<s>[INST] Are there any examples of $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$? I am writing a short discussion of Farey numbers and was wondering if there are any examples of when the mediant function is ever actually equal to the sum of the two fractions in the usual sense? (Not to produce Farey numbers, I just thought it might be an amusing way to introduce Farey addition). 
Explicitly: Are there any examples of fractions $\frac{a}{b}$ and $\frac{c}{d}$ where 
$$\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$$
for positive integers? 
Currently I have found the example 
$$\frac{1}{1}+\frac{1}{i} =\frac{1+1}{1+i}$$
if we remove the integers requirement but it would be nice to find a case that didn't involve complex numbers though! 
 [/INST] Seems like straight forward algebra manipulation
$\frac ab + \frac cd = \frac {a+c}{b+d}$
$(ad+bc)(b+d) = bd(a+c)$
$abd + b^2c + ad^2+bcd = abd +bcd$
$b^2c +ad^2 = 0$
$b^2c = -ad^2$
Wolog $b,d$ are positive.  Wolog $c$ is positive an $a$ negative (I assume you don't want $a=c =0$.)
If we let $c = c'(M^2)$ where $c'$ is square free then $a =-c'N^2$ for some $n$.  We can shuffle the square factors of $b,c,M,N$ around anyway we want.
$c =12$ and $a=-75$ and $b=5$ and $d=2$ would be a solution.  
$\frac {-75}5 + \frac {12}{2} =\frac {-75+12}{5+2}=-9$.
</s>