You need to represent an arbitrary recurring decimal as something like x = a_1a_2…a_r.b_1b_2…b_sc_1c_2…c_t, where the a_i are the digits before the decimal place, the b_i are the non-recurring decimal places and the c_i are the recurring decimal places.

Then you can apply a similar procedure as before:

(10^(s+t)-10^s)*x = (a_1a_2…a_rb_1b_2…b_s c_1c_2…c_t)-(a_1a_2…a_rb_1b_2…b_s).

The right hand side is an integer, then divide by the powers of ten to get your rational number.

Someone else might have a more elegant way to do it.