Question

Show that 0.12 bar +0.1 bar = 0.23

Answer 1

$x = 0.12 \: bar \\ 10x = 1.2121212... \\ 100x = 12.121212... \\ 100x - x = 12.1212... - 0.121212... \\ 99x = 12 \\ x = \frac{12}{99} = \frac{4}{33}$
$x = 0.1111111.. \\ 10x = 1.1111111... \\ 10x - x = 1.11111... - 0.111111... \\ 9x = 1 \\ x = \frac{1}{9} \\ \\ now \: add \: both \\ \frac{4}{33} + \frac{1}{9} \\ = \frac{12 + 11}{99} \\ = \frac{23}{99} \\ on \: dividing \: it \: we \: get \: \\ 0.232323... \\ = 0.23 \: bar$