A steamer goes downstream from one port to another in 9 hours.

Let the speed of the steamer in still water be $\mathrm{x} \mathrm{km} / \mathrm{h} .$

Speed (downstream) $=(x+1) \mathrm{km} / \mathrm{h}$

Speed (upstream) $=(x-1) \mathrm{km} / \mathrm{h}$

Distance covered in 9 hours while going downstream $=9(x+1) \mathrm{km}$

Distance covered in 10 hours while going upstream $=10(x-1) \mathrm{km}$

But both of these distances will be same.

$9(x+1)=10(x-1)$

$\Rightarrow 9 x+9=10 x-10$

$\Rightarrow 9+10=10 x-9 x$

$\Rightarrow 19=x$

$\Rightarrow x=19$

Therefore, the speed of the steamer in still water is $19 \mathrm{~km} / \mathrm{h} .$

Distance between the ports $=9(x+1)=9(19+1)=9 \times 20=180 \mathrm{~km}$