A traveller first goes 20 miles towards the north, then 25 miles towards the east. After that, he travels 40 miles by turning to his right, then 30 miles by turning to his left, then another 12 miles to his left and finally 20 miles towards the north. Approximately how far is he now from his original starting point and in which general direction is he located?

Introduction / Context:
This is a multi step direction sense problem where a person travels in several straight segments, turning right or left at different stages. The objective is to find the approximate straight line distance from the starting point and the overall direction of the final position. Questions like this simulate real life navigation and require systematic tracking of horizontal and vertical movements using simple geometry.

Given Data / Assumptions:

• The traveller starts at an initial point which we take as the origin.
• He moves 20 miles north, then 25 miles east.
• From facing east, a right turn takes him south, and he travels 40 miles.
• From facing south, a left turn takes him east, and he travels 30 miles.
• From facing east, a left turn takes him north, and he travels 12 miles.
• Finally he travels another 20 miles north in the same direction.

Concept / Approach:
We treat the movements along east west and north south axes and use coordinates to represent his position after each leg of the journey. Right and left turns are interpreted relative to the direction he is currently facing. Once all displacements are recorded, we add up all east west components and all north south components to get the net displacement vector. The straight line distance is then found using the Pythagorean theorem, and the direction is described by comparing the net east and north components.

Step-by-Step Solution:
Step 1: Place the starting point at (0, 0) with east as positive x and north as positive y.Step 2: After 20 miles north, his position is (0, 20).Step 3: After 25 miles east, his position is (25, 20), and he is facing east.Step 4: A right turn from east means he now faces south. Travelling 40 miles south takes him to (25, −20).Step 5: A left turn from south means he faces east. Travelling 30 miles east takes him to (55, −20).Step 6: A left turn from east means he faces north. Travelling 12 miles north takes him to (55, −8).Step 7: Finally, travelling another 20 miles north brings him to (55, 12).Step 8: The net displacement from the origin is therefore 55 miles east and 12 miles north.Step 9: The straight line distance d is given by d = √(55^2 + 12^2) = √(3025 + 144) = √3169, which is approximately 56 miles.

Verification / Alternative check:
We can roughly estimate the distance without exact squares by noting that 55^2 is 3025 and 12^2 is 144, so the sum is slightly above 3000. The square root of 3136 is 56, and 3169 is very close to this, so the distance is a little more than 56 miles. Since the net displacement has a large east component and a smaller north component, the direction lies between east and north. It is reasonable to describe this as towards the north east, or as slightly north of east, which justifies the approximate answer choice.

Why Other Options Are Wrong:
The option of approximately 40 miles towards the south west would require both net south and west components, which is opposite to the calculated 55 miles east and 12 miles north. The option of 30 miles towards the east underestimates the journey and ignores the significant northward displacement. The option of 25 miles towards the north is also too small and does not match the net east component. The alternative description of approximately 56 miles towards the east of north is close in magnitude but misstates the dominant direction, since the motion is mainly east with a slight north component, not mainly north.

Common Pitfalls:
Common mistakes include misinterpreting left and right turns, losing track of the current facing direction, and adding distances without considering direction signs. Some learners may incorrectly assume that the traveller returns close to the starting point and choose a small distance by guesswork. Drawing a simple diagram or coordinate table and breaking each leg into x and y components prevents these errors and makes the final Pythagorean calculation straightforward.

Final Answer:
The traveller ends up approximately 56 miles away from his starting point, located mainly to the east with some northward displacement, that is, approximately towards the north-east.