  After introducing the definitions of positive, negative and companion rules,
from a given pair of companion rules we construct a new rule with higher degree
of precision The scheme is generalized giving rise to a transformation which we
call the mean rule. We show that the mean rule is the best approximation, in
the sense of least-squares, obtained from a linear combination of two rules of
the same degree of precision. Finally, we show that a rule of degree 2k+1 can
be constructed as linear combination of k+1 rules of degree one and rational
pseudorandom nodes. Several worked examples are presented.
