Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in TN TET Mathematics. This topic tests your ability to perform arithmetic operations, convert between representations, and solve word problems—skills essential for teaching Classes 1–8. Questions typically appear as direct calculations or embedded in mensuration, ratio, and percentage problems.

Mastery here requires fluency in three areas: operations on fractions (addition, subtraction, multiplication, division), operations on decimals, and seamless conversion between fractions, decimals, and percentages. The pedagogy section also expects you to explain these concepts to young learners using visual models like number lines and area diagrams.

Expect 3–5 questions directly on this topic in Paper I and Paper II Mathematics sections. Speed and accuracy matter—most are calculation-based with close answer options designed to trap careless errors.

Key Concepts

• **Fraction** = Part/Whole. Numerator (top) counts parts taken; denominator (bottom) counts equal parts in the whole.

• **Types of fractions**: Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole number + proper fraction).

• **Equivalent fractions**: Multiply or divide both numerator and denominator by the same non-zero number. Example: 2/3 = 4/6 = 6/9.

• **Like fractions** share the same denominator; **unlike fractions** have different denominators and must be converted before adding/subtracting.

• **Decimal place values**: Tenths (1/10), hundredths (1/100), thousandths (1/1000). Moving decimal right multiplies by 10; moving left divides by 10.

• **Terminating decimals** end after finite digits (1/4 = 0.25). **Non-terminating repeating decimals** have a recurring pattern (1/3 = 0.333...).

• **Conversion rule**: Fraction to decimal—divide numerator by denominator. Decimal to fraction—write decimal over appropriate power of 10 and simplify.

Formulas / Key Facts

| Operation | Rule | |-----------|------| | Adding/Subtracting fractions | Make denominators same (LCM), then add/subtract numerators. | | Multiplying fractions | (a/b) × (c/d) = ac / bd | | Dividing fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) — multiply by reciprocal | | Mixed to improper | Whole × Denominator + Numerator, keep same denominator | | Decimal × 10ⁿ | Shift decimal point n places right | | Decimal ÷ 10ⁿ | Shift decimal point n places left | | Fraction to percentage | Multiply by 100% | | Decimal to percentage | Multiply by 100 |

**Key conversions to memorise:**

• 1/2 = 0.5 = 50%
• 1/4 = 0.25 = 25%
• 3/4 = 0.75 = 75%
• 1/5 = 0.2 = 20%
• 1/8 = 0.125 = 12.5%
• 1/3 ≈ 0.333 ≈ 33.33%

Worked Examples

### Example 1: Adding Unlike Fractions **Problem:** 3/4 + 5/6 = ?

**Solution:** 1. Find LCM of 4 and 6 → LCM = 12 2. Convert: 3/4 = 9/12; 5/6 = 10/12 3. Add numerators: 9 + 10 = 19 4. Result: 19/12 = 1 7/12

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### Example 2: Dividing Fractions **Problem:** 7/8 ÷ 14/16 = ?

**Solution:** 1. Reciprocal of 14/16 = 16/14 2. Multiply: (7/8) × (16/14) = (7 × 16) / (8 × 14) = 112/112 = 1 3. Answer: 1

*Shortcut check:* 14/16 simplifies to 7/8, so you're dividing 7/8 by itself = 1.

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### Example 3: Decimal Multiplication **Problem:** 2.5 × 0.04 = ?

**Solution:** 1. Ignore decimals: 25 × 4 = 100 2. Count total decimal places: 1 (in 2.5) + 2 (in 0.04) = 3 3. Place decimal in product: 100 → 0.100 = 0.1 4. Answer: 0.1

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### Example 4: Converting Recurring Decimal to Fraction **Problem:** Convert 0.666... to a fraction.

**Solution:** 1. Let x = 0.666... 2. Multiply by 10: 10x = 6.666... 3. Subtract: 10x − x = 6.666... − 0.666... → 9x = 6 4. Solve: x = 6/9 = 2/3 5. Answer: 2/3

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### Example 5: Word Problem **Problem:** A tank is 3/5 full. If 1/4 of the water is used, what fraction remains?

**Solution:** 1. Water used = 1/4 of 3/5 = (1/4) × (3/5) = 3/20 2. Water remaining = 3/5 − 3/20 3. Convert 3/5 = 12/20 4. Remaining = 12/20 − 3/20 = 9/20 5. Answer: 9/20

Common Mistakes

• **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 take reciprocal when dividing** → Students write (a/b) ÷ (c/d) = (a/b) × (c/d). Correct: Multiply by reciprocal (d/c).

• **Misplacing decimal point in multiplication/division** → Count decimal places carefully. In 0.3 × 0.3, there are 2 decimal places total, so answer = 0.09, not 0.9.

• **Not simplifying final answers** → 4/8 should be written as 1/2. Always reduce to lowest terms.

• **Confusing "of" with addition** → "1/3 of 12" means (1/3) × 12 = 4, not 1/3 + 12.

• **Wrong LCM calculation** → For 4 and 6, LCM is 12 (not 24). Use prime factorisation when unsure.

Quick Reference

• To add/subtract fractions: same denominator first, then operate on numerators only.

• To multiply fractions: straight across (numerator × numerator, denominator × denominator).

• To divide fractions: flip the second fraction, then multiply.

• Decimal places in product = sum of decimal places in both factors.

• Any fraction with denominator having only 2 and 5 as prime factors gives a terminating decimal.

• 1/3 = 0.333..., 1/6 = 0.1666..., 1/7 = 0.142857142857... (repeating block of 6 digits).