Step 1: Calculate time elapsed on the faulty clock

First, we find the total hours shown by the incorrect clock from the start time to the end time.

• Start Time: 5 a.m. on the first day.
• End Time (Indicated): 10 p.m. on the fourth day.

The duration is:

• From 5 a.m. (Day 1) to 5 a.m. (Day 4) = 3 full days = $3 \times 24 = 72$ hours.
• From 5 a.m. (Day 4) to 10 p.m. (Day 4) = 17 hours.
• Total time shown by faulty clock = 72 + 17 = 89 hours.

Step 2: Relate faulty time to correct time

The clock loses 16 minutes in 24 hours. This means for every 24 hours of correct time, the faulty clock only shows 23 hours and 44 minutes.

• Correct time = 24 hours.
• Faulty time = 23 hours 44 minutes = $23 \frac{44}{60}$ hours = $23 \frac{11}{15}$ hours = $\frac{356}{15}$ hours.

So, $\frac{356}{15}$ hours on the faulty clock is equivalent to 24 hours of correct time.

Step 3: Calculate the actual time passed (correct time)

We can set up a proportion to find the correct duration ('x') corresponding to the 89 hours shown on the faulty clock.

If $\frac{356}{15}$ faulty hours = 24 correct hours

Then, 89 faulty hours = $89 \times \frac{24}{\frac{356}{15}}$ correct hours.

Since $356 = 4 \times 89$, we can simplify:

$x = \frac{89 \times 24 \times 15}{4 \times 89} = \frac{24 \times 15}{4} = 6 \times 15 = 90$ hours.

So, the actual time that has passed is 90 hours.

Step 4: Determine the correct final time

We add the 90 hours of correct time to the initial start time.

• Start Time: 5 a.m. on the first day.
• Time to add: 90 hours = 3 days and 18 hours (since $90 = 3 \times 24 + 18$).

Adding 3 full days to 5 a.m. (Day 1) brings us to 5 a.m. on Day 4.

Now, add the remaining 18 hours to 5 a.m. on Day 4:

5 a.m. + 18 hours = 23:00, which is 11 p.m.

The correct time is 11:00 PM on the fourth day.