A clock is set right at 5 a.m. The clock loses 16 minutes in 24 hours. What will be the correct approximate time when the clock indicates 10 p.m. on 4th day?

The correct option/answer is 11 p.m.

Let us break down the logical and mathematical derivation step-by-step to understand why this is the correct approximate time.

Step 1: Calculate the total time elapsed on the incorrect clock
The clock was set right at 5 a.m. on the 1st day.
We need to find the indicated time at 10 p.m. on the 4th day.
Let us calculate the total hours elapsed from 5 a.m. on Day 1 to 10 p.m. on Day 4:
- From 5 a.m. Day 1 to 5 a.m. Day 4 is exactly 3 days, which is:
$3\times 24=72\text{ hours}$
- From 5 a.m. Day 4 to 10 p.m. Day 4 is:
From 5 a.m. to 5 p.m. is 12 hours, plus 5 more hours to reach 10 p.m., making it:
$12+5=17\text{ hours}$
Therefore, the total time elapsed on the incorrect clock is:
$72+17=89\text{ hours}$

Step 2: Find the relationship between the incorrect clock and a true clock
The clock loses 16 minutes in 24 hours.
This means that in 24 hours of true time, the incorrect clock only displays:
$24\text{ hours}-16\text{ minutes}$
Let's convert 16 minutes to hours:
$\frac{16}{60}=\frac{4}{15}\text{ hours}$
So, 23 hours and 44 minutes in decimal hours is:
$24-\frac{4}{15}=\frac{356}{15}\text{ hours of the incorrect clock}$
Thus, $\frac{356}{15}$ hours of the incorrect clock is equivalent to $24$ hours of a true clock.

Step 3: Calculate the true time elapsed corresponding to 89 hours of the incorrect clock
Using the ratio:
$\text{True Hours}=89\times \left(\frac{24}{356/15}\right)$
Simplifying the expression:
$\text{True Hours}=89\times \frac{24\times 15}{356}$
Notice that $356=4\times 89$. We can cancel 89 from the numerator and denominator:
$\text{True Hours}=\frac{24\times 15}{4}$
$\text{True Hours}=6\times 15=90\text{ hours}$

Step 4: Determine the correct time
The true time elapsed is exactly 90 hours from 5 a.m. on the 1st day.
Since the incorrect clock showed 89 hours elapsed (reaching 10 p.m. on the 4th day), the true clock, having elapsed 90 hours (which is 1 hour more than 89 hours), will indicate:
$10\text{ p.m.}+1\text{ hour}=11\text{ p.m.}$