Infinite Decimals

$\color{#007fff}{(0.111\dots \quad \div \quad 0.222\dots)} \quad \times \quad\color{#007fff}{(0.333\dots \quad \div \quad 0.444\dots)} \quad \times \quad\color{#007fff}{(0.555\dots \quad \div \quad 0.666\dots)} \quad \times \quad\color{#007fff}{(0.777\dots \quad \div \quad 0.888\dots)} \quad \times \quad0.999$ $=\color{#007fff}{\dfrac{1}{2}}\times \color{#007fff}{\dfrac{3}{4}}\times \color{#007fff}{\dfrac{5}{6}}\times \color{#007fff}{\dfrac{7}{8}}\times 0.999$ $=\dfrac{35\times 0.999}{128}=\color{#0C6AC7}{\boxed{\color{#D61F06}{0.273}}}$ $~~~~~~\small\color{#456461}{{\text{(to 3 decimal places)}}}$

$\Rightarrow x=0.111...$ .... $(1)$
$\Rightarrow 10x=1.111...$ .... $(2)$

Subtracting $(1)$ from $(2)$ .

$\Rightarrow 9x=1$
$\Rightarrow x=\color{#BA33D6}{\frac{1}{9}}$

Similarly,

$\Rightarrow 0.111...= \color{#BA33D6}{\frac{1}{9}}$

$\Rightarrow 0.222...=\color{#302B94}{\frac{2}{9}}$

$\Rightarrow 0.333...=\color{#3D99F6}{\frac{3}{9}}$

$\Rightarrow 0.444...=\color{#20A900}{\frac{4}{9}}$

$\Rightarrow 0.555...=\color{#624F41}{\frac{5}{9}}$

$\Rightarrow 0.666...=\color{#EC7300}{\frac{6}{9}}$

$\Rightarrow 0.777...=\color{#D61F06}{\frac{7}{9}}$

$\Rightarrow 0.888...=\color{#CEBB00}{\frac{8}{9}}$

$\Rightarrow 0.999...=\color{magenta}{\frac{9}{9}}$

Now,

$\Rightarrow (\color{#BA33D6}{\frac{1}{9}}×\color{#302B94}{\frac{9}{2}})×(\color{#3D99F6}{\frac{3}{9}}×\color{#20A900}{\frac{9}{4}})×(\color{#624F41}{\frac{5}{9}}×\color{#EC7300}{\frac{9}{6}})×(\color{#D61F06}{\frac{7}{9}}×\color{#CEBB00}{\frac{9}{8}})×\color{magenta}{\frac{9}{9}}$

$\Rightarrow \color{grey}{\frac{3×5×7}{2×4×6×8}}=\color{grey}{\frac{35}{128}}\approx\color{#D61F06}{\boxed{\color{#3D99F6}{0.273}}}$