http://domingogomez.web.officelive.com/rm.aspx

The first paragraph of your write-up claims the
"well known mathematician Cauchy proved" what you
call Mr[V], "the Rational Mean of all elements of":

V = {a_1/b_1,...,a_n/b_n}

where all a_i's,b_i's are real and all b_i's have
the same sign:

Mr[V] = (SUM a_i)/(SUM b_i)

"always produces a mean value between a_1/b_1 and
a_n/b_n":

a_1/b_1 =< Mr[V] =< a_n/b_n

However this cannot be true (without imposing some
further conditions on elements in V or their ordering)
because it implies a_1/b_1 =< a_n/b_n.

You are absolutely right, but the main point is that
the Rational Mean is always  a mean value between a_1/b_1
and a_n/b_n, no matter if  (a_1/b_1)  >= (a_n/b_n)
or (a_1/b_1)  <= (a_n/b_n).
I am fixing that bug which is the result of transferring my old webpages to this new officelive website.

Please consider the set V = {1/2, 3/1, 1/3},
all real and all denominators positive.  The
definition you gave is that the Rational Mean
is Mr[V] = 5/6.  This doesn't fall between 1/2
and 1/3, even if these numbers be transposed.

Perhaps what you want is to impose ascending
order on the elements of V.