Multiplying & Dividing Fractions

 

Multiplying Algebraic Fractions

When multiplying algebraic fractions, the steps are the same as with regular fractions: multiply the numerators and multiply the denominators.

 

General Rule $$\frac{A}{B} \times \frac{C}{D} = \frac{A \cdot C}{B \cdot D}$$

 

Example: Multiplying with Factorisation

Multiply: $$\frac{2x^2}{3} \times \frac{9}{4x}$$

Step-by-Step Solution:

1. Multiply the numerators: $$2x^2 \times 9 = 18x^2$$
2. Multiply the denominators: $$3 \times 4x = 12x$$
3. Write the result: $$\frac{2x^2}{3} \times \frac{9}{4x} = \frac{18x^2}{12x}$$
4. Simplify: $$\frac{18x^2}{12x} = \frac{3x}{2}$$

 

 

 

 

 

Dividing Algebraic Fractions

Dividing algebraic fractions is just like dividing regular fractions: flip the second fraction and multiply.

General Rule $$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$$

 

Example: Basic Division

Divide: $$\frac{5x}{6} \div \frac{10}{3y}$$

Step-by-Step Solution:

1. Flip the second fraction: $$\frac{10}{3y} \to \frac{3y}{10}$$
2. Multiply: $$\frac{5x}{6} \times \frac{3y}{10}$$
3. Multiply the numerators: $$5x \times 3y = 15xy$$
4. Multiply the denominators: $$6 \times 10 = 60$$
5. Write the result: $$\frac{5x}{6} \div \frac{10}{3y} = \frac{15xy}{60}$$
6. Simplify: $$\frac{15xy}{60} = \frac{xy}{4}$$

 

Example: Division with Factorisation

Divide: $$\frac{x^2 - 9}{4x} \div \frac{x - 3}{2x^2}$$

Step-by-Step Solution:

1. Flip the second fraction: $$\frac{x - 3}{2x^2} \to \frac{2x^2}{x - 3}$$
2. Multiply: $$\frac{x^2 - 9}{4x} \times \frac{2x^2}{x - 3}$$
3. Factorise $x^2 - 9$: $$x^2 - 9 = (x - 3)(x + 3) \\ \text{So the expression becomes:} \\ \frac{(x - 3)(x + 3)}{4x} \times \frac{2x^2}{x - 3}$$
4. Cancel $x - 3$ from numerator and denominator: $$\frac{(x + 3)}{4x} \times 2x^2$$
5. Multiply numerators and denominators: $$\frac{(x + 3) \cdot 2x^2}{4x} = \frac{2x^2(x + 3)}{4x}$$
6. Simplify: $$\frac{2x^2(x + 3)}{4x} = \frac{x(x + 3)}{2}$$