A watch gains 5 seconds in every 3 minutes and is set correctly at 8:00 a.m. What time will it show at 10:00 p.m. on the same day?

Introduction / Context:
This problem again involves a fast watch, but now we know the true time interval and must find what the watch will indicate at the end of that interval. The watch gains a fixed amount of time every few minutes. We use this rate of gain to find how many extra minutes and seconds the watch will show after running from 8:00 a.m. to 10:00 p.m. on the same day.

Given Data / Assumptions:

• The watch gains 5 seconds in every 3 minutes.
• It is set correct at 8:00 a.m.
• We consider the true time 10:00 p.m. on the same day.
• We assume the gain rate is uniform.

Concept / Approach:
We first calculate the true time elapsed between 8:00 a.m. and 10:00 p.m. in minutes. We then find how many 3 minute intervals occur in that time and thus how much total gain (in seconds) the watch accumulates. Adding this gain to the true time gives the reading shown by the watch.

Step-by-Step Solution:
Step 1: Compute the true time interval. From 8:00 a.m. to 10:00 p.m. is from hour 8 to hour 22. So the number of hours elapsed = 22 - 8 = 14 hours. Convert to minutes: 14 hours = 14 * 60 = 840 minutes. Step 2: Determine how many 3 minute intervals occur. Number of intervals = 840 / 3 = 280 intervals. Step 3: Compute total gain in seconds. Each 3 minute interval adds 5 seconds. Total gain = 280 * 5 seconds = 1400 seconds. Step 4: Convert gained seconds to minutes and seconds. 1400 seconds / 60 = 23 minutes and 20 seconds (since 23 * 60 = 1380, remainder 20 seconds). So the watch is 23 minutes 20 seconds fast at 10:00 p.m. true time. Step 5: Find the watch reading. True time: 10:00 p.m. Add the gain of 23 minutes 20 seconds. 10:00 p.m. + 23 minutes 20 seconds = 10:23:20 p.m. Therefore, the watch will show 10:23:20 p.m.

Verification / Alternative check:
We can express the rate factor as a ratio: in 3 minutes of real time, the watch shows 3 minutes 5 seconds, which is 185 seconds versus 180 seconds of real time. The factor is 185 / 180. Multiply 840 real minutes by 185 / 180 gives 840 * 185 / 180 = 862 * 1? more precisely 840 * 185 / 180 = 840 * (37 / 36) = 863.333... minutes, which is 14 hours 23 minutes 20 seconds. Adding this to 8:00 a.m. indeed reaches 10:23:20 p.m., confirming the earlier calculation.

Why Other Options Are Wrong:
Options a and b involve times in the morning and do not relate correctly to a gain from 8:00 a.m. to 10:00 p.m. Option c (9:45:15 p.m.) would correspond to a loss rather than a gain. Only option d reflects a time that is ahead of the true time by the correct amount.

Common Pitfalls:
Learners sometimes forget to convert the total gained seconds into minutes and seconds correctly, or they mis count the number of 3 minute intervals in 840 minutes. Another typical mistake is to apply the gain per hour instead of per 3 minutes without a proper conversion, which leads to inaccurate totals.

Final Answer:
At 10:00 p.m. true time, the fast watch will show 10:23:20 p.m.