N10: Terminating & Recurring Decimals
Convert between terminating decimals and fractions; recurring decimals to fractions (Higher)
📋 Key Concepts
Terminating Decimal: A decimal that ends (has a finite number of digits). Example: 0.75, 0.125, 3.4
Recurring Decimal: A decimal where digits repeat forever. Shown with dots or bars. Example: 0.333... = 0.̇3
Key Recurring Decimals:
0.̇3 = 1/3
0.̇6 = 2/3
0.̇1̇4̇2̇8̇5̇7 = 1/7
0.̇0̇9 = 1/11
0.̇3 = 1/3
0.̇6 = 2/3
0.̇1̇4̇2̇8̇5̇7 = 1/7
0.̇0̇9 = 1/11
| Notation | Meaning | Example |
|---|---|---|
| 0.̇3 | Single digit repeats | 0.33333... |
| 0.̇4̇5 | Multiple digits repeat | 0.454545... |
| 0.1̇6 | Starts then repeats | 0.166666... |
📝 Converting Terminating Decimals to Fractions
Method: Write as a fraction over the appropriate power of 10, then simplify.
Example 1
Convert 0.75 to a fraction
Solution:
0.75 = 75/100
Divide by 25: = 3/4
Example 2
Convert 0.125 to a fraction
Solution:
0.125 = 125/1000
Divide by 125: = 1/8
Example 3
Convert 2.4 to a mixed number
Solution:
2.4 = 2 + 0.4 = 2 + 4/10 = 2 2/5
📝 Converting Fractions to Decimals
Method: Divide the numerator by the denominator. Use bus stop division.
Example 4
Convert 3/8 to a decimal
Solution: 3 ÷ 8 = 0.375
Example 5
Convert 5/6 to a decimal
Solution: 5 ÷ 6 = 0.8333... = 0.8̇3
📝 Recurring Decimals to Fractions (Higher)
Algebraic Method: Multiply by powers of 10 to eliminate the recurring part, then subtract.
Example 6
Convert 0.̇7 to a fraction
Solution:
Let x = 0.̇7 = 0.7777...
Multiply by 10: 10x = 7.̇7 = 7.7777...
Subtract: 10x - x = 7
9x = 7
x = 7/9
Example 7
Convert 0.̇4̇5 to a fraction
Solution:
Let x = 0.̇4̇5 = 0.454545...
Multiply by 100: 100x = 45.̇4̇5 = 45.454545...
Subtract: 100x - x = 45
99x = 45
x = 45/99 = 5/11
Example 8
Convert 0.2̇3 to a fraction
Solution:
Let x = 0.2̇3 = 0.23333...
Multiply by 10: 10x = 2.̇3 = 2.3333...
Multiply by 10 again: 100x = 23.̇3 = 23.3333...
Subtract: 100x - 10x = 23.333... - 2.333...
90x = 21
x = 21/90 = 7/30
📝 Which Fractions Recur?
Rule: In simplest form, if the denominator has only 2s and/or 5s as prime factors, the decimal terminates. Otherwise, it recurs.
| Fraction | Denominator factors | Decimal | Type |
|---|---|---|---|
| 1/2 | 2 | 0.5 | Terminating |
| 3/4 | 2² | 0.75 | Terminating |
| 7/8 | 2³ | 0.875 | Terminating |
| 2/5 | 5 | 0.4 | Terminating |
| 1/3 | 3 | 0.̇3 | Recurring |
| 5/6 | 2 × 3 | 0.8̇3 | Recurring |
| 2/7 | 7 | 0.̇2̇8̇5̇7̇1̇4 | Recurring |
🧮 Worked Examples — Exam Style
Terminating decimals become fractions by reading the place value (tenths, hundredths, thousandths) then simplifying. Recurring decimals need the algebra trick.
Convert 0.375 to a fraction in its simplest form.
0.375 = 375/1000 (three decimal places → thousandths)
Simplify: ÷ 125 top and bottom → 3/8
Quick check: 1/8 = 0.125, so 3/8 = 0.375 ✓
M1 for 375/1000, A1 for 3/8. Knowing the eighths (0.125, 0.25, 0.375, 0.5 …) turns this into write-down marks.
Write 7/40 as a decimal.
40 = 5 × 8, so scale to a power of 10: 7/40 = 17.5/100 = 0.175
Or divide: 7 ÷ 40 = 0.175 ✓
B1 for 0.175. Fractions with denominators of 2, 4, 5, 8, 10, 20, 25, 40 are the exam favourites.
Exam wording to expect: "Write … as a fraction in its simplest form" or "Show that 0.7̇2̇ = 8/11" (Higher — the “show that” format means every algebra line is marked).
❓ Practice Questions
Q1: Convert 0.35 to a fraction in its simplest form
Q2: Convert 3/20 to a decimal
Q3: Convert 1/6 to a decimal
Q4 (Higher): Convert 0.̇5 to a fraction
Q5 (Higher): Convert 0.̇1̇2 to a fraction
Q6: Convert 0.125 to a fraction in its simplest form
Q7: Write 5/8 as a decimal
✅ Answers
- 35/100 = 7/20
- 3 ÷ 20 = 0.15
- 1 ÷ 6 = 0.1̇6
- x = 0.̇5, 10x = 5.̇5, 10x - x = 5, x = 5/9
- x = 0.̇1̇2, 100x = 12.̇1̇2, 99x = 12, x = 12/99 = 4/33
🧮 Worked Examples — Harder Exam Style
Show that 0.7̇2̇ can be written as 8/11.
