Question 305634
Well let's take a slightly different problem that can be easily expressed in decimal format.
1/2 divided by 3/4
.50/.75
multiply by 100
50/75 
factor out 25
2/3
What just happened.
We want to know how many times 3/4 can go into 1/2
Well it can't go a complete time  because 3/4 is larger than 1/2
the ratio holds true for 50/75
How many times can 75 go into 50 it can't one whole time. It can go 2/3 of 1 time?

But what is interesting to me about this problem and yours is how we resolve the problem using fractions.
(1/2)/(2/3)=1/2*3/2=/3/4
and 
(1/2) /(3/4) = 1/2*4/3=4/6=2/3 as we learned before.

We used the reciprocals to multiply rather than divide.