t1 - the time the trains are moving to each other.
For time t1, first train will go the distance 70*t.
The second train, for time t will go the distance 60*t.
Both for time t1 - 70t+60(t1)=(70+60)*(t1).
45 - (70+60)*t is the distance between trains that left after t hours.
For the first time they will be 10 miles apart when they are moving to each other
45 - (70+60)*t = 10
45-10=(70+60)*t
35=130*t
t=35/130 h =7/26 h≈ 0.27 h
The train meet when distance=0
45 - (70+60)*t=0
45=130t
t=45/130 h = 9/26 h ≈0.35 h when trains met.
Trains began to move apart from each other after 9/26 h moving,
After t2 time trains will be (70+60)*(t2)=130*(t2) mi apart from each other.
130*(t2)=10
t2=1/13h≈0.08h after they met, they are apart from each other 10 mi.
so after beginning of the movement it will be
time when they met + time when they are moved 10 mi apart from each other
9/26 h+1/13h = 11/26 h≈ 0.42 h