Subtract: $0.\overline{52} - 0.4\overline{6}$

(N/A) Step $1$: Convert $0.\overline{52}$ to a fraction.
Let $x = 0.525252...$ $(i)$
$100x = 52.525252...$ (ii)
Subtracting $(i)$ from (ii): $99x = 52$,so $x = \frac{52}{99}$.
Step $2$: Convert $0.4\overline{6}$ to a fraction.
Let $y = 0.4666...$ (iii)
$10y = 4.666...$ (iv)
$100y = 46.666...$ $(v)$
Subtracting (iv) from $(v)$: $90y = 42$,so $y = \frac{42}{90} = \frac{7}{15}$.
Step $3$: Subtract the fractions.
$\frac{52}{99} - \frac{7}{15} = \frac{52 \times 5 - 7 \times 33}{495} = \frac{260 - 231}{495} = \frac{29}{495}$.
Step $4$: Convert to decimal.
$\frac{29}{495} = 0.0585858... = 0.0\overline{58}$.