A man covers a certain distance from his home to a destination by train at a speed of 50 km/h and returns over the same distance on foot, walking at 5 km/h. If the total time taken for the two-way journey is 5 hours 30 minutes, what is the distance (in kilometres) of one side?

Introduction / Context:
This question is a classic example of travelling the same distance by two different modes of transport at different speeds, one for the onward journey and one for the return journey. The total time for the round trip is known, and we must find the one way distance. It uses the relationship between distance, speed, and time along with basic algebra to solve for the unknown distance.

Given Data / Assumptions:

Speed of the train on the onward journey = 50 km/h.
Speed of walking on the return journey = 5 km/h.
Distance from home to destination (one side) = d kilometres (unknown).
Total two way time = 5 hours 30 minutes = 5.5 hours.
Speeds are constant and there are no breaks during each leg of the journey.

Concept / Approach:
For a given distance, time = distance / speed. There are two legs: going by train and returning on foot. Thus, total time = d / 50 + d / 5. This total time is given as 5.5 hours. Setting up this equation and solving for d gives the required one way distance. Careful conversion of minutes to hours and correct algebraic manipulation are essential.

Step-by-Step Solution:
Step 1: Let the one way distance be d kilometres. Step 2: Time taken by train = d / 50 hours. Step 3: Time taken walking back = d / 5 hours. Step 4: Total time is given as 5.5 hours, so d / 50 + d / 5 = 5.5. Step 5: Express 1 / 5 as 10 / 50 to combine fractions: d / 50 + 10d / 50 = 5.5, so 11d / 50 = 5.5. Step 6: Multiply both sides by 50: 11d = 5.5 * 50 = 275. Step 7: Solve for d: d = 275 / 11 = 25 kilometres.

Verification / Alternative check:
Check the times for d = 25 km. Time by train = 25 / 50 = 0.5 hours (30 minutes). Time walking back = 25 / 5 = 5 hours. Total time = 0.5 + 5 = 5.5 hours, which is exactly 5 hours 30 minutes. This confirms that the one way distance is 25 km.

Why Other Options Are Wrong:
If the distance were 30 km, times would be 0.6 hours by train and 6 hours walking, totaling 6.6 hours, which is too long. For 20 km, total time would be 0.4 + 4 = 4.4 hours, which is too short. Similarly, 18 km or 22 km do not produce a total time of 5.5 hours. Only 25 km satisfies the exact time condition.

Common Pitfalls:
Learners sometimes forget to convert 5 hours 30 minutes into 5.5 hours or mishandle the fraction addition when combining d / 50 and d / 5. Another error is to attempt to use an average speed formula incorrectly instead of summing the individual times. Carefully forming the time equation and solving it systematically leads to the correct distance.

Final Answer:
The distance of one side is 25 kilometres.