Two runners A and B start from two different points and run towards each other along a straight track. Runner A runs at 12 km/h and runner B runs at 16 km/h. If they meet after 1 hour and 30 minutes, what was the distance (in kilometres) between them when they started?

Introduction / Context:
This problem is a straightforward application of the relative speed concept in time and distance. Two runners are moving directly towards each other, and we are given their individual speeds and the time taken to meet. The question asks for the initial separation distance between them, which is a classical and frequently asked type in quantitative exams.

Given Data / Assumptions:

Speed of runner A = 12 km/h.
Speed of runner B = 16 km/h.
They are running towards each other in a straight line.
Time taken to meet = 1 hour 30 minutes = 1.5 hours.
Speeds are assumed to be constant and there are no breaks in their motion.

Concept / Approach:
When two objects move towards each other along the same straight line, their relative speed is equal to the sum of their individual speeds. The total distance between them initially is equal to relative speed multiplied by the time until they meet. Because speeds are in kilometres per hour, we convert the time into hours and then apply distance = speed * time to find the initial separation between the two runners.

Step-by-Step Solution:
Step 1: Convert the meeting time into hours: 1 hour 30 minutes = 1.5 hours. Step 2: Compute the relative speed: relative speed = 12 + 16 = 28 km/h. Step 3: Use distance = speed * time. Here, distance between them initially = relative speed * time taken to meet. Step 4: Distance = 28 * 1.5 = 42 kilometres. Step 5: Therefore, the initial distance between runner A and runner B was 42 kilometres.

Verification / Alternative check:
We can check by calculating how far each runner travels in 1.5 hours. Runner A covers 12 * 1.5 = 18 kilometres. Runner B covers 16 * 1.5 = 24 kilometres. Adding these distances gives 18 + 24 = 42 kilometres, which must equal the original separation between them. This confirms that 42 kilometres is correct.

Why Other Options Are Wrong:
If the distance were 36 kilometres or 40 kilometres, the time required at a combined speed of 28 km/h would be 36 / 28 hours or 40 / 28 hours, which are not equal to 1.5 hours. A distance of 45 kilometres would require 45 / 28 hours, which is more than 1.5 hours. Similarly, 30 kilometres would result in a smaller time. Only 42 kilometres corresponds exactly to 1.5 hours at a relative speed of 28 km/h.

Common Pitfalls:
Common mistakes include subtracting the speeds instead of adding them for motion towards each other, or failing to convert 1 hour 30 minutes into 1.5 hours. Another typical error is mixing up which distance to attribute to each runner instead of summing them. A clear understanding of relative motion and careful unit handling avoids these issues.

Final Answer:
The runners were initially separated by 42 kilometres.