When someone asks: "What is half of 20?", you immediately answer $10$. In mathematics, the word "of" represents multiplication:
$$\frac{1}{2} \text{ of } 20 = \frac{1}{2} \times 20 = 10$$
To multiply two fractions, simply multiply the numerators together and multiply the denominators together:
$$\mathbf{\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}}$$
- Proper $\times$ Proper: The product is smaller than both fractions! E.g., $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ (here $\frac{8}{15} < \frac{2}{3}$ and $\frac{8}{15} < \frac{4}{5}$).
- Fraction $\times$ Improper ($> 1$): The product is larger than the first fraction! E.g., $\frac{2}{3} \times \frac{5}{2} = \frac{10}{6} = \frac{5}{3} > \frac{2}{3}$.
Example: Find the area of a rectangular cardboard sheet of length $3\frac{1}{3}\text{ m}$ and breadth $1\frac{1}{5}\text{ m}$.
Step 1: Convert mixed fractions into improper fractions:
$$\text{Length} = \frac{10}{3}\text{ m}, \quad \text{Breadth} = \frac{6}{5}\text{ m}$$
Step 2: $\text{Area} = \text{Length} \times \text{Breadth} = \frac{10}{3} \times \frac{6}{5}$
Step 3: Cancel common factors before multiplying:
$$\frac{\cancel{10}^2}{\cancel{3}_1} \times \frac{\cancel{6}^2}{\cancel{5}_1} = \frac{2 \times 2}{1 \times 1} = \mathbf{4\text{ sq m}}$$
Students often waste time finding the LCM of denominators when multiplying fractions!
Rule: Common denominators are needed ONLY for addition and subtraction. For multiplication, multiply straight across: top $\times$ top and bottom $\times$ bottom.
Cooking recipes rely on fraction multiplication: if a cake recipe needs $\frac{3}{4}$ cup of sugar, making $\frac{1}{2}$ batch means calculating $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$ cup.