Fractions and Decimals
Overview
Fractions and decimals form the backbone of numerical literacy at the primary level and appear consistently in KTET Mathematics sections across Categories I, II, and III. This topic tests both your computational accuracy and your conceptual understanding of how parts of a whole can be represented in different forms.
For KTET, you must master three core skills: performing operations (addition, subtraction, multiplication, division) on fractions and decimals, converting between fractions and decimals fluently, and applying these concepts to word problems involving money, measurement, and daily-life situations. Questions typically range from straightforward calculations to multi-step problems requiring conversion mid-solution.
Understanding this topic deeply also supports the pedagogy section, where you may need to explain how to teach fraction concepts using visual models, manipulatives, or real-world contexts to young learners.
Key Concepts
- **Fraction as part-whole relationship**: A fraction a/b represents 'a' equal parts out of 'b' total parts. The denominator tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.
- **Types of fractions**: Proper fractions (numerator < denominator, e.g., 3/5), improper fractions (numerator ≥ denominator, e.g., 7/4), and mixed numbers (whole + proper fraction, e.g., 1¾).
- **Equivalent fractions**: Fractions representing the same value, obtained by multiplying or dividing both numerator and denominator by the same non-zero number. Example: 2/3 = 4/6 = 6/9.
- **Decimal as base-10 fraction**: Decimals are fractions with denominators of 10, 100, 1000, etc. The decimal 0.25 means 25/100.
- **Place value in decimals**: Tenths (1/10), hundredths (1/100), thousandths (1/1000) — each place to the right of the decimal point represents division by 10.
- **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators and require a common denominator for addition/subtraction.
- **Reciprocal**: The reciprocal of a/b is b/a. Used when dividing fractions — "invert and multiply."
Formulas / Key Facts
| Operation | Rule | |-----------|------| | Addition/Subtraction of fractions | Make denominators equal (LCM), then add/subtract numerators | | Multiplication of fractions | (a/b) × (c/d) = ac/bd | | Division of fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | Write decimal over appropriate power of 10, then simplify | | Mixed to improper | (whole × denominator) + numerator, over same denominator | | Improper to mixed | Divide numerator by denominator; quotient = whole, remainder = new numerator |
**Key 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... (recurring), 2/3 = 0.666... (recurring)
Worked Examples
**Example 1: Adding unlike fractions**
Find: 2/5 + 3/4
Step 1: Find LCM of 5 and 4 = 20
Step 2: Convert to equivalent fractions
- 2/5 = 8/20
- 3/4 = 15/20
Step 3: Add numerators = 8 + 15 = 23
Answer: 23/20 = 1 3/20
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**Example 2: Dividing fractions**
Find: 3/7 ÷ 2/5
Step 1: Take reciprocal of divisor: 2/5 becomes 5/2
Step 2: Multiply: (3/7) × (5/2) = 15/14
Answer: 15/14 = 1 1/14
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**Example 3: Converting recurring decimal to fraction**
Convert 0.666... to a fraction.
Let x = 0.666...
Multiply by 10: 10x = 6.666...
Subtract: 10x − x = 6.666... − 0.666...
9x = 6
x = 6/9 = 2/3
Answer: 2/3
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**Example 4: Word problem with decimals**
A ribbon is 3.75 metres long. If 1.28 metres is cut off, how much remains?
3.75 − 1.28:
- Align decimal points
- 3.75 − 1.28 = 2.47 metres
Answer: 2.47 metres
Common Mistakes
- **Adding fractions by adding numerators AND denominators separately**: Wrong: 1/2 + 1/3 = 2/5. Correct: Find common denominator first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- **Forgetting to simplify final answers**: Many students leave 4/8 instead of reducing to 1/2. Always check if numerator and denominator share common factors.
- **Misplacing decimal point in multiplication**: When multiplying 0.3 × 0.2, students may write 0.6 instead of 0.06. Count total decimal places in both numbers (1+1=2) and place decimal accordingly.
- **Not converting mixed numbers before multiplying/dividing**: You cannot directly multiply 2½ × 1⅓. First convert: 5/2 × 4/3 = 20/6 = 10/3 = 3⅓.
- **Confusing "of" with addition in word problems**: "1/3 of 60" means multiplication (1/3 × 60 = 20), not addition.
Quick Reference
- To add/subtract fractions → same denominator first (use LCM)
- To multiply fractions → multiply across (numerator × numerator, denominator × denominator)
- To divide fractions → multiply by reciprocal of divisor
- Decimal to fraction → place over 10/100/1000, then reduce
- Fraction to decimal → numerator ÷ denominator
- Mixed to improper → (whole × denominator) + numerator over denominator