Columnar Addition?

Relevant wiki: Converting Fractions into Decimals

$\Large\color{#D61F06}{\text{Method 1}}$

Let $y=\frac{123}{999}$ .

$\Rightarrow y+\dfrac{y}{10}=\dfrac{1}{10}+\dfrac{354}{x}$

$\dfrac{11y-1}{10}=\dfrac{354}{x}$

$x=\dfrac{10}{11y-1}×354$

$x=\dfrac{3540}{11×\frac{123}{99}-1}$

$x=\dfrac{3540×999}{354}$

$x=\boxed{9990}$

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$\Large\color{#D61F06}{\text{Method 2}}$

Converting (0.1354354...) into fraction.

Let $x=0.1354354...$

$\rightarrow 10x=1.354354...$

$\rightarrow 10000x=1354.354354...$

Subtracting these two.

$\rightarrow 9990x=1353$

$\Rightarrow 0.1354354... =\dfrac{1353}{9990}$

And we know that.

$\Rightarrow 0.123123...+0.0123123=0.1354354...$

$\Rightarrow \dfrac{123}{999}+\dfrac{123}{9990}=\dfrac{1353}{9990}$

$\Rightarrow \dfrac{1353}{9990}=\dfrac{1}{10}+\dfrac{354}{9990}=\dfrac{1}{10}+\dfrac{354}{x}$

$\therefore x=\boxed{9990}$