If on a map of scale 1: 500000 the distance between two settlements is 3 cm, then on the ground it will correspond.

At a scale of 1: 500000, a gap of 1 cm on the map corresponds to a gap of 500,000 cm on the ground.

1 m contains 100 cm and 1 km contains 1000 m, which means that 500,000 cm = 5000 m = 5 km, that is, an interval of 1 cm on the map corresponds to an interval of 5 km on the map.

Therefore, multiplying 5 km by 3, we find that the distance of 3 cm on the map corresponds to a distance of 15 km on the ground.

Answer: 15 km