Fractions and Decimals
Overview
Fractions and decimals form the bridge between whole numbers and the more abstract number concepts students encounter later. For UTET Paper I, this topic carries significant weight because it tests both your mathematical understanding and your ability to teach these concepts to Classes I-V students. Examiners frequently draw questions from equivalent fractions, comparison of fractions, and conversion between fractions and decimals.
At the primary level, fractions are introduced through concrete experiences—sharing a roti equally, dividing a chocolate bar, or folding paper. Decimals enter around Class IV-V through money (₹25.50) and measurement (1.5 metres). Your task as a teacher is to connect these abstract symbols to real-life situations children already understand. UTET questions test whether you grasp these foundational ideas clearly enough to explain and apply them.
Mastery here requires understanding why procedures work, not just how to perform them. Questions often present scenarios where students make typical errors, asking you to identify the misconception or suggest the correct teaching approach.
Key Concepts
- **Fraction as part of a whole**: A fraction represents equal parts of a whole. In 3/4, the whole is divided into 4 equal parts, and we consider 3 of them. The whole must be divided into *equal* parts—this is the most fundamental idea.
- **Numerator and denominator**: The numerator (top number) tells how many parts are taken; the denominator (bottom number) tells how many equal parts the whole is divided into. The denominator can never be zero.
- **Equivalent fractions**: Different fractions can represent the same quantity. 1/2 = 2/4 = 3/6 = 4/8. Multiplying or dividing both numerator and denominator by the same non-zero number produces an equivalent fraction.
- **Like and unlike fractions**: Like fractions have the same denominator (2/7, 5/7); unlike fractions have different denominators (2/3, 3/4). Comparison and addition/subtraction of unlike fractions require converting them to like fractions.
- **Proper, improper and mixed fractions**: Proper fraction has numerator < denominator (3/5). Improper fraction has numerator ≥ denominator (7/4). Mixed fraction combines a whole number and proper fraction (1¾).
- **Decimal as an extension of place value**: Decimals use place value to represent parts smaller than one. The places after the decimal point are tenths (1/10), hundredths (1/100), thousandths (1/1000), moving right.
- **Fraction-decimal relationship**: Every fraction can be written as a decimal by dividing numerator by denominator. Common equivalents: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2.
- **Comparison of decimals**: Compare digit by digit from left. 0.45 > 0.39 because 4 tenths > 3 tenths. Adding trailing zeros helps: 0.5 = 0.50, so comparing 0.5 and 0.45 becomes comparing 0.50 and 0.45.
Formulas / Key Facts
| Concept | Key Fact | |---------|----------| | Equivalent fractions | a/b = (a×k)/(b×k) for any non-zero k | | Simplest form | Divide both numerator and denominator by their HCF | | Mixed to improper | a b/c = (a×c + b)/c | | Improper to mixed | Divide numerator by denominator; quotient is whole part, remainder is new numerator | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | 0.25 = 25/100 = 1/4 (write decimal as fraction over power of 10, then simplify) | | Comparing fractions | Cross-multiply: a/b ? c/d → compare a×d with b×c | | Adding unlike fractions | Find LCM of denominators, convert to like fractions, then add numerators | | Place values | Ones . Tenths Hundredths Thousandths (1 . 1/10 . 1/100 . 1/1000) |
Worked Examples
**Example 1: Finding equivalent fractions**
*Find three fractions equivalent to 2/5.*
Multiply numerator and denominator by 2: 2/5 = 4/10 Multiply by 3: 2/5 = 6/15 Multiply by 4: 2/5 = 8/20
Answer: 4/10, 6/15, 8/20
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**Example 2: Converting mixed fraction to improper**
*Convert 3²/₇ to an improper fraction.*
Step 1: Multiply whole number by denominator: 3 × 7 = 21 Step 2: Add the numerator: 21 + 2 = 23 Step 3: Write over original denominator: 23/7
Answer: 23/7
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**Example 3: Converting decimal to fraction**
*Express 0.125 as a fraction in simplest form.*
Step 1: Write as fraction over power of 10. Since three decimal places: 0.125 = 125/1000 Step 2: Find HCF of 125 and 1000. HCF = 125 Step 3: Divide both by 125: 125÷125 = 1, 1000÷125 = 8
Answer: 1/8
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**Example 4: Comparing fractions**
*Which is greater: 5/8 or 7/12?*
Cross-multiply: 5 × 12 = 60 and 8 × 7 = 56 Since 60 > 56, we have 5/8 > 7/12
Answer: 5/8 is greater
Common Mistakes
- **Thinking larger denominator means larger fraction** → Students believe 1/8 > 1/4 because 8 > 4. Correct understanding: larger denominator means smaller pieces, so 1/8 < 1/4. Use visual models to demonstrate.
- **Adding fractions by adding numerators and denominators separately** → Students write 1/2 + 1/3 = 2/5. This is incorrect. The correct method requires finding a common denominator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- **Ignoring place value in decimals** → Students think 0.15 > 0.9 because 15 > 9. Correct approach: compare digit by digit from left, or write 0.9 as 0.90. Then 0.90 > 0.15 is clear.
- **Forgetting to simplify final answers** → Writing 4/8 instead of 1/2 as final answer. Always check if numerator and denominator share common factors.
- **Converting mixed fractions incorrectly** → Adding instead of multiplying: writing 2³/₄ as (2+3)/4 = 5/4 instead of correct (2×4+3)/4 = 11/4. Emphasise the procedure step by step.
- **Misplacing decimal point during conversion** → Writing 3/4 = 0.34 instead of 0.75. Reinforce that fraction means division: 3 ÷ 4 = 0.75.
Quick Reference
- Fraction = numerator/denominator; denominator ≠ 0
- Equivalent fractions: multiply or divide top and bottom by the same number
- Mixed to improper: (whole × denominator + numerator)/denominator
- Decimal places: tenths, hundredths, thousandths (moving right from decimal point)
- To compare unlike fractions: cross-multiply or convert to same denominator
- 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125