Two village women are returning home starting from the same point. The first woman walks 1 kilometre towards the south and then turns to her right and walks 6 kilometres to reach her house. In the meantime, the second woman walks 5 kilometres towards the west, then turns towards the north and walks 4 kilometres, and finally turns to her left and walks 1 kilometre to reach her house. Where is the second woman's house with respect to the first woman's house?

Introduction / Context:
This is a relative position problem in which two people start from the same point but follow different paths to reach their respective homes. We are asked to find where the second woman's house lies relative to the first woman's house. Such questions test whether the candidate can compute the final coordinates of each person and then determine the relative displacement between those two final points rather than between any of them and the origin.

Given Data / Assumptions:

• Both women start from the same common point.
• The first woman walks 1 kilometre south, then turns right and walks 6 kilometres to reach her house.
• The second woman walks 5 kilometres west, then 4 kilometres north, and finally 1 kilometre to her left from that direction to reach her house.
• All directions are along cardinal directions and turns are right angles.
• We must find the location of the second woman's house relative to the first woman's house.

Concept / Approach:
We adopt coordinate geometry. Let the starting point be (0, 0). Take east as positive x, north as positive y, west as negative x and south as negative y. We first compute the coordinates of the first woman's house and then of the second woman's house. The vector from the first house to the second house is found by subtracting their coordinates, which then gives the required distance and direction.

Step-by-Step Solution:
Step 1: The starting point is (0, 0). Step 2: For the first woman: walking 1 kilometre south moves her to (0, -1). Step 3: From facing south, a right turn means facing west. Walking 6 kilometres west moves her to (-6, -1). This is the position of the first house. Step 4: For the second woman: walking 5 kilometres west from (0, 0) moves her to (-5, 0). Step 5: From facing west, turning towards the north means she now faces north. Walking 4 kilometres north moves her to (-5, 4). Step 6: From facing north, a left turn means facing west. Walking 1 kilometre west moves her to (-6, 4). This is the position of the second house. Step 7: The first house is at (-6, -1) and the second house is at (-6, 4). Step 8: To find the second house relative to the first, we subtract: (-6, 4) minus (-6, -1) equals (0, 5). Step 9: A difference of (0, 5) means the second house lies 5 kilometres north of the first house.

Verification / Alternative check:
We can ignore the origin and make a simpler statement: both houses are on the same vertical line x equals minus 6, because each last horizontal move takes them to that line. The first woman's last position has y coordinate minus 1, while the second woman's final y coordinate is plus 4. The second house is therefore directly north of the first by 4 minus minus 1 equals 5 kilometres. This matches the coordinate subtraction result exactly.

Why Other Options Are Wrong:

• Option B, 3 kilometres North, underestimates the difference of 5 kilometres between the y coordinates of the two houses.
• Option C, 5 kilometres South, reverses the direction; the second house is north, not south, of the first house.
• Option D, 3 kilometres South, is wrong in both magnitude and direction.
• Option E, 1 kilometre East, suggests a horizontal separation that does not exist since both houses share the same x coordinate.

Common Pitfalls:
Students sometimes compute each house's position relative to the starting point correctly but then forget that the question asks for relative position between the houses, not between a house and the origin. Another common error is misinterpreting "left" from the north as east instead of west. Using coordinates and carefully subtracting them for relative positions is a robust strategy to avoid confusion.

Final Answer:
The second woman's house is located 5 kilometres North of the first woman's house.