Points A and B are 15 km apart. When travelling towards each other, two persons starting from A and B meet after half an hour. When they travel in the same direction, they meet 2.5 hours later. What is the speed of the faster person (in km/h)?

Introduction / Context:
This question uses relative speed concepts both when two people move towards each other and when they move in the same direction. The same pair of speeds must satisfy both situations, which allows us to form two equations. Solving these equations gives the individual speeds. Identifying which person is faster and translating that into the final answer is the goal. Such problems are classic examples of how relative motion can simplify time and distance calculations.

Given Data / Assumptions:

Distance between A and B = 15 km.
When travelling towards each other, they meet after 0.5 hours.
When travelling in the same direction, they meet after 2.5 hours.
Let the speeds be a km/h and b km/h, where a is the faster speed and b is the slower speed.
They move along a straight line and maintain constant speeds.

Concept / Approach:
When two people move towards each other, their relative speed is the sum of their speeds (a + b). When they move in the same direction, the relative speed is the difference between their speeds (a - b), assuming a is greater than b. Using distance = speed * time, we write two equations: one with a + b and one with a - b, both with the same distance of 15 km. Solving these two equations simultaneously gives us a and b, and we then choose the larger one as the speed of the faster person.

Step-by-Step Solution:
Step 1: Use the case when they travel towards each other. Distance = 15 km, time = 0.5 hours. 15 = (a + b) * 0.5, so a + b = 30. Step 2: Use the case when they travel in the same direction. Distance = 15 km, time = 2.5 hours. 15 = (a - b) * 2.5, so a - b = 15 / 2.5 = 6. Step 3: Solve the system of equations: a + b = 30 and a - b = 6. Add the two equations: 2a = 36, so a = 18 km/h. Substitute back to find b: 18 + b = 30, so b = 12 km/h.

Verification / Alternative check:
Check the opposite direction scenario: relative speed = 18 + 12 = 30 km/h. Time = 15 / 30 = 0.5 hours, correct. Check the same direction scenario: relative speed = 18 - 12 = 6 km/h. Time = 15 / 6 = 2.5 hours, also correct. Both conditions are satisfied, confirming that the speeds are 18 km/h and 12 km/h and that 18 km/h is the faster speed.

Why Other Options Are Wrong:
Options a (15 km/hr), c (10 km/hr) and d (8 km/hr) do not produce the correct pair of relative speeds and times when paired with any other reasonable speed. Only 18 km/hr works smoothly with a second speed of 12 km/hr to satisfy both the head-on and same direction meeting conditions.

Common Pitfalls:
A frequent error is mixing up which equation belongs to which scenario, or incorrectly using a + b for both. Another mistake is to forget that the slower person's speed must be positive, which acts as a quick check on algebraic manipulations. Some students also compute 15 / 2.5 incorrectly. Taking care with basic arithmetic and clearly labelling equations helps avoid these problems.

Final Answer:
The faster person travels at a speed of 18 km/hr.