An auto-rickshaw picks up a passenger and first travels 6.5 km towards the North from the starting point.\nIt then turns left and rides for 1.5 km towards the West, turns again towards the South and rides 2.5 km, and finally turns left once more and rides 1.5 km.\nWhere is the auto now with respect to its starting position?

Introduction / Context:
This direction sense problem involves an auto-rickshaw moving along a set of straight segments with left turns, and we must determine the final position relative to the starting point. The path consists of movements towards North, West, South and finally East (due to the last left turn). By converting the description into horizontal and vertical displacements, we can compute how far and in which direction the auto ends up from its initial location.

Given Data / Assumptions:

• The auto starts from an initial point.
• It travels 6.5 km towards the North.
• It then turns left and rides 1.5 km, which from facing North means towards the West.
• Next, it turns towards the South and rides 2.5 km.
• Finally, it turns left again and rides 1.5 km.
• We assume all turns are 90 degree turns and all segments are along straight roads.
• Standard orientation is used: North is up, East is right, South is down and West is left.

Concept / Approach:
The best approach is to treat each segment as a vector on a coordinate grid. Vertical displacements (North and South) affect the y-coordinate, while horizontal displacements (East and West) affect the x-coordinate. Often, equal movements in opposite directions can cancel each other out. Tracking this step by step allows us to easily find the net position. In this problem, the Westward and Eastward movements are equal, so only the net North–South displacement remains.

Step-by-Step Solution:
Step 1: Place the starting point at coordinates (0, 0). Step 2: The auto first travels 6.5 km North to reach (0, 6.5). Step 3: From facing North, a left turn means facing West. Riding 1.5 km West brings it to (-1.5, 6.5). Step 4: It then turns towards the South and rides 2.5 km, reaching (-1.5, 4.0). The North–South displacement is now 4.0 km above the origin. Step 5: Finally, from facing South, a left turn makes it face East. Riding 1.5 km East brings it to (0, 4.0). Step 6: The final coordinates are (0, 4.0), while the starting point is (0, 0). This shows that the auto is 4 km to the North of the starting point and has no net East–West displacement.

Verification / Alternative check:
We can check by only considering net vertical and horizontal movements. Vertically, it goes 6.5 km North and then 2.5 km South, giving a net of 6.5 − 2.5 = 4 km North. Horizontally, it goes 1.5 km West and later 1.5 km East, so these cancel each other out completely, leaving zero net East–West movement. Therefore, the auto must be directly North of the starting point at a distance of 4 km, which confirms our coordinate-based result.

Why Other Options Are Wrong:
9 km North and 9 km South are impossible because the total distance travelled is not even that large in a single direction, and we have both North and South movements that partially cancel. 4 km South is incorrect because the net vertical movement is clearly towards the North, not the South. 1.5 km West would imply that Westward motion is not cancelled by Eastward motion, but the path includes equal West and East segments of 1.5 km each, resulting in zero horizontal displacement.

Common Pitfalls:
Many candidates lose track of direction when multiple left turns are involved, or they forget that some segments undo earlier movements. Another common mistake is to try to add all distances numerically without considering direction, which is not meaningful for displacement questions. Drawing a diagram with the distances labelled or maintaining a simple table of cumulative North–South and East–West components is an effective way to avoid confusion and arrive at the correct answer quickly.

Final Answer:
The auto is now located 4 km North of its starting position.