A clock is set correctly at 8:00 a.m. It gains 10 minutes in every 24 hours. What is the true time when this clock shows 1:00 p.m. on the following day?

Introduction / Context:
This problem tests the concept of clocks that do not keep perfect time. The clock gains time, meaning it runs faster than a correct clock. We are asked to relate the incorrect reading of the faulty clock to the actual or true time. Such questions frequently appear in aptitude exams to check understanding of proportional reasoning and relative rate of gain or loss of time in clocks and watches.

Given Data / Assumptions:

• The clock is correct at 8:00 a.m. on day one.
• The clock gains 10 minutes in 24 hours.
• We consider the reading when the clock shows 1:00 p.m. on the following day.
• We assume the gain of time is uniform over the whole period.

Concept / Approach:
A correct clock gains zero minutes in 24 hours. Here the faulty clock shows more time than actually passed. In 24 true hours, the faulty clock advances by 24 hours plus 10 minutes. Thus, the rate of the faulty clock relative to true time is: faulty rate factor = (24 hours 10 minutes) / 24 hours Once this factor is known, we can set: shown time elapsed = faulty rate factor * true time elapsed We know the shown time elapsed from 8:00 a.m. to 1:00 p.m. next day and solve for the true time elapsed.

Step-by-Step Solution:
Step 1: Time shown by the faulty clock from day one 8:00 a.m. to day two 1:00 p.m. From 8:00 a.m. day one to 8:00 a.m. next day = 24 hours. From 8:00 a.m. to 1:00 p.m. on next day = 5 hours. Total shown time elapsed = 24 + 5 = 29 hours. Step 2: Faulty rate factor. In 24 true hours, the clock gains 10 minutes, so it shows 24 hours 10 minutes. Convert to minutes: 24 hours = 1440 minutes, gain is 10 minutes, so total = 1450 minutes. Rate factor = 1450 / 1440. Step 3: Let true time elapsed = T hours. Shown time elapsed = rate factor * T. 29 = (1450 / 1440) * T. T = 29 * 1440 / 1450 hours. Compute T in hours: 29 * 1440 / 1450 = 28.8 hours. Step 4: Convert 28.8 hours. 28 hours + 0.8 hour; 0.8 hour = 0.8 * 60 = 48 minutes. True time elapsed from 8:00 a.m. day one is 28 hours 48 minutes. Step 5: Find the actual clock time. Add 24 hours to return to 8:00 a.m. next day, remaining = 4 hours 48 minutes. 8:00 a.m. next day + 4 hours 48 minutes = 12:48 p.m. So the true time is 48 minutes past 12.

Verification / Alternative check:
We can check whether 28 hours 48 minutes of true time, multiplied by the rate factor, does give 29 hours shown: 28.8 * 1450 / 1440 = 29 hours This confirms that our computation and reasoning are consistent.

Why Other Options Are Wrong:
Option b (46 minutes past 12) and option c (45 minutes past 12) underestimate the true elapsed time and would not satisfy the proportional rate equation. Option d (47 minutes past 12) is close but still does not fit the required ratio exactly and is a trap for approximate calculators. Only 48 minutes past 12 satisfies the exact gain rate.

Common Pitfalls:
Common mistakes include directly adding or subtracting 10 minutes without considering the proportion over 24 hours, or assuming that 29 hours of shown time equals 29 hours of true time. Another frequent error is mishandling unit conversions between minutes and hours, or forgetting that the gain is spread over the whole period, not added just once.

Final Answer:
The true time when the clock shows 1:00 p.m. on the following day is 48 minutes past 12, that is 12:48 p.m.