Passing trains......

The relative speed of the two trains when moving in opposite directions is: (500 + 300) /10 =80 meters per second. Their relative speeds when moving in the same direction is:(500 - 300) / 25 =8 meters per second. Therefore the speed of the faster train is:(80 + 8)/2 =44 meters  per second, or 158.4 km/h, and the speed of the slower train is: (80 - 8)/2 =36 meters per second, or 129.6 km/h.

Two trains pass each other in 10 seconds when moving in opposite directions. When moving in the same direction, the swifter train passes the slower one in 25 seconds. What is the speed of each train if their respective lengths are 300 meters and 500 meters? Thank you.

 Well...if by "passing" you mean that the trains completely pass each other.......

Call the rate of the slower train R1   and call the rate of the faster train R2.......and these are in m/sec

Then when passing each other,  the distance that the back of the either train must cover in 10 seconds is  800m....but  either train is covering this distance by the combined rate of both trains

So....we have   

Total distance to be covered / [ combined rates] = total time

800 / [ R2 + R1]  = 10   →    800 = 10R2 + 10R1 →     80  = R2 + R1    (1)

And....when the trainns are going in the same direction we have that the effective rate of the faster train is being slowed by the rate of the slower train

800 / [ R2 - R1] = 25  →    800  =  25R2  -  25R1  →   32 = R2 - R1 (2)

Adding (1)  and (2), we have

112 = 2R2    divide both sides by 2

56  = R2  = 56 m/sec  = 201.6 km/h     ....this is the rate of the faster train

And the slower train moves at  80  = 56 + R1  →  R1 = 24m/sec  = 86.4 km/hr

Proof :

If the slower train were standing still, it would take the back of the faster train   about 14 + 2/7 seconds to cover 800m ....but ...its rate is effectively being "slowed" by the rate of the slower train......so...its effective rate is only [ 56 - 24]  = 32m/s

And 25 sec * 32 m/s  = 800 m