How should one interpret the term x:2a?

Question

I was asked about the right interpretation of $x:2a$. There are two ways to interpret this term $$x:2a=x:(2a)=\frac{x}{2a}$$ and $$x:2a=(x:2)a=\frac{x}{2}\cdot a$$ I am not aware of a convention in mathematics how such a term should be read. So my answer is, that expressions like the one above should be avoided (because they cannot be interpreted uniquely). Am I right with my answer?

Answer 1

The $:$ symbol is not so much a mathematical operator as it is a convenient form of notation. Thus you would never see a symbolic equation using $:$ in place of division, but you might see a ratio written verbally with it (as in your question). Furthermore, the $:$ takes all terms on the left and compares them to all terms on the right. Thus $a+b-c:efg$ is another way of expressing $\frac{a+b-c}{efg}$. Note, however, it is not "$=$" to this expression, since $:$ is not a mathematical operator in that sense (unless you define it as such).

Simply put, there is only one way to interpret the $:$ symbol, because it has either been defined as an operator and thus has a unique interpretation, or it is not a mathematical operator and hence is not "part of an expression," but rather means "the ratio between all terms on the left to all terms on the right."