Question

A passenger train leaves a train depot three hours after a freight train leaves the same depot. The freight train is traveling 16 mph slower than the passenger train. Find the rate of the freight train if the passenger train overtakes the freight train after 7.125h.

Answer 1

From the question:
Passenger Train (PT)
Freight Train (FT)

As at when it is 7.125 hours, when the PT meets up the FT,
If the FT has been traveling for x miles per hour, then
The PT which is faster is traveling (x+16) miles per hour.

Since the PT has traveled 7.125 hours, the FT would have traveled for (7.125 + 3) = 10.125.

As at when both trains meet, the distances traveled by both are equal.

Distance = Speed * time
PT = FT
(x+16)*7.125 = x(10.125)
7.125x + 16*7.125 = 10.125x
7.125x + 114 = 10.125x
114 = 10.125x - 7.125x
114 = 3x
x = 114/3
x = 38

Therefore FT had been traveling speed of 38 miles per hour.
Of course you know the PT is 16 miles per hour faster = 38 + 16 = 54 miles per hour.
I hope this helps.

Answer 2

-- The passenger train is traveling 16 mph faster than the freighter,
so it gains 16 miles on the freighter every hour.

-- If the passenger train overtakes the freighter after 7.125 hours,
then it gained (7.125 x 16) = 114 miles on it in order to overtake it.

-- But that was just the lead that the freighter built up in the first 3 hours.
So the freighter's speed is 114/3 = 38 miles per hour.

And the passenger train's speed is 54 mph.

Check:

-- The passenger train traveled at 54 mph for 7.125 hours = 384.75 miles

-- The freighter traveled at 38 mph for 10.125 hours = 384.75 miles

-- The distances are equal, so that's when one overtook the other, and
they were both in the same place.