In a certain symbolic notation for directions, A x B means A is to the south of B, A + B means A is to the north of B, A % B means A is to the east of B and A - B means A is to the west of B. Using this notation, in the expression P % Q + R - S, in which direction is S located with respect to Q?

Introduction / Context:
This question tests the ability to interpret symbolic relations that encode positions in terms of directions. Instead of directly stating that one point is north, south, east, or west of another, symbols like x, +, %, and - are used. The target is to parse a compound expression involving P, Q, R, and S and then determine the relative direction of S with respect to Q. This type of coding is common in reasoning tests.

Given Data / Assumptions:

• A x B means A is to the south of B.
• A + B means A is to the north of B.
• A % B means A is to the east of B.
• A - B means A is to the west of B.
• The given expression is P % Q + R - S.
• We interpret each pair in the expression separately, maintaining relative positions consistently in a coordinate plane.

Concept / Approach:
The expression P % Q + R - S can be broken into three relationships: P % Q, Q + R, and R - S. From these, we can deduce the positions of P, R, and S relative to Q, treating Q as a reference. Using coordinate style reasoning, we can assign Q any convenient position, for example the origin, and then place the other points accordingly. The final coordinate of S relative to Q will reveal the direction of S with respect to Q.

Step-by-Step Solution:
Step 1: Interpret P % Q. Since A % B means A is to the east of B, P % Q means P is east of Q.Step 2: Interpret Q + R. Since A + B means A is to the north of B, Q + R means Q is north of R, so R is south of Q.Step 3: Interpret R - S. Since A - B means A is to the west of B, R - S means R is west of S, so S is east of R.Step 4: Place Q at the coordinate (0, 0) on a grid, with east as positive x and north as positive y.Step 5: Since R is south of Q, place R at (0, −1). The exact distance is not important, only relative direction matters.Step 6: Since S is east of R, place S at (1, −1).Step 7: The displacement from Q at (0, 0) to S at (1, −1) involves moving one step east and one step south, which is the south east direction.

Verification / Alternative check:
We can also verify without coordinates by reasoning verbally. If R is south of Q and S is east of R, then S is south and east from Q when both relationships are combined. Any consistent picture of Q at the centre, R directly below Q, and S further to the right of R will show S lying diagonally down and to the right of Q. This is the south east direction. Because the expression does not constrain any other orientation, the south east inference is robust.

Why Other Options Are Wrong:
The north west direction would require S to be both north and west of Q, which contradicts the established facts that R is south of Q and S is east of R. Pure east or pure west would require no vertical displacement relative to Q, whereas here S is explicitly south of Q. South west would require the point to be west of Q, which is not consistent with S being east of R and R being directly south of Q. Therefore, only south east matches the combined relational data.

Common Pitfalls:
Many learners misread the compound expression and try to apply all operations from left to right without breaking them into pairwise relations. Another common error is to place R north of Q instead of south when interpreting Q + R. Using a simple coordinate diagram and writing down each relationship helps prevent such mistakes. It is also useful to remember that the first symbol in A x B describes the position of A relative to B and not the other way around.

Final Answer:
In the configuration described by P % Q + R - S, point S lies in the South-East direction with respect to point Q.