  An elementary but useful fact is that the numerator of the difference of two
consecutive Farey fractions is equal to one. For triples of consecutive
fractions the numerators of the differences are well understood and have
applications to several interesting problems. In this paper we investigate
numerators of differences of fractions which are farther apart. We establish
algebraic identities between such differences which then allow us to calculate
their average values by using properties of a measure preserving transformation
of the Farey triangle.
