A clock gains 15 minutes every day. It is set correctly at exactly 12:00 noon. When the actual time is 4:00 a.m. the next day, what time will this fast clock show?

Introduction / Context:
This question tests your understanding of clocks that are not perfectly accurate, specifically a clock that gains time uniformly each day. You must relate the actual elapsed time to the time shown on a faulty clock that gains 15 minutes in 24 hours. Such problems are common in aptitude tests under the topic of clocks and time gain or loss, and they build your skills in proportional reasoning and time calculations.

Given Data / Assumptions:

• The clock gains 15 minutes in every 24 hours (one full day).
• The clock is set correctly at 12:00 noon (actual time = clock time at that moment).
• We are interested in the clock reading when the actual time is 4:00 a.m. the next day.
• From noon of one day to 4:00 a.m. next day is 16 hours of actual elapsed time.
• The gain is uniform over time.

Concept / Approach:
If a clock gains time uniformly, then its rate is slightly faster than the correct clock. The idea is to find how much time the faulty clock shows for a given actual time. The gain of 15 minutes in 24 hours means that when 24 hours of real time pass, the faulty clock shows 24 hours 15 minutes. Using a proportional relationship, we can find what the faulty clock will show after 16 hours of actual time.

Step-by-Step Solution:
Step 1: Gain per day = 15 minutes in 24 hours. Step 2: The rate factor of the faulty clock = (24 hours + 15 minutes) / 24 hours. Step 3: Convert 15 minutes to hours: 15 minutes = 15 / 60 = 0.25 hours. Step 4: So, in 24 hours of real time, the clock shows 24.25 hours. Step 5: Rate factor = 24.25 / 24 = 97/96 (since 24.25 = 24 + 0.25 = 24 + 1/4 = 97/4, and 24 = 96/4). Step 6: Actual elapsed time between noon and 4:00 a.m. next day = 16 hours. Step 7: Time shown by the faulty clock for 16 hours actual = 16 * (97/96) hours. Step 8: 16 * (97/96) = (16 * 97) / 96 = 1552 / 96 = 16 + 10/60 hours = 16 hours 10 minutes. Step 9: 16 hours 10 minutes after 12:00 noon corresponds to 4:10 a.m. on the next day.

Verification / Alternative check:
Think of the gain per hour. The gain is 15 minutes in 24 hours, so gain per hour = 15 / 24 minutes = 0.625 minutes per hour. For 16 hours of real time, total gain = 0.625 * 16 = 10 minutes. Therefore, when the actual time is 4:00 a.m., the faulty clock will be 10 minutes fast and will show 4:10 a.m. This provides a quick alternative check.

Why Other Options Are Wrong:
• 4:20 a.m.: This would imply a gain of 20 minutes in 16 hours, which is larger than the calculated 10 minutes.
• 4:30 a.m.: This suggests a massive gain of 30 minutes in 16 hours, inconsistent with a gain of only 15 minutes per day.
• 4:02 a.m.: This indicates a gain of only 2 minutes in 16 hours, which is far smaller than the proportional gain from 15 minutes per 24 hours.

Common Pitfalls:
A common mistake is to add 15 minutes directly to the next day's time, forgetting that the gain of 15 minutes occurs over a full 24-hour period, not over 16 hours. Another error is miscounting the elapsed time between noon and 4:00 a.m. the next day. Remember that from 12:00 noon to midnight is 12 hours, plus 4 more hours to 4:00 a.m., giving 16 hours in total. Using the correct proportional relation avoids these mistakes.

Final Answer:
When the actual time is 4:00 a.m. the next day, the fast clock will show 4:10 a.m..