A long time ago one of my professors gave me this question. He didn't know the answer and has since passed away.
For which $a,b,c,d \in \mathbb{Z}$ does $\frac{a}{b} + \frac{c}{d} = \frac{a+b}{c+d}$?
It's a cute problem, but I don't have a solution. Does anyone have any ideas?
What about if $a,b,c,d \in \mathbb{R}$?
Here is an answer that may help for a related equation: $$\frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}$$
We will keep careful track of which quantities may be zero. To start, we know that $b, d, b+d \neq 0$ because they are in the denominator.
Let's multiply both sides by $b\cdot d \cdot (b+d)$:
$$\begin{align*}ad(b+d) + bc(b+d) &= (a + c)bd\\ abd + ad^2 + b^2c + bcd &= abd + bcd\\ abd + ad^2 + b^2c + bcd - abd - bcd &= 0\\ \fbox{$ad^2 + b^2 c$ = $0$} \end{align*}$$
Here's an example assignment that satisfies this equation: $$\begin{align*}a &= 5\\b&=3\\c&=-20\\d&=6\\\end{align*} $$
For fun, here is a nomogram that allows you to find such collections of numbers:
You can draw any line passing through, say, the B and C axes, and any line passing through the A and D axes. If the two lines meet at the same point on the central vertical axis, then the corresponding points $(a, b, c, d)$ satisfy the equation.
In the picture, the dashed lines show an example. Here, we see that the equation is satisfied when $a = 4$, $b=3$, $c=-1$, and $d=1.5$.
Also, if you specify any four of the following five variables—a value for $a$, a value for $b$, a value for $c$, a value for $d$, a spot on the central axis— you can "solve" for the remaining point.