A train covers the distance between station A and station B in 45 minutes at its usual speed. If the speed of the train is reduced by 5 km/h, it takes 48 minutes to cover the same distance. What is the distance between stations A and B in kilometres?

Introduction / Context:
This problem explores how changes in speed affect travel time over a fixed distance. It tests the ability to convert minutes into hours, to set up equations based on distance = speed * time, and to solve for the unknown distance when two different speed and time combinations are given.

Given Data / Assumptions:

• At original speed, the train covers the distance in 45 minutes.
• At reduced speed, which is 5 km/h less, it covers the same distance in 48 minutes.
• We must find the distance between stations A and B.
• Speeds are constant in each scenario and the track is straight.

Concept / Approach:
Let the original speed be v km/h and the distance be D kilometres. Time in hours is obtained by dividing minutes by 60. The first equation is D = v * (45 / 60), and the second equation is D = (v - 5) * (48 / 60). Since both expressions equal D, we can equate them and solve for v. After finding v, we substitute back into either expression to determine D.

Step-by-Step Solution:
Step 1: Convert times into hours: 45 minutes = 45 / 60 = 3 / 4 hour, and 48 minutes = 48 / 60 = 4 / 5 hour.Step 2: Let original speed be v km/h and distance be D km.Step 3: From the first condition, D = v * 3 / 4.Step 4: From the second condition, D = (v - 5) * 4 / 5.Step 5: Equate the two expressions for D: v * 3 / 4 = (v - 5) * 4 / 5.Step 6: Cross multiply: 5 * 3v = 4 * 4 * (v - 5), which gives 15v = 16v - 80, so v = 80 km/h.Step 7: Substitute back to find D: D = v * 3 / 4 = 80 * 3 / 4 = 60 km.

Verification / Alternative check:
At 80 km/h, time to travel 60 km = 60 / 80 hours = 3 / 4 hour = 45 minutes, matching the first condition.At reduced speed 75 km/h, time to travel 60 km = 60 / 75 hours = 4 / 5 hour = 48 minutes, matching the second condition.

Why Other Options Are Wrong:
Distances like 32 km, 45 km, 50 km or 80 km do not satisfy both travel times when checked with the given speed changes. If you plug them into the distance = speed * time relationship, at least one of the 45 minute or 48 minute conditions will fail.

Common Pitfalls:
Some learners forget to convert minutes to hours, which leads to incorrect distances. Others incorrectly add 5 km/h instead of subtracting it or mis-handle the cross multiplication step. Writing down both equations clearly and working through the algebra carefully is essential for such questions.

Final Answer:
The distance between stations A and B is 60 km.