Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning at the primary level and appear consistently in TS TET Mathematics. This topic tests both your computational fluency and your ability to explain these concepts to young learners. Questions typically involve operations (addition, subtraction, multiplication, division), conversions between fractions and decimals, and word problems requiring application of these skills.

Mastery here is essential because fractions and decimals connect directly to percentages, ratios, and measurement—topics that build upon this foundation. As a prospective teacher, you must understand not just *how* to solve problems but *why* the procedures work, since pedagogy questions often ask about common student misconceptions and effective teaching strategies.

Expect 3–5 questions on this topic, ranging from straightforward computation to conceptual questions about equivalent forms and ordering.

Key Concepts

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

• **Types of fractions**: Proper fractions (numerator < denominator), improper fractions (numerator ≥ denominator), and mixed numbers (whole number + proper fraction) are interchangeable forms of the same value.

• **Equivalent fractions**: Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent fraction. Example: 2/3 = 4/6 = 6/9.

• **Decimal place value**: Each place to the right of the decimal point represents a power of ten—tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on.

• **Terminating vs repeating decimals**: Fractions with denominators whose only prime factors are 2 and 5 give terminating decimals. Others produce repeating (recurring) decimals.

• **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. Division by a fraction equals multiplication by its reciprocal.

Formulas / Key Facts

| Operation | Rule | |-----------|------| | Addition/Subtraction of fractions | Find LCM of denominators, convert to like fractions, then add/subtract numerators | | Multiplication of fractions | (a/b) × (c/d) = (a×c)/(b×d) | | Division of fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | Write decimal as fraction over power of 10, then simplify | | Mixed to improper | (whole × denominator + numerator)/denominator | | Improper to mixed | Divide numerator by denominator; quotient = whole part, 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... (repeating), 2/3 = 0.666... (repeating)

Worked Examples

**Example 1: Addition of unlike fractions**

Add 3/4 + 2/5.

Step 1: Find LCM of 4 and 5 = 20. Step 2: Convert each fraction.

• 3/4 = (3×5)/(4×5) = 15/20
• 2/5 = (2×4)/(5×4) = 8/20

Step 3: Add numerators: 15 + 8 = 23. Answer: 23/20 or 1 3/20.

---

**Example 2: Division of fractions**

Divide 5/6 by 2/3.

Step 1: Take reciprocal of divisor: 2/3 becomes 3/2. Step 2: Multiply: (5/6) × (3/2) = (5×3)/(6×2) = 15/12. Step 3: Simplify: 15/12 = 5/4 = 1 1/4. Answer: 5/4 or 1.25.

---

**Example 3: Decimal to fraction conversion**

Convert 0.375 to a fraction in lowest terms.

Step 1: Write as fraction over power of 10: 375/1000. Step 2: Find GCD of 375 and 1000. Both divisible by 125. Step 3: Simplify: 375÷125 = 3, 1000÷125 = 8. Answer: 3/8.

---

**Example 4: Ordering fractions and decimals**

Arrange in ascending order: 0.6, 5/8, 3/5.

Step 1: Convert all to decimals.

• 0.6 = 0.6
• 5/8 = 0.625
• 3/5 = 0.6

Step 2: Compare: 0.6 = 0.6 < 0.625. Answer: 3/5 = 0.6 < 5/8.

(Note: 0.6 and 3/5 are equal, so both come before 5/8.)

Common Mistakes

• **Adding fractions by adding numerators and denominators separately** → Wrong: 1/2 + 1/3 ≠ 2/5. Correct fix: Find common denominator first (1/2 + 1/3 = 3/6 + 2/6 = 5/6).

• **Forgetting to simplify the final answer** → Always reduce fractions to lowest terms by dividing by GCD. Exam marking may penalise unsimplified answers.

• **Misplacing the decimal point during multiplication/division** → Count total decimal places in factors for multiplication; adjust correctly when dividing. Example: 0.3 × 0.2 = 0.06 (not 0.6).

• **Confusing reciprocal with inverse sign** → Reciprocal of 3/4 is 4/3 (not −3/4). Reciprocal is about flipping, not changing signs.

• **Assuming all fractions give terminating decimals** → Only fractions whose simplified denominators have prime factors of only 2 and/or 5 terminate. Example: 1/6 = 0.1666... (repeating), not terminating.

• **Incorrectly converting mixed numbers before operations** → Always convert mixed numbers to improper fractions first when multiplying or dividing. Example: 2 1/2 × 3 = (5/2) × 3 = 15/2 = 7 1/2.

Quick Reference

• LCM of denominators is essential for adding/subtracting unlike fractions.
• Multiply fractions straight across: top × top, bottom × bottom.
• To divide fractions: Keep → Change → Flip (KCF method).
• Decimal places: tenths, hundredths, thousandths = 1, 2, 3 digits after point.
• Terminating decimals come from denominators with only 2s and 5s as prime factors.
• Always simplify fractions to lowest terms in final answers.