A compass is incorrectly aligned. For the true direction of west, the compass needle is showing south west. Assuming the same misalignment for all directions, what direction will the compass show when the person is actually facing true north?

Introduction / Context:
This question is about an incorrectly aligned compass that shows wrong directions. We are told how the compass reading differs from the true direction for west and are asked what it would show for true north. The problem deals with the concept of a fixed angular error in the compass reading and requires applying the same error to another cardinal direction.

Given Data / Assumptions:

• For the true direction west, the compass shows south west.
• South west is 45 degrees away from west on a standard compass.
• The misalignment is assumed to be constant for all directions.
• We must determine what the compass will indicate when the user is actually facing true north.
• We use the standard positions: north at 0 degrees, east at 90 degrees, south at 180 degrees, west at 270 degrees, south west at 225 degrees, and north west at 315 degrees.

Concept / Approach:
The key is to understand that the compass reading is rotated by a fixed angle relative to the true direction. Since true west (270 degrees) is shown as south west (225 degrees), the reading is 45 degrees less than the true direction. In other words, the compass subtracts 45 degrees from every true direction. To find what it shows for true north, we subtract the same 45 degrees from the north direction and then map the result back to a compass point, taking care to work modulo 360 degrees.

Step-by-Step Solution:
Step 1: Represent west as 270 degrees and south west as 225 degrees on the compass.Step 2: The difference between the compass reading and the true direction is 225 − 270 = −45 degrees, meaning the reading is rotated 45 degrees anticlockwise from the true direction.Step 3: Assume this error is the same for every direction. For a true direction of north, which is 0 degrees, the compass reading will be 0 − 45 degrees.Step 4: Subtracting 45 degrees from 0 degrees gives −45 degrees, which is equivalent to 315 degrees when we add 360 degrees to stay within a full circle.Step 5: An angle of 315 degrees corresponds to the north west direction on a compass.Step 6: Therefore, when facing true north, the misaligned compass will show north west.

Verification / Alternative check:
We can quickly recheck the logic by applying the same reasoning to another direction. For example, true south at 180 degrees would be shown as 180 − 45 = 135 degrees, which is south east. This is consistent with the idea that every reading is shifted 45 degrees anticlockwise. Returning to north, 0 degrees minus 45 degrees must wrap around the circle, landing at 315 degrees. This corresponds to north west and confirms that our initial calculation is correct.

Why Other Options Are Wrong:
East corresponds to 90 degrees and would be the reading if the compass added 90 degrees rather than subtracting 45 degrees, which contradicts the given misalignment. North east at 45 degrees would require a shift of only 45 degrees from north, but in the wrong direction for the described error. South east and south west lie in the opposite half of the compass and do not follow from applying the same anticlockwise shift that moves west to south west. Only north west matches the computed 315 degree reading.

Common Pitfalls:
Some learners mistakenly think that if west is shown as south west, then north must be shown as north east by moving in the same direction on the compass card, without considering angles. Another error is to add 45 degrees instead of subtracting it when moving from west to its erroneous reading. Being precise about angles and remembering to use a consistent sign for the error helps to avoid these mistakes.

Final Answer:
When the person is actually facing true north, the misaligned compass will show the direction as North-West.