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

(720/15=48)

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 LCM=2x2x2x2x3x5=240

the two clocks must make 240 12hr cycles to both be at 12 o'clock at the same time

240x12=2880 hours

## Comments

1 576 960

2 1152 1920

3 1728

28804 2304 3840

5

288048006 3456 5760

7 4032 6720

8 4608 7680

9 5184 8640