow at 10:00 AM Monday, it must gain exactly 2 minutes to show the correct time.
7. Given the rate of \(\frac{1}{6}\) minute per hour, we'll calculate the hours required to gain 2 minutes:
8. \(\text{Time to gain 2 minutes} = \frac{2 \, \text{minutes}}{\frac{1}{6} \, \text{minute per hour}} = 12 \, \text{hours}\)
9. From 10:00 AM Monday, adding 12 hours gives us 10:00 PM Monday.

Therefore, the watch showed the correct time at 10:00 PM Monday.