MAH MBA CET Clocks Questions

The watch gains 5 seconds in 3 minutes. It means that in 60 minutes, the watch will gain 100 seconds i.e. $$\frac{5}{3}\min$$.
So, for every hour passed, the clock will cover 1hr + $$\frac{5}{3}\min$$ = $$61\ \frac{2}{3}$$ mins.
We need to find the correct time, when the clock has covered 9 hrs 15 mins i.e. quarter past 4 o'clock from 7:00 am ==> 555 mins.
60 mins -----> $$61\ \frac{2}{3}$$ mins 
$$\frac{60}{61\frac{2}{3}}$$ mins ------> 1 min
$$\frac{60}{61\frac{2}{3}}\times\ 555$$ mins -------> 555 mins.
So, the actual time passed ==> $$\frac{60}{61\frac{2}{3}}\times\ 555=\frac{60\times\ 3}{185}\times\ 555=540$$ mins ==> 9 hrs.
So, the actual time is 7:00 am + 9hrs = 4 o'clock.

At 9 o'clock, the smaller angle between hour hand and minute hand is 90 degrees.
As time passes, this smaller angle will increase.
For the hands to point in exact opposite directions, the angle has to become 180 degrees.
It means that the minute hand has to gain 90 degrees more over the hour hand.
Every minute, the minute hand gains 6 - 0.5 = 5.5 degrees over the hour hand.
To gain 90 more degrees, the minute hand will take $$\frac{90}{5.5}$$ mins ==> $$16\ \frac{4}{11}$$ mins.
So, the hands will face exact opposite directions at $$16\ \frac{4}{11}$$ mins past 9.

The smaller angle between the hands at 5 o'clock = 150 degrees.
As the time passes, this smaller angle will decrease.
The relative speed of movement = 6 - 0.5 = 5.5 degrees per minute.
In 15 mins i.e. at 5:15, the distance covered by the minute hand over the hour hand = 15*5.5 = 82.5 degrees.
So, the smaller angle will now be ==> 150 - 82.5 = 67.5 degrees.

A clock gains 15 mins in 24 hrs.
From 12 noon to 4 am, 16 hrs have passed and the clock will gain: $$\frac{16}{24}\times\ 15=10\ $$mins.
Thus, the clock will show 4:10 am.