Multiplying and Dividing Fractions.

Multiplication and division of fractions are actually simpler than addition — you don't need a common denominator.

Multiplying

Multiply the tops, multiply the bottoms. $\frac{a}{b} \cdot \frac{c}{d} \;=\; \frac{a \cdot c}{b \cdot d}$

$\frac{2}{3} \cdot \frac{4}{5} \;=\; \frac{2 \cdot 4}{3 \cdot 5} \;=\; \frac{8}{15}$

Dividing — flip and multiply

Dividing by a fraction is the same as multiplying by its reciprocal (flipped fraction). $\frac{a}{b} \div \frac{c}{d} \;=\; \frac{a}{b} \cdot \frac{d}{c} \;=\; \frac{a \cdot d}{b \cdot c}$

$\frac{2}{3} \div \frac{4}{5} \;=\; \frac{2}{3} \cdot \frac{5}{4} \;=\; \frac{10}{12} \;=\; \frac{5}{6}$

Cancel before multiplying — saves work

If a top and a bottom share a factor, divide both by it before multiplying. $\frac{3}{4} \cdot \frac{8}{9}$ The $3$ on top and the $9$ on bottom share a factor of $3$ → $1$ and $3$. The $4$ on bottom and the $8$ on top share a factor of $4$ → $1$ and $2$. $\frac{1}{1} \cdot \frac{2}{3} \;=\; \frac{2}{3}$

This is exactly the same answer as $\frac{24}{36}$ reduced — but with much less arithmetic.

With whole numbers

A whole number $n$ is just $\frac{n}{1}$. $5 \cdot \frac{3}{4} \;=\; \frac{5}{1} \cdot \frac{3}{4} \;=\; \frac{15}{4}$