A man wishes to cover a distance of $1\text{km}$ in river water. In still water he takes 12 minutes to cover it, but in the flowing river he takes 13 minutes. The speed of the flowing water of the river is:

Let the speed of the man in still water be $v$ and the speed of the river current be $u$.
Given that the man covers $1\text{km}$ in still water in 12 minutes:

$$v=\frac{\text{Distance}}{\text{Time}}=\frac{1\text{km}}{\frac{12}{60}\text{h}}=5\text{km/h}$$

Since it takes more time (13 minutes) to cover the same distance in the flowing river, the man must be moving upstream (against the stream flow).
Upstream speed ($v-u$):

$$v-u=\frac{1\text{km}}{\frac{13}{60}\text{h}}=\frac{60}{13}\text{km/h}$$

Substitute the value of $v=5\text{km/h}$ into the equation:

$$5-u=\frac{60}{13}$$

$$u=5-\frac{60}{13}=\frac{65-60}{13}=\frac{5}{13}\text{km/h}$$