Approximating $\frac{x-y}{w-z}$ in terms of $\frac{x}{w}$ and $\frac{y}{z}$

Question

I have the expression $\frac{x-y}{w-z}$ and I would like to approximate it in two fractions, $x/w$ and $y/z$. Ideally a linear combination of $x/w$ and $y/z$ and their powers. If that is not possible, a non-linear expression will also do.

Answer 1

This is not possible. We can multiply numerator and denominator of $\frac xw$ by $k$ without changing the value of the fraction. As $k\to\infty$, $\frac{kx-y}{kw-z}\to\frac xw$, but as $k\to0$, $\frac{kx-y}{kw-z}\to\frac yz$. That is, $\frac{kx-y}{kw-z}\to\frac xw$ can't be approximated as a fuction of $\frac wx$ and $\frac yz$.

Answer 2

I doubt that you can. Suppose $y = z = 1$. Then you want to approximate $$ A = \frac{x-1}{w-1} $$ using $B = x/w$ and the number $1$. But $A$ can be any value between $1$ and $B$.