Proof of dividing fractions $\frac{a/b}{c/d}=\frac{ad}{bc}$ For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$

There is something assumed about the order of operations in the notation 
$a/b \div c/d$.  I see this as a problem in pedagogy when fractions are first introduced in schools.  Somehow we have to assume that the slash bars are to be done before the division symbol, even though each of those symbols stands for "divide".  If I were you, I'd always stay in the habit of writing fractions as $\frac{a}{b}$ rather than $a/b$.
end rant.
Traditionally, $\frac{a}{b} \div \frac{c}{d}$ means:  Beginning with the quantity $\frac{a}{b}$, divide this by the quantity $\frac{c}{d}$.  So let's think of a small example not dealing with fractions first:
$$ 8 \div 2.$$
Now, we all know the answer is $4$.  But I want to draw your attention to the fact that this expression is the same as:
$$ 8 \cdot \frac{1}{2}.$$
What happened?  Well division by $2$ is the same as multiplication by $\frac{1}{2}$, because $\frac{1}{2}$ is the multiplicative inverse, or reciprocal of $2$.  By defintion, the reciprocal of a number $x$ is a number $y$ such that $x\cdot y = 1$.
Every nonzero number has a reciprocal.  If $x \neq 0$, then its reciprocal is $\frac{1}{x}$ (because $x \cdot \frac{1}{x} = 1$).  So, for $x \neq 0$:
$$ 8 \div x = 8 \cdot \frac{1}{x}.$$
Now, if $x$ is a fraction, such as $\frac{c}{d}$, then the most natural way to write the reciprocal of $x$ is by "flipping" the fraction.  The reciprocal of $\frac{c}{d}$ is $\frac{d}{c}$ (well... because $\frac{c}{d}\cdot \frac{d}{c} = \frac{cd}{cd} = 1$).  Thus:
$$ 8 \div \frac{c}{d} = 8 \cdot \frac{d}{c}. $$
Finally, it does not matter so much what the first factor of the expression is.  It could be another fraction:
$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}.$$
Hope this helps!
