$0 < \frac{a}{b}, \frac{c}{d} < 1$, when do we have $\frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d}$?

Question

Given natural numbers $a,b,c,d$, let $a,b$ be coprime with $b>a$ and let $c,d$ be coprime with $d>c$. Define a function $f:\mathbb{Q}^2 \to \mathbb{Q}$ as $$f\left(\frac xy, \frac zw\right)= \frac{x+z}{y+w}$$ Where $x,y$ are coprime integers, as are $y,w$. I have two questions:

1. Is $f$ in literature? I would like some references for further study.
2. Is it ever the case that $$f\left(\frac ab , \frac cd\right) = \frac ab + \frac cd$$ ? I have tried lots of numerical examples but can’t find any such $a,b,c,d$.

Answer 1

I don't know if this function has any name in the literature, but to your second question, the answer is yes, but only trivially.

To see this, suppose \begin{align*} \frac{a+c}{b+d} =& \frac{a}{b} + \frac{c}{d} \\ \frac{a+c}{b+d} =& \frac{ad+bc}{bd} \\ (a+c)bd =& (b+d)(ad+bc) \\ abd+cbd =& abd + ad^2 +b^2c + cbd \\ 0 =& ad^2 + b^2c \end{align*} The right hand side is clearly positive. Hence, the only solution is $a=c=0$.

Answer 2

the new fraction is called the "mediant" and comes up naturally in contiued fractions. When the two original fractions are distinct, the mediant lies strictly between them.

Given everything is positive integers, the same result occurs for any "partial quotient" $n,$ say, as $$ \frac{x+nz}{y+nw} $$ is also strictly between the two original fractions. Same for $$ \frac{nx+z}{ny+w} $$