Assembing a Rational Number from Constituent Digit Vectors Relative to a Given Base

The Set-up

The given vectors are $u,v,w$, and the given base is $b$. In each case the subscripts on the coordinates of the digit vectors increase from left to right. For example, in base $10$ if $$u=\left[3,6\right],v=\left[5,4\right],\text{and}w=\left[1,2\right]\text{,}$$ then the indicated decimal expansion is $$36.5412121212\dots \text{,}$$ and its representation as a quotient of coprime integers is $$\frac{60293}{1650}\text{.}$$

Formulas:
$$\mathrm{result}=r+s+t\text{where}$$ $$\begin{array}{cccc}& \hfill r& \hfill =\hfill & \sum _{j=1}^k{u}_j{b}^{k-j}\hfill \\ & \hfill s& \hfill =\hfill & \sum _{j=1}^l{v}_j{b}^{-k}\hfill \\ & \hfill t1& \hfill =\hfill & \sum _{j=1}^m{w}_j{b}^{m-j}\hfill \\ & \hfill t2& \hfill =\hfill & \left(b^l\right)\left(b^m-1\right)\hfill \\ & \hfill t& \hfill =\hfill & t1⁄t2\hfill \end{array}$$ with $$k=\mathrm{length}\left(u\right)l=\mathrm{length}\left(v\right)m=\mathrm{length}\left(w\right)\text{.}$$

Exercise

Raise issues you find with the following code proposed for the task:

ratFromVecs := proc (u, v, w) local b, r, s, t, j, k, l, m, c1, c2, n;

u := [u[1],...,u[k]];

v := [v[1],...,v[l]];

w := [w[1],...,w[m]];

k := nops(u);

l := nops(v);

m := nops(w);

r := sum('u[j]*b^(k-j)', 'j' = (1 .. k));

s := Sum('v[j]*b^(-j)', 'j' = (1 .. l));

t1 := sum('w[j]*b^(m-j)', 'j' = (1 .. m));

t2 := b^l*(b^m-1);

t := t1/t2;

for j to k do n := r+s+t; end do;

n;

end;