Village Q is to the North of village P. Village R is to the East of village Q.…

Direction-sense problems are solved by plotting each location on a common grid: fix one point as the origin, then move along the axes for every relative direction given (North/South = vertical, East/West = horizontal). The direction between any two plotted points is then read off from the sign of their horizontal and vertical offset — both offsets negative means the second point lies to the south-west of the first, and so on for the other three combinations.

1. Place village P at the origin, (0, 0).

2. Q is North of P, so plot Q above P on the same vertical line: Q = (0, a) for some a > 0.

3. R is East of Q, so plot R to the right of Q on the same horizontal line: R = (b, a) for some b > 0 — R now sits north of and to the right of P.

4. S is to the left of P, so plot S to the left of P on P's horizontal line: S = (−c, 0) for some c > 0.

Comparing S to R: the horizontal offset is (−c) − b, which is negative, so S lies to the west of R; the vertical offset is 0 − a, which is also negative, so S lies to the south of R. Both offsets negative together give the south-west direction.

The diagram below plots all four villages on the same grid: R sits in the upper-right, while S sits in the lower-left — directly confirming the diagonal south-west relationship found algebraically.

So village S is to the South-West of village R.