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Algebra - Algebraic Fractions

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Simplifying Algebraic Fractions: Factoring the numerator and denominator completely and cancelling common factors.

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Multiplying Fractions: Multiply numerators together and denominators together, simplifying where possible.

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Dividing Fractions: Multiply the first fraction by the reciprocal (flip) of the second fraction.

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Adding and Subtracting Fractions: Find a common denominator (usually the Lowest Common Multiple of the denominators) before combining numerators.

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Solving Algebraic Fraction Equations: Eliminate denominators by multiplying every term by the common denominator or cross-multiplying.

📐Formulae

ac±bc=a±bc\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

ab±cd=ad±bcbd\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}

ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

ab÷cd=ab×dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

If ab=cd, then ad=bc\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } ad = bc

💡Examples

Problem 1:

Simplify x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}

Solution:

x−3x+2\frac{x-3}{x+2}

Explanation:

First, factor both the numerator and denominator. x2−9x^2 - 9 is a difference of two squares: (x−3)(x+3)(x-3)(x+3). The quadratic x2+5x+6x^2 + 5x + 6 factors into (x+2)(x+3)(x+2)(x+3). Cancel the common factor (x+3)(x+3) from both.

Problem 2:

Write as a single fraction: 3x+1−2x−2\frac{3}{x+1} - \frac{2}{x-2}

Solution:

x−8(x+1)(x−2)\frac{x-8}{(x+1)(x-2)}

Explanation:

Find a common denominator, which is (x+1)(x−2)(x+1)(x-2). Rewrite the fractions: 3(x−2)(x+1)(x−2)−2(x+1)(x+1)(x−2)\frac{3(x-2)}{(x+1)(x-2)} - \frac{2(x+1)}{(x+1)(x-2)}. Expand the numerators: 3x−6−(2x+2)=3x−6−2x−2=x−83x - 6 - (2x + 2) = 3x - 6 - 2x - 2 = x - 8.

Problem 3:

Solve 2x+13=4x\frac{2}{x} + \frac{1}{3} = \frac{4}{x}

Solution:

x=6x = 6

Explanation:

Multiply all terms by the common denominator 3x3x to clear the fractions. 3x(2x)+3x(13)=3x(4x)3x(\frac{2}{x}) + 3x(\frac{1}{3}) = 3x(\frac{4}{x}) simplifies to 6+x=126 + x = 12. Subtracting 6 from both sides gives x=6x = 6.