TITLE: Most students asked why is that ${a\over b}\div{c\over d}={ad\over bc}$
QUESTION [7 upvotes]: Most students asked why is that
$${a\over b}\div{c\over d}={ad\over bc}$$
I just told them: inverse the second fraction and multiply. Why? They ask me. I have no idea.
Any logical answers to them kids?
These day teachers just told students at secondary to memorise and no why is allowed in lessons. I think that is wrong. Putting people of studying maths.

REPLY [3 votes]: I hate the $\div$ symbol, and would ban it if I had the power.
Fractions ARE division.  Division IS multiplication by the reciprocal.
No need to learn 3 different notations.
I would teach that division is multiplication by the reciprocal first with whole numbers.  Then introducing division by fractions, you say that is it exactly the same.
Anyway, I learned the flip it over and multiply as rote process, and I did it, and didn't know why it worked, but I was good at following instructions.  It finally clicked when I had:
${\frac ab} \div{\frac cd} = \dfrac {\frac ab}{\frac cd}$ 
How do you get the fraction out of the denominator? You multiply top and bottom by $d.$  And to get the fraction out of the numerator? You multiply top and bottom by $b.$
$\dfrac {ad}{cb}$