This is basically how we were taught to convert decimal fractions with periodic decimal expansions to regular fractions in school. You can even do that without

This is basically how we were taught to convert decimal fractions with periodic decimal expansions to regular fractions in school. You can even do that without that lining up of zeroes and nines, just multiply by the period: if x = 0.090909..., then 100x = 09.090909... (we shift the decimal point by two positions), and since the stuff after the point is the same, after subtracting it cancels out: 100x - x = 9.0, and so 99x = 9, from which we obtain x = 1/11.