Step 1: Determine the Relationship Between the Faulty and Correct Clock

The faulty clock loses 16 minutes every 24 hours. This means:

• When 24 hours of correct time have passed,
• The faulty clock has only shown 23 hours and 44 minutes.

Let's convert the faulty time to hours:
$23 \text{ hours } 44 \text{ minutes } = 23 + \frac{44}{60} \text{ hours } = 23 + \frac{11}{15} \text{ hours } = \frac{356}{15} \text{ hours}$.

So, $\frac{356}{15}$ hours on the faulty clock corresponds to 24 hours on a correct clock.

Step 2: Calculate the Time Elapsed on the Faulty Clock

The clock was set at 5 a.m. on the first day and it now shows 10 p.m. on the fourth day.

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

Step 3: Calculate the Correct Time Elapsed

We need to find out how much correct time has passed when the faulty clock shows 89 hours.

Using the relationship from Step 1:

Correct Time = (Time on Faulty Clock) $\times \frac{\text{Correct Hours}}{\text{Faulty Hours}}$

Correct Time = $89 \times \frac{24}{\frac{356}{15}}$

Correct Time = $89 \times \frac{24 \times 15}{356}$

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

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

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

Step 4: Find the Correct Time

We add the 90 hours of correct time to the starting time.

• Start Time: 5 a.m. (Day 1)
• Time Elapsed: 90 hours = 3 days and 18 hours (since $90 = 3 \times 24 + 18$).

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

Now we add the remaining 18 hours:

5 a.m. (Day 4) + 18 hours = 23:00 hours = 11 p.m.

Therefore, the correct time is 11 p.m. on the fourth day.