Explanation

(a)
$0.234 = 0.2343434\ldots$
$= 0.2 + 0.034 + 0.00034 + \ldots$
$= \frac{2}{10} + 34\left\lbrack \frac{1}{10^{3}} + \frac{1}{10^{5}} + ... \right\rbrack$
$= \frac{2}{10} + 34 \times \frac{1}{1000} \times \frac{100}{99}$
$= \frac{2}{10} + \frac{34}{990} = \frac{232}{990}$