X and Y start from the same point.\nFirst, X walks 40 m towards the North, then turns to his left and walks 80 m towards the West, and finally turns to his right and walks 50 m towards the North.\nAt the same time, Y walks 90 m straight towards the North from the common starting point.\nWhere is Y now with respect to the final position of X?

Introduction / Context:
This direction sense question involves two people, X and Y, starting from the same point but taking different paths. We are asked to compare their final positions. Instead of simply finding where each one ends up relative to the origin, we must determine where Y is located relative to X. Such problems are very common in aptitude tests and require careful tracking of both paths and then comparing the coordinates or relative positions at the end.

Given Data / Assumptions:

• X and Y start from the same initial point.
• X walks 40 m towards the North.
• Then X turns left and walks 80 m, which means he goes towards the West.
• Next, X turns right and walks 50 m, which again takes him towards the North.
• Meanwhile, Y walks 90 m straight towards the North from the starting point.
• All turns are 90 degree turns, and paths are straight.
• We use the usual orientation: North is up, East is right, South is down and West is left.

Concept / Approach:
It is efficient to place the starting point at the origin of a coordinate system and track the movements of X and Y separately. Vertical movements (North–South) change the y-coordinate, while horizontal movements (East–West) change the x-coordinate. Once we know the final coordinates of both X and Y, we can compare them to find the relative displacement of Y with respect to X. The sign of the x-coordinate difference will immediately show whether Y is to the East or West of X and by how many metres.

Step-by-Step Solution:
Step 1: Let the starting point be (0, 0). Step 2: Track X. X walks 40 m North to reach (0, 40). Step 3: From facing North, a left turn takes X towards the West. He walks 80 m West to reach (-80, 40). Step 4: From facing West, a right turn makes him face North again. He walks 50 m North to reach (-80, 90). So X's final coordinates are (-80, 90). Step 5: Track Y. Y walks 90 m North in a straight line from the starting point to reach (0, 90). So Y's final coordinates are (0, 90). Step 6: To find where Y is relative to X, compare coordinates: X is at (-80, 90) and Y is at (0, 90). The y-coordinates are the same, so they are horizontally aligned. Step 7: The x-coordinate of Y (0) is 80 units greater than the x-coordinate of X (-80). This means Y is 80 m to the East of X.

Verification / Alternative check:
We can also interpret the movements qualitatively. From the start, both X and Y eventually reach the same North–South level of 90 m because X moves 40 m North, then an additional 50 m North, and Y moves directly 90 m North. However, X also moves 80 m West, while Y stays on the original vertical line. Therefore, Y must be exactly 80 m to the East of X along a horizontal line. This matches the coordinate method and confirms that the separation between them is purely East–West, with Y lying to the East.

Why Other Options Are Wrong:
30 m East or West is incorrect because the actual horizontal separation between their x-coordinates is 80 m, not 30 m. Saying Y is 80 m to the West of X reverses their relative positions because X has moved West while Y has not, so Y cannot be further West than X. Claiming Y is 10 m to the South of X is wrong because both have the same North–South coordinate (90 m), so there is no vertical difference at all; they are on the same horizontal line.

Common Pitfalls:
A typical mistake is to compare each person's path to the origin but not to each other, which can lead to incorrect inferences about relative locations. Another common error is misinterpreting left and right turns, especially when there are multiple turns for the same person. Drawing a simple diagram and labelling the positions after each move or using coordinates is the best strategy for avoiding confusion and quickly deducing the relative position.

Final Answer:
Y is finally located 80 m to the East of X.