$$ A+\frac{1}{B+\frac{1}{C-9}}=\frac{29}{5} $$, then the value of $$A+B+C$$, is

$$B+\frac{1}{C-9}=\frac{B\left(C-9\right)+1}{C-9}$$
$$\frac{1}{B+\frac{1}{C-9}}=\frac{C-9}{B\left(C-9\right)+1}$$.
Putting this value in the equation to get:
$$A+\frac{C-9}{B\left(C-9\right)+1}=\frac{A\cdot B\cdot\left(C-9\right)+A+\left(C-9\right)}{B\cdot\left(C-9\right)+1}=\frac{29}{5}$$.
This means that B(C-9) + 1 = 5 ==> B(C-9) = 4.
C- 9 can be 1 or 2 or 4.
and, $$A\cdot B\cdot\left(C-9\right)+A+\left(C-9\right)=29$$.
4*A + A + (C-9) = 29
5A + (C-9) = 29
C-9 has to be 4 which gives A = 5 and C = 13.
B*(C-9) = 4 which means B = 1.
A + B + C = 5 + 1 + 13 = 19.