Learning pack · IGCSE handout

Recurring Decimals and Surds

What's in this pack: printable worksheet PDF, matching answers PDF (IGCSE / Secure–Stretch practice). Recurring decimals convert to fractions via powers of 10; surds simplify by extracting square factors.

Resources at the top · full summary · videos with short notes below.

🎒 Resources in this pack

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Scaffolded Learning Sheet — attempt the worksheet, then use the Answers for a full written solution.

📝 Open Learning Sheet ✅ Open answers 📖 Full summary

Files: 47_recurring_decimals_and_surds_handout.pdf · 47_recurring_decimals_and_surds_handout_answers.pdf

Full topic summary

Recurring decimals convert to fractions via powers of 10; surds simplify by extracting square factors.

Key learning points

Let $x=0.\dot{a}\dot{b}\ldots$ , multiply by $10^k$ , subtract to clear the repeat.
Simplify $\sqrt{a}$ : factor out square factors.
Rationalise denominators: multiply by a suitable surd conjugate when needed.

Worked example 1 — Recurring decimal

Express $0.\dot{3}$ as a fraction.

Working

\tfrac{1}{3}

Let $x=0.\dot{3}$ ; $10x-x=3$ .

Worked example 2 — Simplify surd

Simplify $\sqrt{50}$ .

Working

5\sqrt{2}

$\sqrt{25\times2}$ .

Worked example 3 — Rationalise

Rationalise $\tfrac{1}{\sqrt{3}}$ .

Working

\tfrac{\sqrt{3}}{3}

Multiply by $\sqrt{3}/\sqrt{3}$ .

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