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Algebraic fractions

Revise Algebraic fractions for Mathematics D 4024 (O Level) — revision notes, 50 practice questions and instant AI marking.

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  Cambridge O Level Mathematics (Syllabus D) 4024

Algebraic fractions

The big idea: Algebraic fractions obey the same rules as number fractions: a common denominator to add or subtract, multiply tops and bottoms, flip to divide. To simplify, factorise first and cancel only whole factors.

Summary — What This Chapter Covers
  • Add and subtract algebraic fractions with numerical and algebraic denominators.
  • Multiply and divide algebraic fractions.
  • Factorise the numerator and denominator of a rational expression and cancel common factors.
1. Adding and Subtracting with Number Denominators
Syllabus 2.3 · Paper 1 & 2

Common denominator, then brackets on the numerators

Find the lowest common denominator. Multiply each numerator by what its denominator was multiplied by, and keep each numerator in brackets until it is expanded: this protects a minus sign in front of a fraction.

Worked Example

Write as a single fraction in its simplest form: (a) x/3 + (x − 4)/2   (b) 2x/3 − 3(x − 5)/2

2. Multiplying and Dividing
Syllabus 2.3 · Paper 1 & 2

Cancel first, then multiply

To multiply, multiply the numerators together and the denominators together, cancelling common factors first. To divide, multiply by the reciprocal of the second fraction.

Worked Example

Simplify (a) 3a/4 × 9a/10   (b) 3a/4 ÷ 9a/10

3. Algebraic Denominators
Syllabus 2.3 · Paper 1 & 2

Multiply the denominators together

When the denominators are different brackets, the common denominator is their product. Leave the denominator factorised; expand and simplify the numerator.

Worked Example

Write 1/(x − 2) + (x + 1)/(x − 3) as a single fraction.

Practice Question 1

Write 3/(x + 1) − 2/(x − 4) as a single fraction in its simplest form.

4. Simplifying Rational Expressions
Syllabus 2.3 · Paper 1 & 2

Factorise top and bottom, then cancel whole factors

Factorise the numerator and the denominator completely. Cancel only factors (whole brackets or whole terms multiplying everything), never individual terms that are being added.

Worked Example

Simplify (x2 − 2x)/(x2 − 5x + 6).

Practice Question 2

Simplify (a) (2x2 + 5x − 3)/(4x2 − 1)   (b) (x2 − 9)/(2x2 + 5x − 3)

Practice Question 3

2 2 2

Memorise
Add or subtract
Multiply
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