A tube $A$ of radius $r$ is connected to two other tubes $B$ and $C$ with the help of a junction valve. The tubes $B$ and $C$ have radii $\frac{r}{2}$ and $\frac{r}{3}$ and the flow velocity in each is $v$ and $3v$ respectively. If the flow velocity in tube $A$ is $\frac{n}{m}v$, where $n$ and $m$ are integers, then what is the value of $n+m$?