Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical operations at the primary level and are heavily tested in JTET Paper I Mathematics. This topic bridges whole number arithmetic with more advanced concepts like percentages, ratio-proportion, and measurement—all of which appear frequently in the exam.

For JTET, you must master not just mechanical calculations but also conceptual understanding and common student misconceptions. Questions typically test conversion between fractions and decimals, ordering, and word problems involving all four operations. The pedagogy section often asks how to teach these concepts using concrete materials like fraction strips or decimal squares.

A strong grasp here directly supports your performance in Percentage, Profit-Loss, and Mensuration topics. Expect 3–5 direct questions plus indirect applications across the mathematics section.

Key Concepts

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

• **Types of fractions**: Proper fractions (numerator < denominator, e.g., 3/5), improper fractions (numerator ≥ denominator, e.g., 7/4), and mixed numbers (whole + proper fraction, e.g., 1¾).

• **Equivalent fractions**: Fractions that represent the same value—multiply or divide both numerator and denominator by the same non-zero number (2/3 = 4/6 = 6/9).

• **Decimal as extension of place value**: Decimals use places to the right of the decimal point—tenths (1/10), hundredths (1/100), thousandths (1/1000).

• **Fraction-decimal relationship**: Every fraction can be written as a decimal by dividing numerator by denominator. Terminating decimals have denominators with only 2 and 5 as prime factors.

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions require finding LCD (Least Common Denominator) before addition or subtraction.

• **Comparison rule**: For like fractions, compare numerators. For unlike fractions, convert to like fractions first or use cross-multiplication.

Formulas / Key Facts

| Operation | Fractions | Decimals | |-----------|-----------|----------| | **Addition/Subtraction** | Convert to like fractions, then add/subtract numerators | Align decimal points, then add/subtract as whole numbers | | **Multiplication** | (a/b) × (c/d) = ac/bd | Multiply as whole numbers, count total decimal places in both factors | | **Division** | (a/b) ÷ (c/d) = (a/b) × (d/c) | Make divisor a whole number by shifting decimal in both numbers equally |

**Key Conversions:**

• Fraction to decimal: Divide numerator by denominator
• Decimal to fraction: Write decimal as fraction over power of 10, then simplify
• 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125

**LCD Method**: Find LCM of denominators to convert unlike fractions to like fractions.

**Mixed Number Conversion**: a b/c = (a×c + b)/c

Worked Examples

### Example 1: Adding Unlike Fractions **Problem**: Add 2/3 + 3/4

**Solution**:

• Step 1: Find LCD of 3 and 4 → LCM = 12
• Step 2: Convert fractions → 2/3 = 8/12 and 3/4 = 9/12
• Step 3: Add numerators → 8/12 + 9/12 = 17/12
• Step 4: Convert to mixed number → 17/12 = 1 5/12

**Answer**: 1 5/12

### Example 2: Multiplying Decimals **Problem**: Calculate 2.5 × 0.04

**Solution**:

• Step 1: Multiply as whole numbers → 25 × 4 = 100
• Step 2: Count decimal places → 2.5 has 1 place, 0.04 has 2 places → Total = 3 places
• Step 3: Place decimal point 3 places from right → 100 becomes 0.100 = 0.1

**Answer**: 0.1

### Example 3: Word Problem **Problem**: A rope is 4.5 metres long. If 1.75 metres is cut off, and the remaining rope is divided into 5 equal pieces, what is the length of each piece?

**Solution**:

• Step 1: Remaining rope = 4.5 − 1.75 = 2.75 metres
• Step 2: Length of each piece = 2.75 ÷ 5 = 0.55 metres

**Answer**: 0.55 metres (or 55 cm)

Common Mistakes

• **Adding numerators and denominators separately** → Wrong: 1/2 + 1/3 = 2/5. Correct: Find LCD first, then 3/6 + 2/6 = 5/6.

• **Forgetting to simplify the final answer** → Always reduce fractions to lowest terms. 4/8 should be written as 1/2.

• **Misaligning decimal points in addition/subtraction** → Wrong: Adding 3.5 + 0.25 by aligning last digits. Correct: Align decimal points vertically (3.50 + 0.25 = 3.75).

• **Placing decimal incorrectly in multiplication** → Students often place the decimal by guessing. Correct method: Count total decimal places in both factors.

• **Inverting the wrong fraction in division** → In a/b ÷ c/d, invert only the second fraction (divisor), not the first.

• **Treating decimals as separate whole numbers** → Thinking 0.15 > 0.9 because 15 > 9. Correct: Compare place by place—0.9 = 0.90 > 0.15.

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: Keep-Change-Flip (keep first, change ÷ to ×, flip second).
• Decimal places in product = Sum of decimal places in both factors.
• For division by decimal: Move decimal point equally in divisor and dividend.
• Always simplify fractions and remove trailing zeros in decimals (0.50 → 0.5).