Two women A and B are shopping in a mall. They start from the same point in the mall and walk in different directions as described below. Woman A walks 150 m towards the West, then turns to her left and walks 160 m. Woman B walks 140 m towards the East, then turns towards the South and walks 160 m, and then turns to her left and walks 120 m. Based on this information, where is B with respect to A now?

Introduction / Context:
This is a standard directions and distance problem in logical reasoning. Two women start from the same point and walk different paths with several turns. You must carefully track each displacement in terms of East West and North South components and then compute the final relative position of B with respect to A. Such questions test your ability to interpret directional language and perform simple coordinate style calculations.

Given Data / Assumptions:
- Both A and B start from exactly the same point in the mall. - Woman A walks 150 m West, then turns left and walks 160 m. span style="display:block;">- Woman B walks 140 m East, then 160 m South, then turns left and walks 120 m. - Left turns are taken relative to the current facing direction. - We assume standard orientation: North up, South down, East to the right, West to the left.

Concept / Approach:
A convenient approach is to imagine a coordinate system with the starting point as the origin. We treat East as the positive x direction and North as the positive y direction. West becomes negative x and South becomes negative y. Every movement can be broken into horizontal and vertical components. At the end, we subtract A's final coordinates from B's final coordinates to know where B is with respect to A. The magnitude of the difference gives us the distance, while the sign gives us the direction.

Step-by-Step Solution:
Step 1: Start with woman A. From the origin, she walks 150 m West. That corresponds to a movement of minus 150 along the x axis and zero along the y axis. Step 2: A then turns left from West. Facing West, a left turn means she now faces South. She then walks 160 m South, which is a change of minus 160 along the y axis. So A's final coordinates are x = -150 and y = -160. Step 3: Now track woman B. From the origin, she walks 140 m East. That is plus 140 along the x axis. Step 4: B then turns South and walks 160 m. This step changes the y coordinate by minus 160, leaving x at plus 140. Step 5: B then turns left from facing South. A left turn from South makes her face East. She walks 120 m East, which adds 120 to the x coordinate. So B's final coordinates are x = 140 + 120 = 260, y = -160. Step 6: Now compute B's position relative to A. A is at (-150, -160) and B is at (260, -160). The relative displacement from A to B is 260 minus ( -150 ) along x and equal y coordinates, which is 410 m in the East direction. The vertical difference is zero.

Verification / Alternative check:
An alternative way is to compare horizontal movements directly. Across the entire journey, A has gone 150 m West, while B has gone 140 m East plus an additional 120 m East, for a total of 260 m East. The net separation in the East West direction between them is 260 m East plus 150 m East, which gives 410 m East from A to B. Since both moved 160 m South after their first turn, their vertical positions match. Therefore B lies exactly 410 m to the East of A and at the same North South level.

Why Other Options Are Wrong:
- Options suggesting 170 m difference result from incorrect subtraction of East and West movements instead of adding them to find separation. - Options that place B to the West of A ignore the fact that B has moved net East while A has moved net West. - The cannot be determined option is invalid because the distances and directions are fully specified.

Common Pitfalls:
A very common mistake is to misinterpret left turn directions when a person is not facing North initially. For example, from West a left turn takes you South, not North. Another error is mixing up net displacement from the origin with relative displacement between two points. Always track each path systematically and then compute the difference between final positions when the question asks where one person is with respect to another.

Final Answer:
Hence, woman B is located 410 m East of A.