What is $a: b: c$ if $a: b$ is $\frac ab$?

Question

I was teaching my younger brother the concept of ratios and proportions. I told him that a ratio can be expressed as a quotient of its constituents. For example if we have a ratio $a:b$, we can express it as $\displaystyle\frac ab$. The question he asked after this was really a thought-provoking question for me. He asked:

If $a:b$ means $\displaystyle\frac ab$, then what does $a:b:c$ mean?

I know that in this case it can't be expressed as a quotient. So is there no other way to express the ratio $a:b:c$ just as we express $a:b$? What should I tell my brother?

Answer 1

Suppose $$a:b:c=x:y:z$$ Then , $$\frac{a}{b}=\frac{x}{y}$$ And $$\frac{b}{c}=\frac{y}{z}$$

Answer 2

A ratio is not exactly a fraction, but somewhat related. For example saying that $a:b= c:d$ means that there is are constants $\xi\ne 0$ and $\eta\ne 0$(*) such that $\xi a = \eta c$ and $\xi b = \eta d$.

This of course is also true (to some extent) to fractions. If $a/b=c/d$ we have that with $\xi = d$ and $\eta = b$ that $\xi a = \eta c$ and $\xi b= \eta d$.

But when we extend the concept to tripples differences will arise because of the left associativity of division. Fractions are expressions of division while ratios are by definition not.

For example we have that $1:2:3=2:4:6$, but $1/2/3 = (1/2)/3 = 1/6$ and $2/4/6=(2/4)/6 = 1/12$ so $1/2/3\ne 2/4/6$.

You still can express such a ratio as a quotient of it's constituents. That you express $a:b$ as $a/b$ means that the first element is $a/b$ times the second. If you want to express a three-way ratio in quotient form you would need to have two quotients, for example one between the two first and one between the two last - the ratio $1:2:3$ then is represented by the quotients $1/2$ and $2/3$ (the first is $1/2$ of the second which in turn is $2/3$ of the third).

(*) We can allow one of them being $0$, but that complicates things a bit as then equality of ratios are no longer an equivalence relation.