Find the rational number whose decimal expansion is
Question:

Find the rational number whose decimal expansion is $0.4 \overline{23}$.

Solution:

Let the rational number $\mathrm{S}$ be $0.4 \overline{23}$.

$\because \mathrm{S}=0.4 \overline{23}=0.4+0.023+0.00023+0.0000023+\ldots \infty$

$\Rightarrow \mathrm{S}=0.4+0.023\left[1+10^{-2}+10^{-4}+\ldots \infty\right]$

Clearly, $\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-2}$.

$\therefore \mathrm{S}=0.4+0.023\left[\frac{1}{1-10^{-2}}\right]$

$\Rightarrow \mathrm{S}=0.4+\frac{2.3}{99}$

$\Rightarrow \mathrm{S}=\frac{419}{990}$

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