Algebraic Fractions
Algebraic fractions are fractions that include algebraic expressions. They are common in Higher GCSE Maths and often require factorising before simplifying.
Examples of algebraic fractions:
x/3
5/(x + 2)
(x² + 5x)/(x)
Simplifying algebraic fractions
To simplify an algebraic fraction, look for common factors in the top and bottom. If needed, factorise first.
Example 1: Simplify a simple algebraic fraction
Simplify 6x/9.
The highest common factor of 6 and 9 is 3.
6x/9 = 2x/3
Example 2: Cancel a common algebraic factor
Simplify 8x²/12x.
Cancel the common factor 4x:
8x²/12x = 2x/3
Factorising before simplifying
Many algebraic fractions cannot be simplified until you factorise the numerator or denominator.
Example 3: Factorise then simplify
Simplify (x² + 5x)/x.
Factorise the numerator:
x² + 5x = x(x + 5)
Now simplify:
x(x + 5)/x = x + 5
Example 4: Simplify using a bracket factor
Simplify (x² - 9)/(x + 3).
Factorise the difference of two squares:
x² - 9 = (x - 3)(x + 3)
Cancel the common factor x + 3:
(x - 3)(x + 3)/(x + 3) = x - 3
Multiplying algebraic fractions
Multiply the numerators together and multiply the denominators together. Simplify before or after multiplying.
Example 5: Multiply algebraic fractions
Simplify (3x/4) × (8/9).
(3x × 8)/(4 × 9) = 24x/36
24x/36 = 2x/3
Dividing algebraic fractions
Dividing by a fraction means multiplying by its reciprocal.
Example 6: Divide algebraic fractions
Simplify (x/5) ÷ (2/15).
Multiply by the reciprocal:
(x/5) × (15/2)
Simplify:
15x/10 = 3x/2
Adding algebraic fractions
To add algebraic fractions, use a common denominator.
Example 7: Add algebraic fractions
Simplify x/3 + x/6.
The common denominator is 6.
x/3 = 2x/6
2x/6 + x/6 = 3x/6 = x/2
Solving equations with algebraic fractions
Multiply through by the denominator to remove the fraction.
Example 8: Solve an equation
Solve x/4 = 7.
Multiply both sides by 4:
x = 28
A common mistake is cancelling terms instead of factors. You can cancel common factors, but you cannot cancel part of an addition or subtraction unless it is factorised first.
If you see a quadratic expression in an algebraic fraction, try factorising it first. This often reveals a common bracket that can be cancelled.
Video explanation
A short Worthing Maths Tutor video explanation for algebraic fractions can be embedded here later to improve student engagement and time on page.
Practice questions
- Simplify 10x/15.
- Simplify 12x²/18x.
- Simplify (x² + 3x)/x.
- Simplify (x² - 16)/(x + 4).
- Simplify x/2 + x/4.
Answers
- 2x/3
- 2x/3
- x + 3
- x - 4
- 3x/4
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