  Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan
continued fraction. Two reciprocity formulas each are also associated with the
Ramanujan--G\"ollnitz--Gordon continued fraction and a level-13 analog of the
Roger-Ramanujan continued fraction. We show that all eight reciprocity formulas
are related to a pair of quadratic equations. The solution to the first
equation generalizes the golden ratio and is used to set the value of a
coefficient in the second equation; and the solution to the second equation
gives a pair of values for a continued fraction. We relate the coefficients of
the quadratic equations to important formulas obtained by Ramanujan, examine
the pattern of the relation between a continued fraction and its parameters,
and use the reciprocity formulas to obtain close approximations for all values
of the continued fraction. We highlight patterns in the expressions for certain
explicit values of the Rogers-Ramanujan continued fraction by expressing them
in terms of the golden ratio. We extend the results to reciprocity formulas for
Ramanujan's cubic continued fraction and the Ramanujan-Selberg continued
fraction.
