In triangle $PQR$, the length of side $QR$ is less than twice the length of side $PQ$ by $3\text{cm}$. The length of side $PR$ exceeds the length of side $PQ$ by $14\text{cm}$. If the perimeter is $55\text{cm}$, find the length of the smallest side of the triangle $PQR$.

Let the length of side $PQ$ be $x\text{cm}$.
According to the problem statements:

* Length of side $QR=2x-3$
* Length of side $PR=x+14$

The perimeter of triangle $PQR$ is the sum of its three sides:

$$\text{Perimeter}=PQ+QR+PR=55\text{cm}$$

$$x+(2x-3)+(x+14)=55$$

$$4x+11=55$$

$$4x=55-11=44$$

$$x=11\text{cm}$$

Now, find the lengths of all three sides:

* $PQ=x=11\text{cm}$
* $QR=2(11)-3=22-3=19\text{cm}$
* $PR=11+14=25\text{cm}$

Comparing the side lengths ($11\text{cm}$, $19\text{cm}$, and $25\text{cm}$), the length of the smallest side is $11\text{cm}$.