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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?

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

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  • $\begingroup$ Notice how $\frac{a}{\frac{b}{c}} = \frac{a*c}{b}$ $\endgroup$ – John Lou Apr 30 '17 at 17:33
  • $\begingroup$ Is that what you're asking? $\endgroup$ – John Lou Apr 30 '17 at 17:34
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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 $\;b,c,d\neq0\;$

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  • $\begingroup$ Is the bottom part of the of my question I added on similarly true, I understand your answer, but in the his explanation he seemed to do what I've edited onto my question. @DonAntonio $\endgroup$ – user429273 Apr 30 '17 at 17:41
  • $\begingroup$ @Skidushe It's exactly the same as explained in my answer...! In what you added, $\;a=8,\,b=-2,\,c=5\;$ , and, of course, $\;\frac{ab}c=\frac ac\cdot b\;$ ... $\endgroup$ – DonAntonio Apr 30 '17 at 17:46
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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.

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