Dividing Fractions
Fractions & DecimalsTo 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). This is often called "keep, change, flip."
Example
$3/4 \div 1/2$: keep $3/4$, change $\div$ to $\times$, flip $1/2$ to $2/1$. Now multiply: $3/4 \times 2/1 = 6/4 = 3/2 = 1$ and $1/2$.
Key Insight
Dividing by $1/2$ is the same as multiplying by $2$ - you end up with twice as much. Dividing by a fraction smaller than $1$ always gives a bigger answer. This makes sense: you are asking "how many half-pieces fit into my amount?"
Definition
Division of fractions is defined as multiplication by the reciprocal: $(a/b)/(c/d) = (a/b) \times (d/c) = ad/(bc)$, provided $c \neq 0$. This follows from the definition of division as the inverse of multiplication: if $(c/d) \times x = a/b$, then $x = (a/b) \times (d/c)$.
Example
$(5/6)/(10/9) = (5/6) \times (9/10) = 45/60 = 3/4$. Check: $(10/9) \times (3/4) = 30/36 = 5/6$. Cross-cancellation: $5$ and $10$ share $5$; $9$ and $6$ share $3 \to (1 \times 3)/(2 \times 2) = 3/4$.
Key Insight
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 $A/B = A \times (1/B)$, which is multiplication by the reciprocal.
Definition
Division in a field $F$ is defined by $a/b = a \cdot b^{-1}$ for $b \neq 0$. In $\mathbb{Q}$, $(a/b)/(c/d) = (a/b) \times (c/d)^{-1} = (a/b) \times (d/c) = (ad)/(bc)$. Division is not a primitive operation in field axioms; it is derived from multiplication and the existence of multiplicative inverses.
Example
Complex fraction simplification: $((x+1)/x)/((x^2-1)/x^2) = ((x+1)/x) \times (x^2/(x^2-1)) = x^2(x+1)/(x(x+1)(x-1)) = x/(x-1)$ for $x \notin \{0, 1, -1\}$.
Key Insight
The fact that division by fractions yields a larger result (for fractions $< 1$) is a consequence of the ordering on $\mathbb{Q}$: if $0 < c/d < 1$, then $d/c > 1$, so multiplying by $d/c$ increases the value. This connects the arithmetic of fractions to the ordered field structure of $\mathbb{Q}$.