Partial Fractions summary

Key learning points

• Linear factors → $\frac{A}{x-a}$; repeated → include $\frac{B}{(x-a)^2}$.
• Irreducible quadratics get a linear numerator.
• Equate coefficients or substitute convenient $x$ values.
• Divide first if the fraction is improper.

Worked example 1 — Equivalent fractions

Write three fractions equivalent to $\tfrac{2}{5}$.

Working

Multiply numerator and denominator by the same integer.

Worked example 2 — Simplifying

Simplify $\tfrac{16}{20}$.

Working

Divide by the HCF of 16 and 20, which is 4.

Worked example 3 — Addition

Calculate $\tfrac{2}{3}+\tfrac{1}{4}$.

Working

LCD of 3 and 4 is 12.

Worked example 4 — Subtraction

Calculate $\tfrac{5}{6}-\tfrac{1}{4}$.

Working

LCD of 6 and 4 is 12.

Worked example 5 — Multiplication

Calculate $\tfrac{2}{5}\times\tfrac{3}{4}$.

Working

Multiply numerators and denominators, then simplify.

Worked example 6 — Division

Calculate $\tfrac{3}{4}\div\tfrac{2}{5}$.

Working

Multiply by the reciprocal of the divisor.

Worked example 7 — Mixed numbers

Calculate $1\tfrac{1}{2}+2\tfrac{1}{3}$.

Working

Convert to improper fractions before adding.

Worked example 8 — Fraction of an amount

Find $\tfrac{3}{5}$ of $£40$.

Working

Divide by the denominator, then multiply by the numerator.

Worked example 9 — Linear factors

Decompose $\frac{5x+3}{(x+1)(x-1)}$.

Working

Clear denominators and solve for the constants.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.