Two cyclists A and B start cycling towards each other at speeds of 21 km/h and 24 km/h respectively. They meet after 1 hour 12 minutes. How far apart (in km) were they when they started?

Introduction / Context:
This is another relative speed question involving two cyclists moving towards each other. Given their individual speeds and the time until they meet, we are asked to determine the initial distance between them. This type of question routinely appears in time and distance sections of aptitude tests and illustrates how relative motion simplifies the calculation of encounter times and distances.

Given Data / Assumptions:

Speed of cyclist A = 21 km/h.
Speed of cyclist B = 24 km/h.
Meeting time = 1 hour 12 minutes = 1.2 hours (since 12 minutes = 12 / 60 hours = 0.2 hours).
They ride directly towards each other along a straight line.
We assume both speeds are constant until they meet.

Concept / Approach:
When two bodies move towards each other, their relative speed is the sum of their speeds. The gap between them closes at this relative speed. The distance between them at the start is therefore equal to relative speed multiplied by the time taken to meet. So, we compute relative speed = 21 + 24 km/h and then multiply by 1.2 hours to obtain the initial separation in kilometres.

Step-by-Step Solution:
Step 1: Compute relative speed. relative speed = 21 + 24 = 45 km/h. Step 2: Convert time to hours. Time = 1 hour 12 minutes = 1 + 12 / 60 = 1.2 hours. Step 3: Use distance = speed * time. distance = 45 * 1.2 km. Compute: 45 * 1.2 = 54 km. Therefore, they were 54 km apart initially.

Verification / Alternative check:
With a starting distance of 54 km and relative speed 45 km/h, the time to meet is distance divided by relative speed = 54 / 45 hours. This equals 1.2 hours, which matches 1 hour 12 minutes. Thus, the initial distance of 54 km is consistent with the given speeds and meeting time.

Why Other Options Are Wrong:
Option a (48 km) would give a meeting time of 48 / 45 = 1.0667 hours (about 1 hour 4 minutes), not 1 hour 12 minutes. Option b (42 km) yields 42 / 45 = 0.9333 hours (about 56 minutes). Option d (36 km) gives 36 / 45 = 0.8 hours (48 minutes). None of these match the provided 1 hour 12 minutes. Only 54 km fits the data correctly.

Common Pitfalls:
Possible mistakes include misreading 1 hour 12 minutes as 1.12 hours (which is wrong), or forgetting to convert minutes to a fraction of an hour. Another error is subtracting speeds instead of adding them, which is appropriate only when moving in the same direction. Always check whether the objects are moving towards or away from each other to choose the correct relative speed expression.

Final Answer:
The cyclists were initially 54 km apart.