Fractions and Decimals
Overview
Fractions and decimals form the backbone of numerical reasoning in primary mathematics and appear consistently in GTET Paper-I and Paper-II. This topic tests your ability to perform arithmetic operations, convert between representations, and apply these concepts to word problems involving money, measurement, and daily-life situations.
For GTET, expect questions that combine multiple operations (e.g., adding fractions then converting to decimals), comparison problems, and application-based questions. Mastery here also supports topics like percentage, ratio-proportion, and mensuration. Focus on speed and accuracy—most errors come from careless mistakes in finding common denominators or misplacing decimal points.
Key Concepts
- **Fraction fundamentals**: A fraction a/b represents 'a' parts out of 'b' equal parts. The numerator (top) counts parts; the denominator (bottom) names the size of each part.
- **Types of fractions**: Proper fractions (numerator < denominator, e.g., 3/5), improper fractions (numerator ≥ denominator, e.g., 7/4), and mixed numbers (whole + fraction, e.g., 1¾).
- **Equivalent fractions**: Fractions that represent the same value (e.g., 2/4 = 1/2 = 3/6). Multiply or divide both numerator and denominator by the same non-zero number.
- **Lowest terms**: A fraction is in lowest terms when HCF of numerator and denominator is 1. Always simplify final answers.
- **Decimal place value**: Each position after the decimal point represents tenths, hundredths, thousandths, etc. In 0.375: 3 tenths + 7 hundredths + 5 thousandths.
- **Terminating vs recurring decimals**: Fractions with denominators having only 2 and 5 as prime factors give terminating decimals (e.g., 1/8 = 0.125). Others give recurring decimals (e.g., 1/3 = 0.333...).
- **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators and require conversion before addition/subtraction.
Formulas / Key Facts
| Operation | Formula/Method | |-----------|----------------| | Adding like fractions | a/c + b/c = (a+b)/c | | Subtracting like fractions | a/c − b/c = (a−b)/c | | Adding unlike fractions | Find LCM of denominators, convert, then add numerators | | Multiplying fractions | (a/b) × (c/d) = (a×c)/(b×d) | | Dividing fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) — multiply by reciprocal | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | Write decimal over appropriate power of 10, then simplify | | Mixed to improper | a b/c = (a×c + b)/c | | Improper to mixed | Divide numerator by denominator; quotient = whole part, remainder = new numerator |
**Quick conversions to memorise**:
- 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
- 1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8
- 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875
- 1/3 ≈ 0.333, 2/3 ≈ 0.667
Worked Examples
### Example 1: Adding Unlike Fractions **Problem**: Find 2/3 + 3/5
**Solution**: 1. Find LCM of 3 and 5 = 15 2. Convert: 2/3 = 10/15 (multiply by 5); 3/5 = 9/15 (multiply by 3) 3. Add: 10/15 + 9/15 = 19/15 4. Convert to mixed number: 19 ÷ 15 = 1 remainder 4 → **1 4/15**
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### Example 2: Multiplying and Dividing Fractions **Problem**: Simplify (3/4 × 2/5) ÷ 1/2
**Solution**: 1. First multiply: 3/4 × 2/5 = 6/20 = 3/10 2. Then divide: 3/10 ÷ 1/2 = 3/10 × 2/1 = 6/10 = **3/5**
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### Example 3: Decimal to Fraction Conversion **Problem**: Convert 0.375 to a fraction in lowest terms
**Solution**: 1. Write as fraction: 375/1000 2. Find HCF of 375 and 1000 = 125 3. Divide both: 375 ÷ 125 = 3; 1000 ÷ 125 = 8 4. Answer: **3/8**
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### Example 4: Word Problem **Problem**: A rope is 4.5 metres long. If 1 3/4 metres is cut off, what length remains? Express in decimals.
**Solution**: 1. Convert 1 3/4 to decimal: 1 + 0.75 = 1.75 m 2. Subtract: 4.5 − 1.75 = **2.75 metres**
Alternatively: 1. Convert 4.5 to fraction: 4 1/2 = 9/2 2. 1 3/4 = 7/4 3. Find common denominator (4): 9/2 = 18/4 4. Subtract: 18/4 − 7/4 = 11/4 = 2 3/4 = **2.75 m**
Common Mistakes
| Wrong Thinking | Correct Fix | |----------------|-------------| | Adding fractions by adding numerators AND denominators separately (2/3 + 1/4 = 3/7) | You must find a common denominator first. 2/3 + 1/4 = 8/12 + 3/12 = 11/12 | | Forgetting to take reciprocal when dividing fractions | Division means "multiply by the reciprocal." 2/3 ÷ 4/5 = 2/3 × 5/4, not 2/3 × 4/5 | | Misaligning decimal points during addition/subtraction | Always write decimals vertically with decimal points aligned. Add zeros as placeholders if needed (e.g., 3.5 + 2.75 → write 3.50 + 2.75) | | Converting 0.25 to 25/10 instead of 25/100 | Count decimal places: 2 places = hundredths. 0.25 = 25/100 = 1/4 | | Not simplifying final answers | Always reduce fractions to lowest terms. Check if numerator and denominator share common factors | | Confusing mixed number conversion | For 3 2/5: multiply 3×5=15, add 2 to get 17, keep denominator 5 → 17/5 (not 32/5) |
Quick Reference
- **Add/subtract fractions**: Same denominator required—use LCM
- **Multiply fractions**: Straight across (top × top, bottom × bottom), then simplify
- **Divide fractions**: Keep-Change-Flip (keep first, change ÷ to ×, flip second)
- **Decimal → Fraction**: Count decimal places, write over 10/100/1000, simplify
- **Fraction → Decimal**: Divide numerator by denominator
- **Mixed → Improper**: (whole × denominator) + numerator, keep denominator
- **Comparison trick**: Convert all to decimals or find common denominator to compare quickly