A watch which gains 5 seconds in 3 minutes was set right at 7:00 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is

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.