Apassenger 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.

-- 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.