Multiplication of Fractions

Theorem

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

Then:

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

Proof

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

\(=\)

\(\ds a \times \dfrac 1 b \times c \times \dfrac 1 d\)

\(\ds \)

\(=\)

\(\ds a \times c \times \dfrac 1 b \times \dfrac 1 d\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

$\blacksquare$

Examples

Example: $\frac 2 3 \times \frac 4 7$

$\dfrac 2 3 \times \dfrac 4 7 = \dfrac 8 {21}$

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