A train travels from Station A to Station B at an average speed of 75 km/h. On its return journey from Station B to Station A, the train increases its speed by 25% for the first half of the distance and then reduces its speed by 20% for the remaining half of the distance. If the total time taken for the round trip is 19 hours, find the distance between Station A and Station B.

Let the one-way distance be D, making the onward time D divided by 75 hours. For the return journey, the new speeds are calculated as 75 multiplied by 1.25 for the first half, yielding 93.75 km/h, and 93.75 multiplied by 0.80 for the second half, returning to 75 km/h. The total return time is (D/2) divided by 93.75 plus (D/2) divided by 75, which simplifies to D divided by 187.5 plus D divided by 150, or D multiplied by the sum of (4/750 + 5/750). This equals D multiplied by 9/750, or D divided by 83.33; setting the total time of (D/75) + (D/83.33) equal to 19 hours gives D = 750 km.

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