A clock is set correctly at 5:00 a.m. It loses 16 minutes in every 24 hours. What is the true time when the clock indicates 10:00 p.m. on the fourth day?

Introduction / Context:
This problem asks us to relate the reading of a slow clock to the actual time. The clock loses time, meaning that in a true 24 hour day it shows less than 24 hours. We are given the starting moment when it is correct and a later moment according to the faulty clock, and we need to deduce the true time at that later reading. These questions check understanding of proportional change and how errors accumulate uniformly over several days.

Given Data / Assumptions:

• The clock is correct at 5:00 a.m. on the starting day.
• The clock loses 16 minutes in 24 hours, so it runs slow.
• The clock indicates 10:00 p.m. on the fourth day after it was set right.
• The loss is uniform (constant rate) over time.

Concept / Approach:
In 24 true hours, the clock shows 24 hours minus 16 minutes. Thus its speed as a fraction of true time is: rate factor = (1440 - 16) / 1440 = 1424 / 1440. If we let T be the true time elapsed since 5:00 a.m., then: shown time elapsed = rate factor * T. We know the shown time elapsed from 5:00 a.m. on day one to 10:00 p.m. on day four, so we can solve for the true elapsed time and hence the true clock time.

Step-by-Step Solution:
Step 1: Compute shown time elapsed. From 5:00 a.m. day one to 5:00 a.m. day four = 3 full days = 72 hours. From 5:00 a.m. to 10:00 p.m. on the same day = 17 hours. Total shown elapsed time = 72 + 17 = 89 hours. Step 2: Rate factor of the slow clock. In 24 hours, it shows 24 hours minus 16 minutes = 1440 - 16 = 1424 minutes. Rate factor = 1424 / 1440. Step 3: Relate shown time and true time. Let T be true hours elapsed since 5:00 a.m. shown time = rate factor * T. 89 = (1424 / 1440) * T. T = 89 * 1440 / 1424 hours. Compute T: simplifying gives T = 90 hours. Step 4: Find the true clock time. From 5:00 a.m. day one add 72 hours (3 days) to reach 5:00 a.m. day four. Remaining time = 90 - 72 = 18 hours. 5:00 a.m. day four + 18 hours = 11:00 p.m. day four. So when the slow clock shows 10:00 p.m. on the fourth day, the true time is 11:00 p.m.

Verification / Alternative check:
Check the loss: in 90 true hours, how much does the clock show? shown = rate factor * 90 = (1424 / 1440) * 90 = 89 hours. Thus in 90 true hours it shows 89 hours, which matches the given 89 hour reading from 5:00 a.m. to 10:00 p.m. on the fourth day. The result is consistent.

Why Other Options Are Wrong:
Option b (12 p.m.) refers to noon and is far earlier than the computed true time. Option c (1 p.m.) and option d (2 p.m.) also correspond to much smaller true elapsed times and do not match the proportional relationship between shown and true time given by the rate factor. Only 11 p.m. satisfies the calculation.

Common Pitfalls:
Learners sometimes incorrectly subtract 16 minutes only once instead of considering the effect over multiple days. Others mistake 10:00 p.m. on the fourth day as 4 days exactly instead of computing accurately from the starting time. Confusing 12 p.m. and 12 a.m. is another small but common source of error in clock questions.

Final Answer:
The true time when the clock indicates 10:00 p.m. on the fourth day is 11:00 p.m.