A ship sails out to an island at the rate of 15 km/h and sails back to the starting point at 10 km/h.

Let the distance from the starting point to the island be x km.
Speed of the ship sailing out to the island = 15 km/h
Speed of the ship sailing back to the starting point = 10 km/h
We know:

Time $=\frac{\text { Distance }}{\text { Speed }}$

Time taken by the ship to travel from the starting point to the island $=\frac{x}{15} \mathrm{~h}$

Time taken by the ship to travel from the island to the starting point $=\frac{x}{10} \mathrm{~h}$

Average speed $=\frac{\text { Total distance travelled }}{\text { Total time taken }}$

$=\frac{x+x}{\frac{x}{15}+\frac{x}{10}}$

$=\frac{2 x}{\frac{2 x+3 x}{30}}$

$=\frac{2 x}{\frac{5 x}{30}}$

$=\frac{60}{5}$

$=12 \mathrm{~km} / \mathrm{h}$

Therefore, the average speed of the ship in the whole journey was 12 km/h.