3. Multiplying & Dividing Fractions

Multiplying fractions is the easiest operation — no common denominator needed. Multiply straight across:
\[ \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}. \]
Intuition: \(\frac{2}{3} \times \frac{4}{5}\) means 'two-thirds of four-fifths.' The word of signals multiplication.

Dividing fractions uses a trick: 'keep, change, flip.' Keep the first fraction, change \(\div\) to \(\times\), and flip the second fraction upside down (its reciprocal).
\[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}. \]

Why flip? Dividing by \(\frac{2}{5}\) asks 'how many two-fifths fit into three-quarters?' Multiplying by the reciprocal answers exactly that.

💡 Tip: always simplify your final answer, and convert improper fractions like \(\frac{15}{8}\) to a mixed number (\(1\frac{7}{8}\)) if the question expects it.