Dividing by a fraction asks “how many of this fraction fit inside that one?”
There is a beautifully simple rule:
to divide by a fraction, flip it over (take its
reciprocal) and multiply.
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a\,d}{b\,c}
For example, how many quarters fit in a half?
\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2
Two quarters fit in a half, so the answer is 2. Notice that
dividing by a number smaller than 1 makes the answer
bigger than what you started with — splitting into small pieces
gives you lots of them.