G.P

Let, $A=0.4\overline{23}=0.4+x$ , where $x=0.0\overline{23}$ .

Then, $10x=0. \overline{23}=\frac{23}{99}$ $\Rightarrow x=\frac{23}{990}$

And, $A=\frac{4}{10}+\frac{23}{990}=\boxed{\frac{419}{990}}$ Therefore, the answer is, $419+990=\color{#0C6AC7} {\boxed {1409}}$

$.4\ddot { 23 } = 0.4 23 23 23 23 23 23 ...$

\frac { 4 }{ 10 } +\frac { 23 }{ 1000 } \frac { 23 }{ 100000 } \frac { 23 }{ 10000000 } \frac { 23 }{ 1000000000 } \ \frac { 4 }{ 10 } +\frac { 23 }{ 1000 } (1+\frac { 1 }{ { 10 }^{ 2 } } +\frac { 1 }{ { 10 }^{ 4 } } +\quad ...\quad )\ \frac { 4 }{ 10 } +\frac { 23 }{ 1000 } (\frac { 1 }{ 1-\frac { 1 }{ { 10 }^{ 2 } } } )\quad \Longrightarrow \quad \frac { 4 }{ 10 } +\quad \frac { 23 }{ 1000 } *\frac { 100 }{ 99 } \ =\quad \frac { 419 }{ 990 } \ \therefore \quad 419\quad +\quad 990\quad =\boxed { 1409 }