To multiply fractions, multiply the numerators together and the denominators together. No common denominator is needed.
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Example: $$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$$
Always simplify your answer. You can also cancel (cross-cancel) before multiplying to keep numbers smaller: $$\frac{3}{4} \times \frac{8}{9} = \frac{3}{4} \times \frac{8}{9} \rightarrow \frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$$
Multiplying mixed numbers: convert to improper fractions first. $$1\frac{1}{2} \times 2\frac{2}{3} = \frac{3}{2} \times \frac{8}{3} = \frac{24}{6} = 4$$
Common error: adding the denominators instead of multiplying them, e.g. writing $\frac{1}{2} \times \frac{1}{3} = \frac{1}{5}$.