A man covers a certain one-way distance by train at a speed of 40 km/hr and returns over the same distance on foot at a speed of 8 km/hr. If the total time for the entire journey (going and coming back) is 12 hours, what is the one-way distance in km?

Introduction / Context:
This is a classic average speed and total time question involving different speeds on the onward and return journeys. The man travels one way by train and returns on foot at a lower speed. By expressing the total time as the sum of the two individual travel times, we can form an equation in terms of the distance and solve it.

Given Data / Assumptions:
- One-way distance between the two points = D km (unknown).
- Speed of the train for the onward trip = 40 km/hr.
- Speed of walking for the return trip = 8 km/hr.
- Total time for the round trip = 12 hours.
- The route and distance are the same in both directions, and speeds are constant.

Concept / Approach:
The total journey time equals the sum of the times for the onward and return journeys. For any leg, time = distance / speed. Hence the onward time is D / 40 hours and the return time is D / 8 hours. Setting their sum equal to 12 hours gives a linear equation in D, which we can solve easily.

Step-by-Step Solution:
Step 1: Let the one-way distance be D km.Step 2: Time taken by train (onward) = D / 40 hours.Step 3: Time taken by walking (return) = D / 8 hours.Step 4: Total time for the round trip is given as 12 hours, so D / 40 + D / 8 = 12.Step 5: Find a common denominator. The least common multiple of 40 and 8 is 40, so rewrite as D / 40 + 5D / 40 = 12.Step 6: Combine terms: (D + 5D) / 40 = 6D / 40 = 12.Step 7: Simplify the fraction: 6D / 40 = 3D / 20, so 3D / 20 = 12.Step 8: Multiply both sides by 20: 3D = 12 * 20 = 240.Step 9: Divide by 3: D = 240 / 3 = 80 km.

Verification / Alternative check:
Check the individual times with D = 80 km. Onward by train: 80 / 40 = 2 hours. Return on foot: 80 / 8 = 10 hours. Total time = 2 + 10 = 12 hours, which matches the given total journey time. Therefore, the distance calculation is consistent and correct.

Why Other Options Are Wrong:
If D = 60 km, total time would be 60 / 40 + 60 / 8 = 1.5 + 7.5 = 9 hours, not 12. For 90 km, total time becomes 90 / 40 + 90 / 8 = 2.25 + 11.25 = 13.5 hours. For 100 km, time is 2.5 + 12.5 = 15 hours. 72 km gives 72 / 40 + 72 / 8 = 1.8 + 9 = 10.8 hours. None of these equal 12 hours.

Common Pitfalls:
Learners sometimes average the speeds (40 and 8) and directly multiply by total time, which is incorrect because time at each speed is not equal. Others may forget that both legs share the same distance and mistakenly use different D values for onward and return. Always express the total time as the sum of distance / speed for each leg.

Final Answer:
The one-way distance between the two points is 80 km.