Two cyclists P and Q cycle towards each other from some distance apart. P's speed is 20 km/h and Q's speed is 16 km/h. They meet after 40 minutes. What was the distance between them (in km) when they started?

Introduction / Context:
This problem uses the idea of relative speed when two objects move towards each other. The combined effect of their motions makes them close the gap between them faster than either one alone. By knowing how long it takes for them to meet, we can find the original separation. This sort of question appears frequently in time and distance topics and is fundamental for understanding relative motion.

Given Data / Assumptions:

Speed of cyclist P = 20 km/h.
Speed of cyclist Q = 16 km/h.
They move towards each other and meet after 40 minutes.
40 minutes = 40 / 60 hours = 2 / 3 hours.
We assume both ride at constant speeds along a straight path.

Concept / Approach:
When two objects move directly towards each other, their relative speed is the sum of their individual speeds. The distance between them shrinks at this combined speed. If we know the relative speed and the time until they meet, the initial distance is simply distance = relative speed * time. Working in hours and km/h keeps the units consistent and the calculation straightforward.

Step-by-Step Solution:
Step 1: Compute relative speed. relative speed = 20 + 16 = 36 km/h. Step 2: Convert meeting time into hours. Time = 40 minutes = 40 / 60 = 2 / 3 hours. Step 3: Use distance = speed * time. distance = 36 * (2 / 3) km. distance = 36 * 2 / 3 = 72 / 3 = 24 km. Therefore, the cyclists started 24 km apart.

Verification / Alternative check:
If the initial distance is 24 km and the relative speed is 36 km/h, then the time to meet should be distance / speed = 24 / 36 = 2 / 3 hours, which is 40 minutes. This matches the given condition exactly, so the computation is correct. Both a forward calculation and a reverse check agree on the value 24 km.

Why Other Options Are Wrong:
Options a (36), b (30), and c (25) would imply different meeting times if combined with a relative speed of 36 km/h. For example, 36 km at 36 km/h means 1 hour to meet, not 40 minutes. Similarly, 30 km at 36 km/h gives 50 minutes, and 25 km gives about 41.7 minutes. Only 24 km corresponds precisely to 40 minutes.

Common Pitfalls:
A typical mistake is to subtract the speeds instead of adding them, which would be appropriate only if the cyclists were moving in the same direction. Another error is to keep time in minutes while using speed in km/h without converting units, leading to incorrect distance values. Always ensure that speed and time are expressed in compatible units, usually km/h with hours or m/s with seconds.

Final Answer:
The starting distance between the two cyclists was 24 km.