Fractions and Decimals
Overview
Fractions and decimals form the backbone of arithmetic competency at the primary level. For CG TET Paper I, this topic tests both your conceptual understanding and your ability to teach these ideas to Classes I–V students. Questions typically involve performing operations (addition, subtraction, multiplication, division) on fractions and decimals, converting between the two forms, and understanding word problems.
Mastery here is non-negotiable because fractions and decimals connect directly to percentage, ratio-proportion, and measurement—topics that appear throughout the syllabus.
Your goal: perform operations quickly and accurately, spot common student errors, and know age-appropriate teaching strategies.
Key Concepts
- Fraction as part of a whole: A fraction a/b means 'a' equal parts out of 'b' total equal parts. The denominator tells how many parts the whole is divided into; the numerator tells how many parts are taken.
- Types of fractions: Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole number + proper fraction). Example: 3/4 is proper; 7/4 is improper; 1¾ is mixed.
- Equivalent fractions: Fractions that represent the same value. Multiply or divide both numerator and denominator by the same non-zero number. Example: 2/3 = 4/6 = 6/9.
- Like and unlike fractions: Like fractions share the same denominator; unlike fractions do not. Converting unlike to like fractions requires finding the LCM of denominators.
- Decimal place value: Positions after the decimal point represent tenths (1/10), hundredths (1/100), thousandths (1/1000), etc. Example: 0.35 = 3 tenths + 5 hundredths.
- Fraction-decimal conversion: Divide numerator by denominator to get decimal. To convert decimal to fraction, place digits over the appropriate power of 10 and simplify.
- Comparing fractions/decimals: Convert to like fractions or to decimals, then compare numerators or digit values.
Formulas / Key Facts
| Operation | Rule | Example |
|---|---|---|
| Adding like fractions | a/c + b/c = (a+b)/c | 2/7 + 3/7 = 5/7 |
| Subtracting like fractions | a/c − b/c = (a−b)/c | 5/9 − 2/9 = 3/9 = 1/3 |
| Adding unlike fractions | Find LCM of denominators, convert, then add | 1/4 + 2/3 → LCM=12 → 3/12 + 8/12 = 11/12 |
| Multiplying fractions | (a/b) × (c/d) = ac/bd | 2/5 × 3/4 = 6/20 = 3/10 |
| Dividing fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) | 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 |
| Adding decimals | Align decimal points, add column-wise | 2.35 + 1.4 = 3.75 |
| Subtracting decimals | Align decimal points, borrow as needed | 5.20 − 2.75 = 2.45 |
| Multiplying decimals | Ignore decimals, multiply, count total decimal places in factors, place decimal in product | 1.2 × 0.3 = 0.36 (1+1=2 places) |
| Dividing decimals | Move decimal in divisor to make it whole; shift same places in dividend; divide | 4.5 ÷ 0.5 → 45 ÷ 5 = 9 |
Conversion shortcuts:
- 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125
- To convert 0.625 to fraction: 625/1000 = 5/8
Worked Examples
Example 1: Adding unlike fractions
Problem: 2/5 + 3/4 = ?
Solution:
- Find LCM of 5 and 4 → LCM = 20
- Convert: 2/5 = 8/20; 3/4 = 15/20
- Add: 8/20 + 15/20 = 23/20 = 1 3/20
Example 2: Multiplying decimals
Problem: 2.5 × 1.4 = ?
Solution:
- Ignore decimals: 25 × 14 = 350
- Count decimal places in factors: 1 + 1 = 2
- Place decimal: 3.50 → 3.5
Example 3: Dividing a fraction by a decimal
Problem: 3/4 ÷ 0.5 = ?
Solution:
- Convert 0.5 to fraction: 0.5 = 1/2
- Apply division rule: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 or 1.5
Example 4: Word problem
Problem: A rope is 4.8 m long. If 1.25 m is cut off, what length remains?
Solution:
- Align decimals: 4.80 − 1.25
- Subtract: 4.80 − 1.25 = 3.55 m
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Adding numerators and denominators directly (2/3 + 1/4 = 3/7) | Find LCM first, convert to like fractions, then add only numerators. |
| Forgetting to simplify the final answer | Always reduce fractions to lowest terms; check if GCD > 1. |
| Misaligning decimal points during addition/subtraction | Write numbers vertically with decimal points in a straight line; pad zeros if needed. |
| Placing decimal incorrectly in multiplication | Count total decimal places in both factors and mark from the right in the product. |
| Confusing "dividing by a fraction" with "multiplying by a fraction" | Remember: dividing by a/b means multiplying by b/a (reciprocal). |
Quick Reference
- Like fractions: same denominator → add/subtract numerators directly.
- Unlike fractions: convert via LCM before adding/subtracting.
- Multiply fractions: multiply tops, multiply bottoms, simplify.
- Divide fractions: multiply by the reciprocal of the divisor.
- Decimal multiplication: total decimal places in factors = decimal places in product.
- Decimal division: shift decimals equally in divisor and dividend to make divisor whole.
Pedagogy tip for CG TET: Use fraction strips, pizza models, and number lines to build conceptual understanding before procedural fluency—questions may ask which manipulative is best suited for a given learning objective.