Let x = 0.727272… (72 repeating)
Two repeating digits → multiply by 100: 100x = 72.7272…
Subtract: 100x − x = 72.72… − 0.72… = 72, so 99x = 72
x = 72/99 = ÷9 → 8/11 ✓
M1 for forming 100x, M1 for the subtraction 99x = 72, A1 for 8/11. In a 'show that' question the final line must END at the given answer — stop exactly there.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 0.35 = 35/100 → ÷5 → 7/20
Q2. 3 ÷ 20: 20 × 0.1 = 2, remainder 1 → 10/20 = 0.05 → total 0.15
Q3. 1 ÷ 6 = 0.1666… = 0.16 recurring (the 6 repeats)
Q4. Let x = 0.555… Then 10x = 5.555…, so 10x − x = 5, 9x = 5 → x = 5/9. The rule: one repeating digit → denominator 9.
Q5. Let x = 0.121212… Then 100x = 12.1212…, so 99x = 12 → x = 12/99 → ÷3 → 4/33. Two repeating digits → denominator 99.
Q6. 0.125 = 125/1000 → ÷125 → 1/8
Q7. 5 ÷ 8 = 0.625
📌 Key Facts to Memorise
- 0.5 = 1/2 · 0.25 = 1/4 · 0.75 = 3/4 · 0.2 = 1/5 · 0.1 = 1/10
- Eighths: 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875
- To convert: tenths → /10, hundredths → /100, thousandths → /1000, then simplify
- Recurring with one repeating digit → /9; two digits → /99 (0.5̇ = 5/9, 0.1̇2̇ = 12/99)
- "Simplest form" = divide top and bottom by the HCF
🗺️ In the Exam
Conversions: 1–2 marks on Paper 1, and the eighth-values are assumed inside fractions-of-amounts and percentages questions on all papers.
Links to: ordering (N1), percentages (N12/R9), recurring decimals (Higher N10).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always simplify fractions fully
- For terminating decimals: count decimal places, use that power of 10
- For recurring decimals: the number of repeating digits tells you what power of 10 to use
- Check: denominators with only 2s and 5s give terminating decimals
- Memorise common conversions: 1/3 = 0.̇3, 1/8 = 0.125, 1/9 = 0.̇1
🧠 Problem-Solving Strategies
Problem-Solving
When converting recurring decimals, set up an algebraic equation: let x equal the decimal, multiply by 10, 100 or 1000 (depending on repeat length), then subtract the original equation. For deciding if a fraction terminates, check the simplified denominator's prime factors — only 2s and 5s mean it terminates.
Multi-Step Problem
Convert 0.2̇7̇ to a fraction in its simplest form.
Solution: Let x = 0.272727... Multiply by 100: 100x = 27.2727... Subtract: 100x − x = 27, so 99x = 27, x = 27/99 = 3/11.
⚠️ Common Errors
Watch Out!
1. Wrong: 0.̇9 = 0.999... ≈ 1 (treating it as approximately 1) Correct: 0.̇9 = 1 exactly (let x = 0.̇9, 10x = 9.̇9, 9x = 9, x = 1)
2. Wrong: 0.1̇6 = 16/100 when converting (misidentifying the recurring part) Correct: 0.1̇6 means 0.1666..., so multiply by 10 to get 1.̇6, then by 10 again to get 16.̇6. Subtract: 90x = 15, x = 1/6
3. Wrong: 3/7 terminates because 7 is prime Correct: 7 is not a factor of 2 or 5, so 3/7 produces a recurring decimal
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Convert 0.̇3̇8 to a fraction in its simplest form. (b) Prove that 0.̇3̇8 is equal to 38/99. (c) Using your answer, or otherwise, convert 1.̇3̇8 to a mixed number.
(a) Let x = 0.̇3̇8. 100x = 38.̇3̇8. 100x − x = 38, so 99x = 38, x = 38/99. Check: 38 and 99 share no common factors (99 = 9 × 11, 38 = 2 × 19), so 38/99 is already simplest form.
(b) Shown above — 0.̇3̇8 = 38/99 by the algebraic method.
(c) 1.̇3̇8 = 1 + 0.̇3̇8 = 1 + 38/99 = 99/99 + 38/99 = 137/99 = 1 38/99.
Mark scheme: 2 marks for (a) with correct algebraic steps, 2 marks for (b) proof, 2 marks for (c) including conversion to mixed number
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student is trying to predict which fractions will produce terminating decimals.
(a) Will 7/40 produce a terminating decimal? Explain your reasoning.
(b) Give an example of a fraction with denominator 12 that recurs, and explain why.
(c) A student says "Any fraction with an even denominator will terminate." Is this claim correct? Justify your answer with an example.
Answers: (a) 40 = 2³ × 5. Only prime factors are 2 and 5, so yes, 7/40 terminates (= 0.175). (b) 5/12 recurs because 12 = 2² × 3, and the factor of 3 causes recurrence. 5/12 = 0.41̇6. (c) No — 1/6 has denominator 6 (which is even), but 6 = 2 × 3, and the factor of 3 causes it to recur (1/6 = 0.1̇6). Even denominators can still have prime factors other than 2 and 5.
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