Meaning of Algebraic Fractions

An algebraic fraction is a fraction that contains algebraic expressions (letters, numbers, or both) in its numerator and/or denominator.

Examples:
\frac{x}{y}, \quad \frac{3x + 1}{2x}, \quad \frac{x^2 + 2x + 1}{x - 3}

2. Simplifying Algebraic Fractions

To simplify algebraic fractions, factorize the numerator and denominator, then cancel common factors.

Example 1:
Simplify (\frac{x^2 + 3x}{x})

Solution:
\frac{x(x + 3)}{x} = x + 3

Example 2:
Simplify (\frac{x^2 - 9}{x - 3})

Solution:
x^2 - 9 = (x - 3)(x + 3)
So,
\frac{(x - 3)(x + 3)}{x - 3} = x + 3

3. Algebraic Fractions with Common Denominators

When the denominators are the same, add or subtract the numerators directly.

Example 3:
Simplify (\frac{3x}{5} + \frac{2x}{5})

Solution:
\frac{3x + 2x}{5} = \frac{5x}{5} = x

4. Algebraic Fractions with Different Denominators

Find the Lowest Common Denominator (LCD), then combine.

Example 4:
Simplify (\frac{x}{3} + \frac{2}{5})

Solution:
LCD = 15
\frac{5x}{15} + \frac{6}{15} = \frac{5x + 6}{15}

5. Subtraction of Algebraic Fractions

Example 5:
Simplify (\frac{3}{x} - \frac{2}{x + 1})

Solution:
LCD = (x(x + 1))
\frac{3(x + 1) - 2x}{x(x + 1)} = \frac{3x + 3 - 2x}{x(x + 1)} = \frac{x + 3}{x(x + 1)}

6. Multiplication of Algebraic Fractions

Multiply numerators together and denominators together, then simplify.

Example 6:
Simplify (\frac{2x}{3y} \times \frac{9y}{4x})

Solution:
\frac{2x \times 9y}{3y \times 4x} = \frac{18xy}{12xy} = \frac{3}{2}

7. Division of Algebraic Fractions

To divide algebraic fractions, multiply by the reciprocal.

Example 7:
Simplify (\frac{x^2}{y} ÷ \frac{x}{y^2})

Solution:
\frac{x^2}{y} \times \frac{y^2}{x} = \frac{x^2 y^2}{xy} = xy

8. Formation of Algebraic Fractions

Forming algebraic fractions involves expressing real-world or word problems in fraction form using variables.

Example 8:
The ratio of boys to girls in a class is (x : y).
The fraction of boys in the class = (\frac{x}{x + y})