Ruchir walks at 20 km/h and Rukma cycles at 25 km/h directly towards each other. If they start at the same time and meet after 48 minutes, what was the distance between them at the start of the journey?

Introduction / Context:
This question tests the fundamental concept of relative speed in time and distance problems. When two people move directly towards each other, their speeds add up, and this combined speed helps us find the initial distance between them. Such problems are very common in aptitude exams and are excellent for revising the link between speed, distance, and time.

Given Data / Assumptions:
Ruchir walks at a constant speed of 20 km/h.
Rukma cycles at a constant speed of 25 km/h towards Ruchir.
They start at the same time and meet after 48 minutes.
Both are assumed to move in a straight line without stopping or changing speed.
We are asked to find the distance between them at the moment they started moving.

Concept / Approach:
The key idea is the formula: distance = speed * time. When two objects move towards each other, the effective or relative speed is the sum of their individual speeds. Once we know the relative speed and the time taken to meet, we can directly compute the starting distance. We must also convert the time from minutes to hours so that the units match the speed in km/h.

Step-by-Step Solution:
Step 1: Relative speed = speed of Ruchir + speed of Rukma.Relative speed = 20 km/h + 25 km/h = 45 km/h.Step 2: Convert 48 minutes to hours: 48 minutes = 48 / 60 hours.48 / 60 = 4 / 5 hours.Step 3: Use distance = speed * time with the relative speed.Distance = 45 * (4 / 5) km.Distance = 45 * 4 / 5 = 180 / 5 = 36 km.So the initial distance between Ruchir and Rukma was 36 km.

Verification / Alternative check:
We can check by working backward. If the distance is 36 km, in 4 / 5 hours, Ruchir covers 20 * 4 / 5 = 16 km and Rukma covers 25 * 4 / 5 = 20 km. The total covered is 16 + 20 = 36 km, which matches the starting separation. Hence the calculation is consistent and confirms the answer.

Why Other Options Are Wrong:
54 km would require a much larger time or higher speeds. At a combined speed of 45 km/h, 54 km would need 54 / 45 hours, which is 1.2 hours or 72 minutes, not 48 minutes.
48 km would require 48 / 45 hours, which is more than 1 hour, again not equal to 48 minutes.
30 km would mean 30 / 45 hours, that is 40 minutes, which is less than 48 minutes.
42 km would need 42 / 45 hours, which is 56 minutes, not 48 minutes.

Common Pitfalls:
Students sometimes subtract the speeds instead of adding them when two people move towards each other. Another common mistake is to forget converting minutes to hours before using the speed in km/h, which leads to incorrect results. Also, some learners directly pick an option that feels roughly close without checking with the exact formula, which is risky in competitive exams.

Final Answer:
The initial distance between Ruchir and Rukma was 36 km.