After their day's work two village women return home from the same starting point. The first woman walks 1.5 km towards the west and then turns right and walks 3 km to reach her house. The second woman walks 1 km towards the north, then turns east and walks 3.5 km, then turns left and walks 2 km to reach her house. Where is the first woman's house with respect to the second woman's house?

Introduction / Context:
This question describes the final positions of two women returning home from a common starting point after their day's work. We are asked to locate the first woman's house with respect to the second woman's house. The problem involves calculating coordinates for each house and then determining the relative horizontal distance and direction between them.

Given Data / Assumptions:
- Both women start from the same origin point. - First woman: walks 1.5 km west, then turns right from west (towards north) and walks 3 km to her house. - Second woman: walks 1 km north, then turns east and walks 3.5 km, then turns left from east (towards north) and walks 2 km to her house. - All movements are along straight segments with right angle turns.

Concept / Approach:
We place the starting point at coordinate (0, 0) and use a standard coordinate system: east as positive x, north as positive y. For each woman, we track the final coordinates by incrementing or decrementing x and y according to each described move. To find where the first woman's house is with respect to the second's, we compute the vector from the second house to the first house. The sign of this difference tells us the direction (east or west), and the magnitude tells us the distance.

Step-by-Step Solution:
1. First woman: start at (0, 0) and go 1.5 km west: position (-1.5, 0). 2. From facing west, a right turn means north. She walks 3 km north: first house at (-1.5, 3). 3. Second woman: start at (0, 0) and walk 1 km north: position (0, 1). 4. She turns east and walks 3.5 km: position (3.5, 1). 5. From facing east, a left turn means north. She walks 2 km north: second house at (3.5, 3). 6. To find the first house relative to the second, subtract coordinates: (-1.5, 3) minus (3.5, 3) = (-5, 0). 7. The result (-5, 0) means the first house is 5 km directly west of the second house.

Verification / Alternative check:
The vertical coordinates of both houses are equal (y = 3 km). This shows they lie on the same east west line. Horizontally, the first house is 1.5 km west of the origin, while the second house is 3.5 km east of the origin. The separation between them is 1.5 + 3.5 = 5 km. Since the first is on the west side and the second on the east side, the first must be located 5 km west of the second. A diagram with both paths drawn from the common starting point visually confirms this reasoning.

Why Other Options Are Wrong:
- 2 km west or 2 km east underestimate the horizontal gap and do not match the coordinate difference. - 5 km east reverses the correct direction; the first house is to the west, not east, of the second. - At the same location ignores the clear horizontal separation between the coordinates.

Common Pitfalls:
Candidates may misinterpret right and left turns relative to the current facing direction rather than the compass, leading to incorrect final coordinates. Another error is to compare each house with the origin instead of directly comparing the two houses. Being systematic in tracking coordinates and carefully interpreting right and left from the walker's perspective is essential to avoid such mistakes.

Final Answer:
The first woman's house is 5 km west of the second woman's house.