If ratios like 4:0 and 2:0 are defined, then how can we determine if they are equivalent? According to answer in this post ratios with zero on either side of them are defined: Are ratios with zero defined?
But it raises another problem. How can we know if ratios 4:0 and 2:0 are equivalent?
Normally we claim that ratios a:b and c:d are equivalent if fractions a/b and c/d are equal.
It works when a=0 and c=0, but breaks down when b=0 and d=0 due to division by zero.
So what do we do? Intuitively it seems that all a:0 and c:0 must be equivalent, but we lack formal criterion to back up this intuition.
 A: Really the mathematical object that models ratios, in this sense, most closely is real projective space. The $n$-dimensional real projective space is defined to be the set of $(n+1)$-tuples of real numbers, other than $(0,\dots,0)$, under the equivalence relation that $(a_0,\dots,a_n) \sim (b_0,\dots,b_n)$ if and only if there exists a (nonzero) real number $\lambda$ such that $a_j=\lambda b_j$ for all $0\le j\le n$. This is the space of lines through the origin in $\Bbb R^{n+1}$.
For example, $1$-dimensional real projective space, called the real projective line, is the set of all (equivalence classes of) ordered pairs $(x,y)\ne(0,0)$ under the equivalence relation $(x,y)\sim(\lambda x,\lambda y)$. There is one equivalence class for each real number $m$ (the slope of the line), namely all points of the form $(\lambda,m\lambda)$; there is an additional equivalence class (the "line of infinite slope") of all points of the form $(0,\lambda)$. (In both cases $\lambda\ne0$.) This is how we want ratios to behave: the ratio $10:2$ is the same as the ratio $5:1$.
A: 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 $0:2$.
