A watch gains 5 seconds in every 3 minutes and is set correctly at 7:00 a.m. On the same day, when the watch shows 4:15 p.m., what is the true time?

Introduction / Context:
In this problem, a watch is running fast and gaining a fixed number of seconds every few minutes. We know the time at which it was set correctly and a later time that it shows. The task is to find the actual correct time at that later reading. This tests understanding of rate of gain, conversion between seconds and minutes, and proportional reasoning.

Given Data / Assumptions:

• The watch gains 5 seconds every 3 minutes.
• The watch is correct at 7:00 a.m.
• Later in the afternoon, it shows 4:15 p.m.
• The gain is uniform over time.

Concept / Approach:
We first find how fast the watch runs relative to true time. In 3 minutes of true time, the watch gains 5 seconds. So in 3 minutes (180 seconds) of real time, it shows 185 seconds. The speed factor is therefore: rate factor = 185 / 180. If T is the true time elapsed from 7:00 a.m., then: shown elapsed time = rate factor * T. We know the shown elapsed time between 7:00 a.m. and 4:15 p.m., so we can solve for T.

Step-by-Step Solution:
Step 1: Compute shown time elapsed. From 7:00 a.m. to 4:15 p.m. is 9 hours 15 minutes. Convert to minutes: 9 hours = 540 minutes, plus 15 gives 555 minutes. Step 2: Rate factor of the fast watch. In 3 minutes of true time, watch shows 3 minutes 5 seconds = 3 minutes + 5/60 minute. In seconds: 3 minutes = 180 seconds, plus 5 = 185 seconds. Rate factor = 185 / 180. Step 3: Set up relation between shown time and true time. Let T be true minutes elapsed since 7:00 a.m. Shown minutes = (185 / 180) * T. Shown minutes are 555, so: 555 = (185 / 180) * T. T = 555 * 180 / 185. Simplify: 555 / 185 = 3, so T = 3 * 180 = 540 minutes. Step 4: Convert true elapsed time to hours. 540 minutes = 9 hours. Step 5: Add to starting time. True time = 7:00 a.m. + 9 hours = 4:00 p.m. Thus, when the watch shows 4:15 p.m., the correct time is 4:00 p.m.

Verification / Alternative check:
From 7:00 a.m. to 4:00 p.m. is 9 hours or 540 minutes of true time. The watch gains 5 seconds every 3 minutes. Number of 3 minute intervals: 540 / 3 = 180 intervals. Total gain = 180 * 5 seconds = 900 seconds = 15 minutes. So the watch will show 9 hours 15 minutes after 7:00 a.m., that is 4:15 p.m., exactly matching the given reading.

Why Other Options Are Wrong:
Options b, c, and d (5:00 p.m., 6:00 p.m., 7:00 p.m.) would require much larger gains or different rate relationships not supported by the data. Only 4:00 p.m. is consistent with the given gain rate and displayed time.

Common Pitfalls:
Common errors include treating the gain of 5 seconds as per minute instead of per 3 minutes, or failing to convert seconds properly into minutes. Another mistake is to add 15 minutes directly to the true time instead of reversing the effect of the gain. Careful setup of the proportional relation avoids these problems.

Final Answer:
The true time when the watch indicates 4:15 p.m. is 4:00 p.m.