Question

A clock gains 15 seconds per day and another loses 45 seconds per day. after what length of time will the first clock 12 hrs ahead of the second

Answer 1

since a clock gains 15 seconds the other loses 45 seconds a day, the total difference per day is 1 minute.
for a 12 hours difference

12×60
720 minutes
it will take 720 days for the 1st clock to be 12 hrs ahead of the second one

Answer 2

Let the time of two clocks be x
Time of 1st Clock (AP Manner)=x, x+15,x+30
time of 2nd clock (AP manner)=x-45,x-90
tn-t'n=12
(a+(n-1)d)-(a+(n-1)d')=12x60x60
(x+(n-1) 15)-(x+(n-1)-45) = 43200
x +15n-15-x+45n-45=43200
60n-60=43200
n = 720