Fractions, Decimals and Percentages
O Level and IGCSE Mathematics Cheat Sheet – 1.4 Fractions, Decimals and Percentages
Key Definitions
| Term | Definition | Example |
|---|---|---|
| Proper Fraction | Numerator < Denominator | 3/4 |
| Improper Fraction | Numerator ≥ Denominator | 9/4 |
| Mixed Number | Whole number + proper fraction | 2 ¹/₂ |
| Decimal | Number with a decimal point | 3.75 |
| Percentage (%) | Number out of 100 | 25% = 25/100 |
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Conversions Between Forms
| From → To | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide numerator by denominator | 3/4 = 0.75 |
| Decimal → Fraction | Place over 10, 100, 1000, simplify | 0.6 = 6/10 = 3/5 |
| Fraction → Percentage | Multiply by 100% | 3/4 × 100% = 75% |
| Percentage → Fraction | Divide by 100, simplify | 80% = 80/100 = 4/5 |
| Decimal → Percentage | Multiply by 100% | 0.25 × 100% = 25% |
| Percentage → Decimal | Divide by 100 | 45% = 0.45 |
Recurring Decimal Notation
| Decimal | Fraction Form |
|---|---|
| 0.1 recurring (0.‾1) | 1/9 |
| 0.‾7 | 7/9 |
| 0.‾3 | 1/3 |
| 0.‾123 | 123/999 = 41/333 |
| 0.1‾7 (only 7 repeats) | (17 − 1) / 90 = 16/90 = 8/45 |
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Converting Recurring Decimal to Fraction – Method
Example: Convert 0.‾7 to a fraction.
Let x = 0.‾7
Multiply by 10 → 10x = 7.‾7
Subtract: (10x − x) = 7.‾7 − 0.‾7
9x = 7
x = 7/9
Simplifying Fractions
- Divide numerator and denominator by their HCF.
Example: 18/24 → ÷6 → 3/4
Mixed Numbers and Improper Fractions
- Mixed → Improper: (Whole × Denominator + Numerator) / Denominator
Example: 2 ¹/₂ = (2 × 2 + 1)/2 = 5/2 - Improper → Mixed: Divide numerator by denominator.
Example: 9/4 = 2 ¹/₄
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Exam Special Tips
- Always simplify fractions to lowest terms unless told otherwise.
- For recurring decimals, identify repeating part clearly before converting.
- Use fraction form for exact answers in non-calculator papers.
- When percentage >100%, convert normally (e.g., 250% = 2.5).
Common Mistakes & Confusions
- Writing 1.5 as 1 ½ (wrong) — one is decimal, the other is mixed fraction.
- Forgetting to simplify fraction answers.
- Confusing 0.‾6 (which is 2/3) with 0.6 (which is 3/5).
- Using percentage sign twice (e.g., writing 0.5% as 0.005%).