A and B start from the same point. A cycles 10 km towards the south and then turns right and cycles 9 km. B cycles 2 km towards the north, then turns left and cycles 15 km and finally turns left again and cycles 12 km. Where is B with respect to A now?

Introduction / Context:
This question compares the final positions of two cyclists, A and B, who start from the same point but follow different routes. The goal is to find where B is located relative to A after all movements. It combines direction sense, relative position and basic coordinate reasoning, which are often tested in competitive examinations.

Given Data / Assumptions:
- A cycles 10 km towards the south. - A then turns right from facing south, which means towards the west, and cycles 9 km. - B cycles 2 km towards the north. - B then turns left from facing north, which means towards the west, and cycles 15 km. - B again turns left from facing west, which means towards the south, and cycles 12 km. - Both start from the same origin on a flat plane.

Concept / Approach:
We treat the starting point as (0, 0). Using the standard convention, north is positive y, south is negative y, east is positive x and west is negative x. We compute the final coordinates of A and B separately by updating x and y for each leg. After that, we subtract A's coordinates from B's coordinates to find B's relative position. The sign and magnitude of the difference tell us the direction and distance from A.

Step-by-Step Solution:
1. A: start at (0, 0), move 10 km south to (0, -10). 2. A turns right from south to west and moves 9 km: A ends at (-9, -10). 3. B: start at (0, 0), move 2 km north to (0, 2). 4. B turns left from north to west and moves 15 km: B reaches (-15, 2). 5. B then turns left from west to south and moves 12 km: B ends at (-15, -10). 6. Final coordinates: A at (-9, -10) and B at (-15, -10). 7. The y coordinates are equal, but B's x coordinate is 6 units less, so B is 6 km further west than A.

Verification / Alternative check:
We can reason without full coordinates by focusing on relative motion. Both A and B ultimately reach the same south level of -10 km relative to the start because A goes 10 south and B goes 2 north then 12 south (net 10 south). Horizontally, A goes 9 km west from the original vertical line, while B goes 15 km west. The difference between 15 and 9 is 6, so B is 6 km further to the west than A. A simple sketch with parallel horizontal lines confirms this relationship.

Why Other Options Are Wrong:
- 6 km East of A reverses the actual direction; B is further west, not east. - 24 km West or East misinterpret the sums of legs and do not match the correct horizontal difference of 6 km. - Directly north of A would require different y coordinates, but both share the same y value.

Common Pitfalls:
Learners often get confused by multiple turns and may misidentify the new facing direction after each turn. Another pitfall is to compare each person's position with the origin separately instead of directly comparing A and B. Writing down the facing direction after each turn and calculating net north south and east west movements systematically avoids such confusion.

Final Answer:
B is 6 km to the west of A.