Dividing Fractions
Turn the second fraction upside down, then multiply.
There are 3 Simple Steps to Divide Fractions:
Step 1. Turn the second fraction (the one you want to divide by) upside down Step 2. Multiply the first fraction by that reciprocal 
Example:
Example:
\frac{1}{2} ÷ \frac{1}{6}
Step 1. Turn the second fraction upside down (it becomes a reciprocal):
\frac{1}{6} becomes \frac{6}{1}
Step 2. Multiply the first fraction by that reciprocal:
(multiply tops ...)
\frac{1}{2} × \frac{6}{1} = \frac{1 × 6}{2 × 1} = \frac{6}{2}
(... multiply bottoms)
Step 3. Simplify the fraction:
\frac{6}{2} = 3
With Pen and Paper
And here is how to do it with a pen and paper (press the play button):
To help you remember:
♫ "Dividing fractions, as easy as pie,
Flip the second fraction, then multiply.
And don't forget to simplify,
Before it's time to say goodbye" ♫
Another way to remember is: "leave me, change me, turn me over" 
How Many?
20 divided by 5 is asking "how many 5s in 20?" (=4) and so:
\frac{1}{2} ÷ \frac{1}{6} is really asking:
how many \frac{1}{6}s in \frac{1}{2} ?
Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?
How many  in  ?  Answer: 3 
So now you can see why \frac{1}{2} ÷ \frac{1}{6} = 3
Another Example:
\frac{1}{8} ÷ \frac{1}{4}
Step 1. Turn the second fraction upside down (the reciprocal):
\frac{1}{4} becomes \frac{4}{1}
Step 2. Multiply the first fraction by that reciprocal:
\frac{1}{8} × \frac{4}{1} = \frac{1 × 4}{8 × 1} = \frac{4}{8}
Step 3. Simplify the fraction:
\frac{4}{8} = \frac{1}{2}
Fractions and Whole Numbers
What about division with fractions and whole numbers?
Make the whole number a fraction, by putting it over 1.
Example: 5 is also \frac{5}{1}
Then continue as before.
Example:
\frac{2}{3} ÷ 5
Make 5 into \frac{5}{1} :
\frac{2}{3} ÷ \frac{5}{1}
Then continue as before.
Step 1. Turn the second fraction upside down (the reciprocal):
\frac{5}{1} becomes \frac{1}{5}
Step 2. Multiply the first fraction by that reciprocal:
\frac{2}{3} × \frac{1}{5} = \frac{2 × 1}{3 × 5} = \frac{2}{15}
Step 3. Simplify the fraction:
The fraction is already as simple as it can be.
Answer = \frac{2}{15}
Example:
3 ÷ \frac{1}{4}
Make 3 into \frac{3}{1} :
\frac{3}{1} ÷ \frac{1}{4}
Then continue as before.
Step 1. Turn the second fraction upside down (the reciprocal):
\frac{1}{4} becomes \frac{4}{1}
Step 2. Multiply the first fraction by that reciprocal:
\frac{3}{1} × \frac{4}{1} = \frac{3 × 4}{1 × 1} = \frac{12}{1}
Step 3. Simplify the fraction:
\frac{12}{1} = 12
And Remember ...
You can rewrite a question like "20 divided by 5" into "how many 5s in 20"
So you can also rewrite "3 divided by ¼" into "how many ¼s in 3" (=12)
Why Turn the Fraction Upside Down?
Because dividing is the opposite of multiplying!
A fraction says to:  

But for DIVISION we:
 divide by the top number
 multiply by the bottom number
Example: dividing by ^{5}/_{2} is the same as multiplying by ^{2}/_{5}
So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.