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I read in a textbook that the explanation to this rule lies in the fact that division is the reverse operation to multiplication. Unfortunately, the author did not elaborate on this point. Based on this, can someone help me to understand why

$\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{a}{b}$ * $\frac{d}{c}$?

s.xw
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  • Simply put, because division is defined this way. Division is multiplication, except multiplying by the reciprocal. It also explains why you can distribute denominators over sums, because multiplication can be distributed over sums. – Kaynex Nov 24 '16 at 18:03

2 Answers2

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Let us say you have a number $X$ that you multiply by $Y$. Then you got $XY$. If you want to revert that operation, you have to divide by $Y$:

$$\frac{XY}{Y}$$

But that is the same as multiplying by $\frac1Y$:

$$\frac{XY}{Y} = XY \cdot \frac1Y$$

That means, like you pointed out, that division is the reverse of multiplication. But division by $Y$ is just multiplication by $\frac1Y$.

Let us say that both $X$ and $Y$ are fractions.

That is,

$$X = \frac{a}{b},\ Y = \frac{c}{d}$$

Let us also note that $\frac{1}{Y}$ now becomes $$\dfrac1{\frac{c}{d}} = \frac{d}{c}$$

Therefore, $$X / Y = X \cdot \frac1Y = \frac{a}{b}\cdot\frac{d}{c}$$

RGS
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  • But is the understanding that division is the reverse operation to multiplication a requirement in explaining the 'invert and multiply' rule? – s.xw Nov 24 '16 at 18:35
  • @FrankShang I would not say it is a requirement, but it sure enables that it makes sense. One could always say "you invert and multiply" because it works. Nonetheless, I believe that the process of showing why it works and where it comes from is much more educative than saying "just use it because it works" – RGS Nov 24 '16 at 18:38
  • So, can 'division is the reverse operation to multiplication' mean that multiplying xy by the reciprocal of y is the reverse operation to multiplying xy by y? – s.xw Nov 24 '16 at 19:13
  • @FrankShang yes it can. Actually, "division is the reverse operation to multiplication", means that multiplying any number (not just xy) by the recyprocal of y is the reverse operation to multiplying the same number by y. That is because multiplying by 1/y *undoes* what multiplying by y did. That is why they are the reverse of eachother. – RGS Nov 24 '16 at 19:34
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If you accept that you can multiply the numerator and denominator by the same nonzero number without changing the value of the fraction, then just multiply the numerator and denominator by $d/c$: $$ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b}\cdot \frac{d}{c}}{\frac{c}{d}\cdot \frac{d}{c}} = \frac{\frac{ad}{bc}}{1} = \frac{ad}{bc}. $$

littleO
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