It depends on what a ratio is.
By that, I mean it depends on what the mathematical definition of the word ratio is in your case.
Let me explain. Remember, we are talking mathematics here. And mathematics deals with statements about mathematical objects, and mathematical objects have strict definitions.
For example, we can talk about a fraction $\frac{a}{b}$, because the expression "$\frac{a}{b}$" has a definition that we all agree on. And we all know that the definition does not cover the case when $b=0$, which means that, by definition, the fraction $\frac{a}0$ does not exist.
Your question is about when two ratios are equivalent and when they are not. Before you ask this question mathematically, you need to determine two things:
- What a ratio is.
- What "this ratio and that ratio are equivalent" means.
Now, point 1 is easy. A ratio is an expression of the type $a:b$, where $a$ and $b$ are two real numbers.
How about point two? For point two, we must determine a rigorous definition of when $a:b$ and $c:d$ are equivalent. Formally, this means defining an equivalence relation on the set of all possible ratios.
The typical definition is that $a:b$ is equivalent to $c:d$ if $\frac{a}{b}=\frac{c}{d}$.
This definition works well when none of the numbers is zero, however, as you corretly pointed out, it fails when $b=d=0$. In that case, the definition, as usually written out, technically says that the two ratios are not equivalent.
What's weirder, the definition claims that $0:a$ is equivalend to $0:c$, but $a:0$ is not equivalent to $c:0$.
The conclusion you should draw from the above is that the typically stated definition of ratio equivalence is, in a sense, "not good". It works fine for nonzero cases, but for zero cases, it returns strange results. Note that the definition is not, mathematically speaking, incorrect (mathematical definitions cannot be incorrect), but it is not useful. It does not model the concept of ratio that we want it to model.
So, a better definition of when two ratios are equivalent is needed. The best (also pointed out by @GregMartin in his answer is to say that
- $0:0$ is not a ratio
- $a:b$ is equivalent to $c:d$ if there exists $\lambda\in\mathbb R$ such that $c=\lambda a$ and $d=\lambda b$.
You can easily see that using this definition, $0:4$ is equivalent to $0:2$.
Best Answer
Let $$\frac ab=\frac cd=\frac ef=k$$
$\implies Xa+Yc+Ze=k(Xb+Yd+Zd)$