The line of reasoning should be as follows:-
It takes 12 hours for the hour hand to make a complete 360° rotation, i.e.
12 hours-------> 360°
==> 1 hour -------> 30°
==> 60 min -------> 30°
==> 1 min -------> 0.5°
==> 20.5 min -------> 10.25°
So, in a time period of 20.5 minutes, the hour hand would have traversed 10.25°.
Now, when the clock shows 10 A.M exactly, the hour hand has traversed 300° (since for each hour, the hour hand traverses 30°. So, after 10 hours exactly, it has traversed 300°). And in a further 20 minutes and 30 seconds, i.e. 20.5 minutes, it would have traversed 10.25° further, as we have just calculated above. So, the total angular distance traversed by the hour hand is (300+10.25)° = 310.25°.
Now, the second hand is at the exact position '6' in the clock (since it travels for a period of 30 seconds), and hence has traversed exactly half of the clock, thus traveling an angular distance of 180°.
Therefore, the angle between the hour hand and the second hand would be (310.25 - 180)° = 130.25°.
Hope that makes it clear!
