A clock is set to the right time at 4:00 AM on Thursday. If it gains 20 seconds in every 3 hours, then what is the time shown on the clock at 8:30 PM on Friday night?

The clock was correct at 4:00 AM on Thursday and gains 20 seconds every 3 hours.

Step 1 - Compute the real time elapsed between 4:00 AM Thursday and 8:30 PM Friday.
Thursday 4:00 AM → Friday 4:00 AM = 24 h
Friday 4:00 AM → Friday 8:30 PM = 16.5 h
Total real time elapsed $$=24+16.5=40.5\text{ h}$$

Convert this to seconds because the gain is given in seconds:
$$40.5\text{ h}\times 3600\text{ s h}^{-1}=145\,800\text{ s}$$

Step 2 - Find how many seconds the clock gains in this interval.
Gain rate: 20 s per 3 h $$=20\text{ s per }3\times3600=20\text{ s per }10\,800\text{ s}$$
That is, the clock gains $$\frac{20}{10\,800}=\frac{1}{540}$$ second for every real second.

Total gain $$=145\,800\text{ s}\times\frac{1}{540}=270\text{ s}$$

Step 3 - Convert the gain to minutes and seconds:
$$270\text{ s}=4\text{ min }30\text{ s}$$

Step 4 - Add the gain to the real time 8:30 PM Friday.
8:30 PM + 4 min 30 s = 8:34 min 30 s PM

Therefore, at 8:30 PM real time on Friday, the clock will show 8 hours 34 minutes 30 seconds PM.

Option D which is: 8 hours 34 minutes 30 seconds PM