Mental division of two fractions?

Question

I've got a non-calc paper coming up, and when going through a test, this fraction came up:

$$ \frac{8}{-0.4} \equiv \frac{8}{\big(\frac{-2}{5}\big)} $$

Going through the answers he says: $$8/2=4$$ I then assume he did -(4*5) so: $$\frac{8}{\big(\frac{-2}{5}\big)} = -20$$

I can see what he's done, but I don't see what's happening mathematically?

$$\frac{a}{\frac{b}{c}} \equiv \frac{a}{b}\cdot c$$

Answer 1

We have that

$$\frac{\frac ab}{\frac cd}=\frac ab\div\frac cd=\frac ab\color{red}\cdot\color{blue}{\frac dc}=\frac {ad}{bc}\implies \frac8{-\frac 25}=\frac{\frac81}{-\frac 25}=\frac{8\cdot(-5)}{2\cdot1}=-\frac{40}2=-20$$

For, of course any real numbers such that $\;b,c,d\neq0\;$

Answer 2

What he is using is that $\frac{a}{\frac{1}{b}} = a*b$. That way he could transform a division involving decimals into simple integer multiplication.