A question about neighboring fractions. I have purchased I.M. Gelfand's Algebra for my soon-to-be high school student son, but I am embarrassed to admit that I am unable to answer seemingly simple questions myself.
For example, this one:

Problem 42. Fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are called neighbor fractions if their difference $\dfrac{ad - bc}{bd}$ has numerator $\pm1$, that is, $ad - bc = \pm 1$.
Prove that
    (a.) in this case neither fraction can be simplified (that is, neither has any common factors in numerator and denominator);
    (b.) if $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are neighbor fractions then $\dfrac{a + b}{c + d}$ is between them and is a niehgbor fraction for both $\dfrac{a}{b}$ and $\dfrac{c}{d}$; moreover, ...

Here is the snapshot from the book online (click on Look Inside on the Amazon page):

So, (a) is simple, but I have no idea how to prove (b). It just does not seem right to me. Embarrassing. Any help is appreciated.

There appears to be a slight misunderstanding in the above discussion of betweenness in the Gelfand exercise.
The fact that $a+c\over b+d$ is between $a\over b$ and $c\over d$ does not depend on the two fractions being neighbor fractions.  It is true for any (positive) fractions (as stated by Gelfand himself on the previous page) and is easily proven algebraically without assuming the fractions are neighbors (a stipulation which only complicates the proof).
An outline of such an algebraic proof would be:
If ${a\over b}\lt {c\over d}$, then $ad\lt bc$.
Add $ab$ to both sides, and it follows that ${a\over b}\lt{a+c\over b+d}$.  The other half of the betweenness follows similarly.
This betweenness property also follows immediately from a logical view of averaging.  If a group of $b$ people has $a$ apples, and a group of $d$ people has $c$ apples, merging the groups will give each person in the new merged group an average number of apples which benefits the people in the "poorer" group and hurts the people in the "richer" group. [Note that Gelfand makes this very argument on the previous page, to explain the betweenness property.]
So in Gelfand's part (b), the fact of betweenness is true always, while the fact that the resulting fraction is a neighbor to both original fractions depends, of course, from the assumption that the original fractions were neighboring.
-Jonathan
