When sum of fraction is the same as the fraction made by the sum of numerators and sum of denominators

Question

My students naturally want to add fractions adding numerators and denominators. I say many times it does not rule like this, but is there a (small) set of integers which this rule work? That is

Where is $$\{ (x, y, z, t ) \in \mathbb{Z}^4 \, | \, \frac{x}{y} + \frac{z}{t} = \frac{x+z}{y+t}\}$$

Answer 1

This can be written as,

$$\frac{xt+yz}{yt} = \frac{x+z}{y+t} \implies xty + y^2z + yzt +xt^2 = ytx + yzt$$

$$y^2z+xt^2 = 0$$

Answer 2

I got the equation $$y^2z+xt^2=0$$

Answer 3

Constructive solution: given a fraction $\frac xy$, choose a non-zero number $k$. Then define $z=-k^2x$ and $t=ky$, thus $$\frac zt =\frac{-k^2x}{ky}.$$ Then either way their sum will be $(1-k)\frac xy$.

An example with small numbers would be $x=1$, $y=2$ and $k=2$ which gives $$\frac12+\frac{-4}{4}=-\frac12.$$

NB: You may even construct more "exotic" ones: $x=1/2$, $y=2/3$ and $k=2$ gives $$\frac{\frac12}{\frac23}+\frac{-2}{\frac43}=-\frac34$$ either way. You should also give $$\frac12+\frac{-2}{2\sqrt2}$$ a try.