How to Add, Subtract, Multiply and Divide Fractions

Fraction arithmetic underpins the whole number section of Edexcel IGCSE Maths, and it turns up in almost every paper. This page covers simplifying a fraction by dividing through by the HCF, adding and subtracting using a common denominator, and multiplying and dividing with cross-cancelling and Keep–Change–Flip. The worked examples show every step, including how to cancel before you multiply rather than wrestling with big numbers afterwards. Every practice question below is auto-marked and comes with a model solution, so scroll down and test each skill in turn.

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Prior Knowledge

You should be confident with times tables and basic number facts before starting. Working with mixed numbers? That's covered separately in Mixed Numbers Arithmetic.

⇩ Jump to Practice Questions ⇩

Core Ideas

Simplify

Divide top and bottom by their HCF. Always give final answers in simplest form.

Add & Subtract

Find a common denominator first. Only add or subtract the numerators — never the denominators.

Multiply

Multiply tops together and bottoms together. No common denominator needed. Simplify after.

Divide (Keep–Change–Flip)

Keep the first fraction, Change ÷ to ×, Flip the second. Then multiply.

Cross-Cancel

Before multiplying, cancel common factors diagonally across the fractions to keep numbers small.

Simplifying Fractions

💡 Example 1 — HCF Method

  1. Find the HCF of numerator and denominator.
  2. Divide both by the HCF.
\[ \begin{aligned} &\tfrac{18}{24}\\[4pt] &= \tfrac{18\div6}{24\div6}\\[4pt] &= \tfrac{3}{4} \end{aligned} \quad(\text{HCF}=6) \]
✅ Always simplify fully — check no further common factors remain.

💡 Example 2 — Splitting into Factors

  1. Write both as products to expose the common factor.
  2. Cancel the common factor.
\[ \begin{aligned} &\tfrac{18}{24}\\[4pt] &= \tfrac{6\times3}{6\times4}\\[4pt] &= \tfrac{3}{4} \end{aligned} \]
✅ Factoring first makes large cancellations visible immediately.

Adding and Subtracting Fractions

💡 Example 3 — Adding

  1. Find the LCM of the denominators.
  2. Convert each fraction to the common denominator.
  3. Add numerators; keep denominator.
  4. Simplify.
\[ \begin{aligned} &\tfrac{2}{3}+\tfrac{1}{4}\\[4pt] &= \tfrac{8}{12}+\tfrac{3}{12}\\[4pt] &= \tfrac{11}{12} \end{aligned} \]
✅ The LCM gives the most efficient common denominator.

💡 Example 4 — Subtracting

  1. Find a common denominator.
  2. Convert each fraction.
  3. Subtract numerators; keep denominator.
  4. Simplify if possible.
\[ \begin{aligned} &\tfrac{5}{6}-\tfrac{1}{4}\\[4pt] &= \tfrac{10}{12}-\tfrac{3}{12}\\[4pt] &= \tfrac{7}{12} \end{aligned} \]
✅ Watch signs carefully — especially subtracting a negative.

Multiplying and Dividing Fractions

💡 Example 5 — Multiplying

  1. Multiply numerators together.
  2. Multiply denominators together.
  3. Simplify.
\[ \begin{aligned} &\tfrac{2}{5}\times\tfrac{3}{4}\\[4pt] &= \tfrac{6}{20}\\[4pt] &= \tfrac{3}{10} \end{aligned} \]
✅ No common denominator needed to multiply.

💡 Example 6 — Dividing (Keep–Change–Flip)

  1. Keep the first fraction.
  2. Change ÷ to ×.
  3. Flip the second fraction.
  4. Multiply and simplify.
\[ \begin{aligned} &\tfrac{3}{4}\div\tfrac{2}{5}\\[4pt] &= \tfrac{3}{4}\times\tfrac{5}{2}\\[4pt] &= \tfrac{15}{8} \end{aligned} \]
✅ No common denominator needed to divide either.

💡 Example 7 — Cross-Cancelling

  1. Look for common factors diagonally across the fractions.
  2. Cancel before multiplying — keeps numbers small.
\[ \begin{aligned} &\tfrac{6}{8}\times\tfrac{4}{9}\\[4pt] &\xrightarrow{\div3}\ \tfrac{6}{8}\times\tfrac{4}{3}\\[4pt] &\xrightarrow{\div4}\ \tfrac{1}{2}\times\tfrac{1}{3}\\[4pt] &= \tfrac{1}{6} \end{aligned} \]
✅ Cross-cancel works for division too — flip first, then cancel.

💡 Example 8 — Dividing with Cross-Cancel

  1. Keep–Change–Flip first.
  2. Cross-cancel, then multiply.
\[ \begin{aligned} &\tfrac{4}{9}\div\tfrac{8}{3}\\[4pt] &= \tfrac{4}{9}\times\tfrac{3}{8}\\[4pt] &\xrightarrow{\div4}\ \tfrac{1}{9}\times\tfrac{3}{2}\\[4pt] &\xrightarrow{\div3}\ \tfrac{1}{3}\times\tfrac{1}{2}\\[4pt] &= \tfrac{1}{6} \end{aligned} \]
✅ Flip before you cancel — never cancel before flipping.

🔑 Key Ideas

  • Add & Subtract always need a common denominator.
  • Multiply & Divide do not need a common denominator.
  • Keep–Change–Flip for division; then treat as multiplication.
  • Cross-cancel before multiplying to keep numbers small.
  • Always give answers in simplest form.

⚠️ Common Pitfalls

  • Adding denominators directly — e.g. \(\tfrac{2}{3}+\tfrac{1}{4}=\tfrac{3}{7}\) is wrong.
  • Using a common denominator when multiplying or dividing — not needed.
  • Flipping the wrong fraction when dividing — always flip the second.
  • Cross-cancelling before flipping in a division — flip first.
  • Leaving answers unsimplified.
⇩ Jump to Practice Questions ⇩

Ready to practise? Try the automatic questions below, or move on to the next topic on Mixed Numbers.

Next: Mixed Numbers →

GCSE & IGCSE Fractions — Practice Rooms

Practise randomly generated, auto-marked fraction questions designed to help you master adding fractions, subtracting fractions, multiplying fractions, dividing fractions and simplifying fractions. Each time you open a room, new values are created automatically so no two sets are the same.

Enter answers as a/b with no spaces, and always write your final answer in its simplest form. For the Factors room, list all factors separated by commas (for example 1,2,4,8).

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