Two watches are together at 12 o'clock. If one gains 75 seconds each hour , and the other loses 45 seconds each hour, when will they be together again at 12?
let's look at the problem from a different point of view
they both are at 12 o'clock
how many hours must go by so that both clocks are at 12 o'clock again ?
the fast clock makes a 12 hour cycle but gains 1.25 minutes times 12=15 minutes, so it is not at 12 o'clock
the slow clock makes a 12 hour cycle but loses 0.75 minutes times 12=9 minutes so it is not at 12 0'clock
in 12 hours there are 720 minutes, so the fast clock must go through 48 12 hour cycles to be at 12 0'clock
the slow clock must go through 80 12 hour cycles to be at 12 o'clock (720/9=80)
when the fast clock goes through 48 12hr cycles it gains 720 minutes (one 12 hour cycle)
when the slow clock goes through 80 12hr cycles it loses 720 minutes (one 12 hour cycle)
now find the LCM of 48 and 80
48=2x2x2x2x3 and 80=2x2x2x2x5
the two clocks must make 240 12hr cycles to both be at 12 o'clock at the same time