Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in primary and upper-primary mathematics. For UPTET, this topic appears consistently across both Paper I (Classes 1–5) and Paper II (Classes 6–8), testing your ability to perform operations, convert between forms, and solve word problems. Questions typically range from straightforward computation to application-based problems involving money, measurement, and comparison.

Mastery here is non-negotiable because fractions and decimals underpin later topics—ratio and proportion, percentage, profit-loss, and mensuration. A teacher must understand not just the procedures but also the conceptual models (part-whole, division interpretation) to explain these ideas effectively to children. Expect 3–5 direct questions in the Mathematics section, plus indirect use in other problems.

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Key Concepts

• **Fraction as part-whole**: A fraction a/b represents 'a' equal parts out of 'b' total equal parts of a whole. The denominator tells how many equal parts; the numerator tells how many are taken.

• **Proper fraction**: Numerator < Denominator (e.g., 3/7). Value is always less than 1.

• **Improper fraction**: Numerator ≥ Denominator (e.g., 9/4). Value is 1 or greater.

• **Mixed fraction**: Combination of a whole number and a proper fraction (e.g., 2¼). Every improper fraction can be written as a mixed fraction and vice versa.

• **Equivalent fractions**: Fractions representing the same value (e.g., 1/2 = 2/4 = 3/6). Obtained by multiplying or dividing numerator and denominator by the same non-zero number.

• **Decimal as a fraction with denominator 10, 100, 1000, etc.**: 0.3 = 3/10; 0.47 = 47/100; 2.135 = 2135/1000.

• **Place value in decimals**: Tenths (1/10), hundredths (1/100), thousandths (1/1000) moving right from the decimal point.

• **Like and unlike fractions**: Like fractions have the same denominator; unlike fractions have different denominators. Converting to like fractions is essential before adding or subtracting.

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Formulas / Key Facts

| Operation | Rule | |-----------|------| | **Converting improper to mixed** | Divide numerator by denominator → Quotient = whole part, Remainder = numerator of fractional part. E.g., 17/5 = 3 and 2/5. | | **Converting mixed to improper** | (Whole × Denominator) + Numerator, keep same denominator. E.g., 4 and 3/7 = (4×7+3)/7 = 31/7. | | **Addition/Subtraction of fractions** | Make denominators equal (LCM), then add/subtract numerators. | | **Multiplication of fractions** | (a/b) × (c/d) = ac/bd. Simplify before or after multiplying. | | **Division of fractions** | (a/b) ÷ (c/d) = (a/b) × (d/c). Multiply by the reciprocal. | | **Decimal to fraction** | Write digits after point over 10, 100, etc., then simplify. 0.75 = 75/100 = 3/4. | | **Fraction to decimal** | Divide numerator by denominator. 3/8 = 0.375. | | **Adding/Subtracting decimals** | Align decimal points, then add/subtract column-wise. | | **Multiplying decimals** | Multiply as whole numbers, count total decimal places in both factors, place decimal in product accordingly. | | **Dividing decimals** | Shift decimal in divisor to make it whole; shift same places in dividend; then divide. |

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Worked Examples

### Example 1: Convert and Compare

**Problem**: Arrange in ascending order: 5/6, 7/9, 3/4.

**Solution**: 1. Find LCM of denominators 6, 9, 4 → LCM = 36. 2. Convert each: 5/6 = 30/36; 7/9 = 28/36; 3/4 = 27/36. 3. Compare numerators: 27 < 28 < 30. 4. **Ascending order**: 3/4 < 7/9 < 5/6.

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### Example 2: Mixed Fraction Operations

**Problem**: Simplify 3 and 2/5 + 2 and 3/4.

**Solution**: 1. Convert to improper fractions: 3 and 2/5 = 17/5; 2 and 3/4 = 11/4. 2. LCM of 5 and 4 = 20. 3. 17/5 = 68/20; 11/4 = 55/20. 4. Add: 68/20 + 55/20 = 123/20. 5. Convert back: 123 ÷ 20 = 6 remainder 3 → **6 and 3/20**.

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### Example 3: Decimal Division

**Problem**: Divide 4.56 by 0.8.

**Solution**: 1. Make divisor a whole number: multiply both by 10 → 45.6 ÷ 8. 2. Divide: 45.6 ÷ 8 = 5.7. 3. **Answer**: 5.7.

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### Example 4: Word Problem

**Problem**: A rope 8.4 m long is cut into pieces of 0.7 m each. How many pieces?

**Solution**: 1. Number of pieces = 8.4 ÷ 0.7. 2. Shift decimals: 84 ÷ 7 = 12. 3. **Answer**: 12 pieces.

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Adding fractions by adding numerators AND denominators separately (e.g., 1/2 + 1/3 = 2/5). | Find LCM of denominators first, convert to like fractions, then add only numerators. Correct: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. | | Forgetting to simplify final answers. | Always reduce fractions to lowest terms by dividing by HCF. | | Placing decimal incorrectly in multiplication (counting places in only one factor). | Count decimal places in BOTH factors combined and place decimal that many places from the right in the product. | | Treating 0.5 as smaller than 0.45 because "45 > 5". | Compare digit by digit from left OR convert to same number of decimal places: 0.50 vs 0.45 → 50 > 45, so 0.5 > 0.45. | | In division, forgetting to shift decimal in dividend when shifting in divisor. | Always shift BOTH by the same number of places to keep the quotient unchanged. |

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Quick Reference

• **Proper**: numerator < denominator; **Improper**: numerator ≥ denominator.
• Mixed to improper: (Whole × Denom) + Num over Denom.
• Fraction division = multiply by reciprocal.
• Decimal places rule: 0.3 × 0.02 = 0.006 (1 + 2 = 3 decimal places).
• To compare fractions, convert to like fractions or to decimals.
• 1/2 = 0.5; 1/4 = 0.25; 1/5 = 0.2; 3/4 = 0.75 — memorise these for speed.