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?

Calculate the total elapsed time on the slow clock:

The clock was set correctly at 5 a.m. on Day 1. It indicates 10 p.m. on Day 4.

From 5 a.m. Day 1 to 5 a.m. Day 4: $3$ days $= 3 \times 24 = 72$ hours.

From 5 a.m. Day 4 to 10 p.m. Day 4: $17$ hours.

Total elapsed time (Slow Clock): $72 + 17 = 89$ hours.

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Determine the relationship between True Time and Slow Clock Time:

The clock loses 16 minutes every 24 hours.

$24$ hours of True Time $= 24 \text{ hours} - 16 \text{ minutes}$ of Slow Clock Time.

$24$ hours $= 1440$ minutes.

Slow Clock Time in 24 true hours $= 1440 - 16 = 1424$ minutes.

The ratio of True Time to Slow Clock Time is:

$\frac{\text{True Time}}{\text{Slow Clock Time}} = \frac{1440}{1424} = \frac{90}{89}$

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Calculate the True Elapsed Time:

$\text{True Elapsed Time} = \text{Slow Elapsed Time} \times \frac{90}{89}$

$\text{True Elapsed Time} = 89 \text{ hours} \times \frac{90}{89} = 90 \text{ hours}$

We add the true elapsed time (90 hours) to the starting time (5 a.m. Day 1):

$90 \text{ hours} = 3 \text{ days} (72 \text{ hours}) + 18 \text{ hours}$.

3 days from 5 a.m. Day 1: 5 a.m. on Day 4.

18 hours after 5 a.m. on Day 4:

$5 \text{ a.m.} + 12 \text{ hours} = 5 \text{ p.m.}$

$5 \text{ p.m.} + 6 \text{ hours} = 11 \text{ p.m.}$

The correct approximate time is 11 p.m.