A man wants to cover a distance of $1\text{km}$ in a river. It takes him $12\text{minutes}$ to cover this distance in still water, but it takes $13\text{minutes}$ in the flowing river. Find the speed of the flowing water of the river.

Let the speed of the man in still water be $u$ and the speed of the river stream be $v$.
**Step 1: Find the speed of the man in still water ($u$)**
Given, distance = $1\text{km}$, time = $12\text{minutes}=\frac{12}{60}\text{hours}=\frac{1}{5}\text{hours}$.

$$u=\frac{\text{Distance}}{\text{Time}}=\frac{1}{\frac{1}{5}}=5\text{km/h}$$

Step 2: Formulate for the flowing river
Since it takes more time ($13\text{minutes}$) to cover the same distance in the flowing river, the man must be swimming against the flow of the river (upstream).
Time upstream = $13\text{minutes}=\frac{13}{60}\text{hours}$.

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

**Step 3: Solve for $v$ **
Substitute $u=5\text{km/h}$ into the upstream speed equation:

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

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

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

Therefore, the speed of the flowing water is $\frac{5}{13}\text{km/h}$.