How do you convert $(12.0251)_6$ into fractions? How do you convert $(12.0251)_6$ (in base 6) into fractions?
I know how to convert a fraction into base $x$ by constantly multiplying the fraction by $x$ and simplifying, but I'm not sure how to go the other way?

You could use continued fractions.  
                       Cf     A   /  B
             12.0251           1  /  0      In the left, we produce a 
    1.0000  -12.0000    8      8  /  1      continued fraction from 
   -0.5420    0.0251   12     97  / 12      12.0521 and 1.0000, in base 6
    0.0140   -0.0140    1    105  / 13        There are Cf of the positive 
   -0.0111    0.0111    1    202  / 25      number, which gives the negative
    0.0025   -0.0054    2    511  / 63      number.
   -0.0013    0.0013    1    713  / 88        The right is Cf(r-1) + Cf(r-2)
              &c                            starting at 1 in A and 0,1 in B.

You stop when the error gets precise enough, or you get a zero in either column.  
