Division of Fractions

Theorem

Let $a, b, c, d \in \Z$ such that $b c d \ne 0$.

Then:

$\dfrac a b \div \dfrac c d = \dfrac {a d} {b c}$

Proof

\(\ds \dfrac a b \div \dfrac c d\)

\(=\)

\(\ds \dfrac a b \times \dfrac 1 {c / d}\)

\(\ds \)

\(=\)

\(\ds \dfrac a b \times \dfrac d c\)

\(\ds \)

\(=\)

\(\ds \dfrac {a d} {b c}\)

Multiplication of Fractions

$\blacksquare$

Examples

Example: $\frac 1 3 \div \frac 3 4$

$\dfrac 1 3 \div \dfrac 3 4 = \dfrac 4 9$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fraction
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fraction