Dividing Fractions
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
Formula
\left(\frac{a}{b}\right) \div \left(\frac{c}{d}\right) = \left(\frac{a}{b}\right) \times \left(\frac{d}{c}\right) = \frac{ad}{bc}
Definition
To divide by a fraction, keep the first fraction, change the division sign to multiplication, and flip the second fraction (use its reciprocal), a method often called "keep, change, flip." Formally, division of fractions is defined as multiplication by the reciprocal: $(a/b)/(c/d) = (a/b) \times (d/c) = ad/(bc)$, provided $c \neq 0$, which follows from the definition of division as the inverse of multiplication. In a field $F$, division is defined by $a/b = a \cdot b^{-1}$ for $b \neq 0$; division is not a primitive operation in the field axioms, it is derived from multiplication and the existence of multiplicative inverses.
Example
$3/4 \div 1/2$: keep $3/4$, change $\div$ to $\times$, flip $1/2$ to $2/1$, giving $3/4 \times 2/1 = 6/4 = 3/2 = 1$ and $1/2$. Similarly, $(5/6)/(10/9) = (5/6) \times (9/10) = 45/60 = 3/4$ (check: $(10/9) \times (3/4) = 30/36 = 5/6$). The same rule handles complex fractions: $((x+1)/x)/((x^2-1)/x^2) = ((x+1)/x) \times (x^2/(x^2-1)) = x^2(x+1)/(x(x+1)(x-1)) = x/(x-1)$ for $x \notin \{0, 1, -1\}$.
Key Insight
Dividing by $1/2$ is the same as multiplying by $2$, you end up with twice as much, and dividing by a fraction smaller than $1$ always gives a bigger answer: you are really asking "how many half-pieces fit into my amount?" The keep-change-flip rule is not magic, it is a direct consequence of the definition of division: any time you ask "how many times does $B$ fit into $A$," you get $A/B = A \times (1/B)$, multiplication by the reciprocal. This also connects to the ordered field structure of $\mathbb{Q}$: if $0 < c/d < 1$, then $d/c > 1$, so multiplying by $d/c$ increases the value, explaining why dividing by a small fraction increases the result.