Usually $$\frac{a}{\frac{b}{c}} = \frac{ac}{b}$$ i.e. $b/c$ is seen as the denominator, and $a$ is the numerator.
If you have $a/b/c/d$, what do you choose to take as the denominator? $$\frac{b}{\frac{c}{d}} , \frac{c}{d},\text{ or } d\text{?}$$ and why?
And what about for $a/b/c/d/e$?
On
As @almagest points out, the convention is that operations at the same level of precedence (e.g. multiplication/division) are carried out from left to right. So $a/b/c/d$ is interpreted as $((a/b)/c)/d$, and similarly $1-2-3-4$ is interpreted as $((1-2)-3)-4$. It is best to use parenthesis in any case, or even better, to use prefix or postfix notation.
$a/(b/c)$ or $\frac{a}{\frac{b}{c}}$ is indeed $ac/b$. We also have $$\frac{a}{\frac{b}{\frac{c}{d}}} = a/(b/(c/d)) = a/(bd/c) = ac/bd$$ and so on.
The trouble is $$a/(b/c) \ne (a/b)/c$$since $a/(b/c) = \frac{{ac}}{b}$ and $(a/b)/c = \frac{a}{{bc}}$. So, without brackets, the notation is ambiguous. Introducing brackets, for example, we could choose between $$a/b/c/d=a{(b/c/d)^{ - 1}}$$ and $$a/b/c/d=(a{/b/c)d^{ - 1}}$$The first leads to $$a{(b{(c/d)^{ - 1}})^{ - 1}} = a{(b{c^{ - 1}}d)^{ - 1}} = \frac{{ac}}{{bd}}$$The second leads to $$(a/b){c^{ - 1}}{d^{ - 1}} = a{b^{ - 1}}{c^{ - 1}}{d^{ - 1}} = \frac{a}{{bcd}}$$ which is less interesting, it seems to me. The first expression leads to $$a/b/c/d/e = a{(b(c/d/e))^{ - 1}} = a{\left( {\frac{{bd}}{{ce}}} \right)^{ - 1}} = \frac{{ace}}{{bd}}$$ Collecting results, the following interesting pattern emerges:$$a/b = a{b^{ - 1}} = \frac{a}{b}$$ $$a/b/c = a{b^{ - 1}}c = \frac{{ac}}{b}$$ $$a/b/c/d = a{b^{ - 1}}c{d^{ - 1}} = \frac{{ac}}{{bd}}$$ $$a/b/c/d/e = a{b^{ - 1}}c{d^{ - 1}}e = \frac{{ace}}{{bd}}$$ $$a/b/c/d/e/f = a{b^{ - 1}}c{d^{ - 1}}e{f^{ - 1}} = \frac{{ace}}{{bdf}}...$$ and so on.