Train a and b are traveling in the same direction?

Train A and B are traveling in the same direction on parallel tracks. Train A is traveling at 40 miles per hour and train B is traveling at 60 miles per hour. Train A passes a station at 2:10 am If train B passes the same station at 2:20 am, at what time will train B catch up to train A?

Consider at time 2:20,

Train B arrives at stn, but train A is 10mins ahead = 40x10/60 = 6.66 miles ahead.

say it takes time t for train B to catch up with A

60t = 6.66 + 40t
t = 6.66/20 = 0.333 hrs = 20mins

so time when train B passes train A = 2:20+ 20 mins = 2:40

Let

t = time when train B catches up with train A after 2:10 AM

At this time "t", both trains would have travelled the same distance from the same station that they passed (train A
@ 2:10 AM and train B @ 2:20 AM).

For train A, Distance = 40t
For train B, Distance = 60 (t - 10)

Since these two distances are equal, then

40t = 60(t - 10)

40t = 60t - 600

Combining terms,

60t - 40t = 600

20t = 600

t = 30 minutes

Train B will catch up with Train A at 2:40 AM.


CHECK:

Distance travelled by train A = 40 (30/60) = 20 miles

Distance travelled by train B = 60 (30 - 10)/60 = 20 miles

If we consider 2:10 to be t=0 or ti (initial time); and
tf = final time = the time where train b cathes up with train a; then you have the following equations:

Train a passes the station at tia=0 and b passes the station at time tib=tia+10 min.

Distance traveled by train a, at the time b catches up with a:

da = va*(tf - ti)

Distance traveled by b:
db = vb*(tf - (ti+10min)) ; or db = vb*(tf - 10 min)
because ti=0

when b catches up with a; da = db; so

va*(tf - ti) = vb* (tf - (ti + 10 min)); ti = 0; so
va*tf = vb*tf - vb*10min
va*tf - vb*tf = -vb*10min
tf(va-vb) = -vb*10min
tf = (-vb*10min)/(va-vb) ; 10 min = (10/60) hr
tf = (-60mi/hr * (1/6)hr)/(40-60 mi/hr)
tf = (-10 mi) / (-20 mi/hr) (watch the units analysis)
tf = (1/2) hr = 30 min.

Train b cathes up with a 30 minutes after ti; which is 2:10; so it catches up with a at 2:40

Check the distances:
da = 40 mi/hr * (30/60)hr = 20 mi

db = 60 mi/hr * (30-10)/60hr = 20 mi