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:
- Find LCM of 3 and 5 = 15
- Convert: 2/3 = 10/15 (multiply by 5); 3/5 = 9/15 (multiply by 3)
- Add: 10/15 + 9/15 = 19/15
- Convert to mixed number: 19 ÷ 15 = 1 remainder 4 → 1 4/15
Example 2: Multiplying and Dividing Fractions
Problem: Simplify (3/4 × 2/5) ÷ 1/2
Solution:
- First multiply: 3/4 × 2/5 = 6/20 = 3/10
- Then divide: 3/10 ÷ 1/2 = 3/10 × 2/1 = 6/10 = 3/5
Example 3: Decimal to Fraction Conversion
Problem: Convert 0.375 to a fraction in lowest terms
Solution:
- Write as fraction: 375/1000
- Find HCF of 375 and 1000 = 125
- Divide both: 375 ÷ 125 = 3; 1000 ÷ 125 = 8
- Answer: 3/8
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:
- Convert 1 3/4 to decimal: 1 + 0.75 = 1.75 m
- Subtract: 4.5 − 1.75 = 2.75 metres
Alternatively:
- Convert 4.5 to fraction: 4 1/2 = 9/2
- 1 3/4 = 7/4
- Find common denominator (4): 9/2 = 18/4
- 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