Asking when will the trains meet.

Question

A train leaves a station and travels north at a speed of 
30 mph. 
Five hours later, a second train leaves on a parallel track and travels north at
105 mph. How far from the station will they meet?

Answer 1

distance = Rate x (final time - start time) = D

 

Rate1 = 1st train rate = 30 mph              D1 = Rate1*(final time1 - start time1)    set start time1 = 0 hr

 

Rate2 = 2nd train rate = 105 mph           D2 = Rate2*(final time2 - start time2)     start time2 = 5 hr  (relative to time1)

 

 

Both trains leave from same starting point; therefore, the distance and final time for each when they meet is the same.

So set both distances equal or D = D1 = D2  and final time = final time1 = final time2

 

and solve for final time (when both trains meet):

 

Rate1*(final time1 - start time1)  =  Rate2*(final time2- start time2)

 

Rate1*(final time - start time1) = Rate2*(final time - start time2)

 

since start time1 = 0,

 

 

final time = (Rate2/Rate1)*(final time - start time2)

 

final time*[1 - (Rate2/Rate1)] =   -(Rate2/Rate1)*(start time2)

 

final time = -(Rate2/Rate1)*(start time2)/[1 - (Rate2/Rate1)]   = -(105 mph/30 mph)*(5 hr)/[1 - (105 mph/30 mph)]

 

final time = -3.5*(5 hr)/(1 - 3.5)  =  -17.5 hr/-2.5 =  17.5 hr/2.5 =  7 hr

 

Now solve for the distance:  D = D1 = Rate1*(final time1 - start time1) = 30 mph*(7 hr - 0 hr) =  210 miles from the station when both trains meet