Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Decimals & Recurring Decimals

In one sentence. Multiplying and dividing decimals without a wobble, and the Higher-tier trick that turns any recurring decimal into an exact fraction.

Number — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with John Napier. Learn Decimals & Recurring Decimals, teach it, then defend it against John Napier.

Where this sits in the specification

Specification 4MA1, 1.3 Decimals (recurring → fraction: Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 1.3 — Decimals (recurring → fraction: Higher)

1 · Understand it

What it is

Decimal arithmetic is whole-number arithmetic plus place-value bookkeeping: 4.7 × 0.06 is 47 × 6 with the decimal point placed by counting decimal places. Recurring decimals (0.454545…) are fractions in disguise, and an algebraic shift-and-subtract converts them exactly.

Why it works

Multiplying by 0.06 is multiplying by 6 then dividing by 100 — that's all 'counting decimal places' secretly is. The recurring trick works because multiplying by 10 shifts every digit left one place: if x = 0.4545… then 100x = 45.4545… has the SAME tail, so subtracting kills the tail exactly and leaves whole numbers.

When a question is asking for it

Money and measurement arithmetic without a calculator; 'given 468 × 23 = 10 764, write down…' questions (pure place-value); and any 'write 0.4̇5̇ as a fraction' question — a Higher-tier regular that is pure method marks once you know the shift-and-subtract.

How to hold it in your head

Ignore the points, multiply the whole numbers, then ask: roughly how big should this be? 4.7 × 0.06 is 'about 5 × 0.06 ≈ 0.3', so 282 must become 0.282. Estimation places the point more reliably than digit-counting under pressure.

2 · The mistakes that cost marks

  • "Multiplying always makes numbers bigger." Multiplying by anything below 1 SHRINKS: 4.7 × 0.06 is far less than 4.7.
  • Placing the decimal point by feel instead of counting places (or estimating). 4.7 × 0.06 has 1 + 2 = 3 decimal places: 0.282.
  • Writing 0.4̇5̇ as 45/100. That's 0.45 exactly — the recurring tail means it's actually 45/99 = 5/11, slightly more.
  • Multiplying by 10 when two digits recur. The shift must match the block length: one digit → ×10, two digits → ×100.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Given that 468 × 23 = 10 764, write down 4.68 × 2.3.

  1. 4.68 is 468 ÷ 100 and 2.3 is 23 ÷ 10.
  2. So the product is 10 764 ÷ 1000.

Answer: 10.764

Write 0.4̇5̇ (= 0.454545…) as a fraction in its simplest form.

  1. Let x = 0.454545…
  2. Two digits recur, so multiply by 100: 100x = 45.454545…
  3. Subtract: 100x − x = 45.4545… − 0.4545… → 99x = 45.
  4. x = 45/99 = 5/11.

Answer: 5/11

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Work out 4.7 × 0.06.
  2. Grade 7 — Given that 468 × 23 = 10 764, write down the value of 46.8 × 0.23.
  3. Grade 8 — Write 7/40 as a decimal.
  4. Grade 9 — Write the recurring decimal 2.3̇6̇ (= 2.363636…) as an improper fraction in its simplest form.
  5. 9+ — Write the recurring decimal 0.32̇ (= 0.3222…) as a fraction in its simplest form.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Decimals & Recurring Decimals, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Decimals & Recurring Decimals works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Decimals & Recurring Decimals against John Napier — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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