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SQA National 5 Mathematics

Algebraic fractions

Simplifying, multiplying and adding fractions with algebraic terms.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Factorise numerators and denominators where possible.
  • State excluded values when a denominator could be zero.
  • Use common denominators for adding and subtracting.
National 5 Mathematics lesson

Explanation

Algebraic fractions behave like number fractions, but the top and bottom contain algebra.

Simplifying usually starts by factorising, then cancelling common factors. You may cancel factors, not individual terms.

When adding or subtracting, use a common denominator before combining the numerator.

Visual support

Cancel factors, not terms

(x² − 9) / (x − 3)

= (x − 3)(x + 3) / (x − 3)

= x + 3

x ≠ 3

Key formulae and rules

  • ab + cd = (ad + bc) / bd
  • (x² − 9) / (x − 3) = x + 3, x ≠ 3

Watch out

Cancelling terms across addition or subtraction.

Check

Check cancelled parts are whole factors, not terms joined by plus or minus signs.

Exam tip

Only cancel common factors. If there is a plus or minus sign, factorise before cancelling.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Simplify by factorising

Simplify (x² − 9) / (x − 3)

  1. Factorise the numerator: x² − 9 = (x − 3)(x + 3)
  2. Write the fraction as (x − 3)(x + 3) / (x − 3)
  3. Cancel the common factor x − 3.

Answer: x + 3, where x ≠ 3

Worked example 2

Multiply algebraic fractions

Simplify 3x4 × 86x

  1. Multiply numerators and denominators.
  2. Numerator: 3x × 8 = 24x
  3. Denominator: 4 × 6x = 24x
  4. 24x24x = 1, assuming x ≠ 0

So: 1.

Worked example 3

Add with a common denominator

Simplify 2x + 3x + 1

  1. Common denominator is x(x + 1).
  2. 2x = 2(x + 1) / x(x + 1)
  3. 3x + 1 = 3x / x(x + 1)
  4. Add numerators: 2x + 2 + 3x.

Answer: (5x + 2) / x(x + 1)

Watch out

  • Cancelling terms across addition or subtraction.
  • Forgetting to factorise first.
  • Not using brackets around a whole numerator.