property: a/b / c/d = a/b * d/c
description: when dividing fractions invert the divisor and multiply.
can someone explain this to me with proof? i don't understand why it works.
$\begin {align*}
&\dfrac{\dfrac a b}{\dfrac c d} = \\ \\
&\dfrac{\dfrac a b}{\dfrac c d} \cdot \dfrac d d = \\ \\
&\dfrac{\dfrac{a d}{b}}{\dfrac {c d}{d}} = \\ \\
&\dfrac{\dfrac{ad}{b}}{c} = \\ \\
&\dfrac{ad}{bc}
\end{align*}$
I would demonstrate this similarly, but slightly differently.
First, it is important to know that anything divided by itself is 1.
So, $\dfrac{a}{a} = 1$ for all $a\neq 0$. We can make this more complicated.
$\dfrac{blah}{blah} = 1$
$\dfrac{\tfrac{a}{b}}{\tfrac{a}{b}} = 1$
So long as the numerator and denominator are equal, you get 1.
Next, any number times 1 gives the original number.
$a = a\cdot 1$
Any number divided by 1 gives the original number:
$\dfrac{a}{1} = a$
And finally, multiplication is commutative:
$ab = ba$
Commutativity is true as part of a fraction, as well:
$\dfrac{ab}{cd} = \dfrac{ba}{cd} = \dfrac{ab}{dc} = \dfrac{ba}{dc}$
$\begin{align*}\left( \dfrac{\tfrac{a}{b}}{\tfrac{c}{d}}\right) & = \left( \dfrac{\tfrac{a}{b}}{\tfrac{c}{d}}\cdot 1 \right) \\ & = \left( \dfrac{\tfrac{a}{b}}{\tfrac{c}{d}}\cdot \dfrac{\tfrac{d}{c}}{\tfrac{d}{c}} \right) \\ & = \left( \dfrac{\tfrac{a}{b}\cdot \tfrac{d}{c}}{\tfrac{c}{d}\cdot \tfrac{d}{c}} \right) \\ & = \left( \dfrac{\tfrac{ad}{bc}}{\tfrac{cd}{dc}} \right) \\ & = \left( \dfrac{\tfrac{ad}{bc}}{\tfrac{cd}{cd}} \right) \\ & = \left( \dfrac{\tfrac{ad}{bc}}{1} \right) \\ & = \left( \dfrac{ad}{bc} \right) \\ & = \left( \dfrac{a}{b}\cdot \dfrac{d}{c} \right) \end{align*}$
I went through a lot more steps to clarify in a bit more detail what is happening at each step, but romsek's approach is correct, as well.