Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical operations at the primary level and are heavily tested in Bihar TET Paper I Mathematics. This topic bridges whole-number arithmetic with more advanced concepts like ratios, percentages, and measurement—making it essential for both content knowledge and classroom teaching.

For the Bihar TET exam, you must demonstrate two competencies: solving problems involving fraction and decimal operations accurately, and understanding how children develop conceptual understanding of these number forms. Questions typically test conversion between fractions and decimals, ordering, and the four basic operations. Mastery here directly supports topics like percentage, ratio-proportion, and money calculations that appear elsewhere in the syllabus.

Students often struggle with fractions because they require a shift from counting discrete objects to understanding parts of a whole. Your role as a teacher—and as a TET candidate—is to understand both the mathematical procedures and the conceptual hurdles children face.

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 equal parts the whole is divided into; the numerator tells how many parts are taken.

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

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

• **Decimal as base-10 fraction**: Decimals extend the place-value system to parts smaller than one. Each place to the right of the decimal point represents division by 10 (tenths, hundredths, thousandths).

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

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators and require conversion before addition or subtraction.

• **Comparing fractions and decimals**: Convert to common denominators (for fractions) or align decimal places (for decimals) to compare values accurately.

Formulas / Key Facts

| Operation | Rule | |-----------|------| | **Adding like fractions** | a/c + b/c = (a+b)/c | | **Subtracting like fractions** | a/c − b/c = (a−b)/c | | **Adding unlike fractions** | Find LCM of denominators, convert, then add numerators | | **Multiplying fractions** | a/b × c/d = (a×c)/(b×d) | | **Dividing fractions** | a/b ÷ c/d = a/b × d/c (multiply by reciprocal) | | **Decimal to fraction** | Write digits as numerator; denominator = 10, 100, 1000 based on decimal places; simplify | | **Fraction to decimal** | Divide numerator by denominator | | **Adding/subtracting decimals** | Align decimal points, then add/subtract as whole numbers | | **Multiplying decimals** | Multiply as whole numbers; count total decimal places in both factors; place decimal in product | | **Dividing decimals** | Make divisor a whole number by shifting decimal; shift same places in dividend; divide |

**Key equivalences to memorize**:

• 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, 1/10 = 0.1, 1/100 = 0.01

Worked Examples

**Example 1: Adding unlike fractions** Find: 2/3 + 3/4

Step 1: Find LCM of 3 and 4 → LCM = 12 Step 2: Convert to equivalent fractions

• 2/3 = 8/12 (multiply both by 4)
• 3/4 = 9/12 (multiply both by 3)

Step 3: Add numerators → 8/12 + 9/12 = 17/12 Step 4: Convert to mixed number → 1 5/12

**Answer: 1 5/12 or 17/12**

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**Example 2: Multiplying a fraction by a decimal** Find: 3/5 × 0.4

Method 1 (Convert decimal to fraction):

• 0.4 = 4/10 = 2/5
• 3/5 × 2/5 = 6/25

Method 2 (Convert fraction to decimal):

• 3/5 = 0.6
• 0.6 × 0.4 = 0.24
• Verify: 0.24 = 24/100 = 6/25 ✓

**Answer: 6/25 or 0.24**

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**Example 3: Dividing decimals** Find: 4.56 ÷ 0.08

Step 1: Make divisor a whole number by multiplying both by 100

• 4.56 × 100 = 456
• 0.08 × 100 = 8

Step 2: Divide → 456 ÷ 8 = 57

**Answer: 57**

Common Mistakes

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

• **Forgetting to find the reciprocal when dividing fractions** → Wrong: 2/3 ÷ 4/5 = 8/15. Correct: Flip the second fraction and multiply. (2/3 × 5/4 = 10/12 = 5/6)

• **Misaligning decimal points during addition/subtraction** → Wrong: 3.5 + 0.25 = 3.75 by placing digits incorrectly. Correct: Write as 3.50 + 0.25 = 3.75, aligning the decimal points vertically.

• **Incorrect decimal placement in multiplication** → When multiplying 0.3 × 0.2, students write 0.6 instead of 0.06. Correct: Count total decimal places (1+1=2) and place decimal accordingly.

• **Confusing "of" with addition in word problems** → "1/2 of 24" means multiplication (1/2 × 24 = 12), not addition. The word "of" signals multiplication with fractions.

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 multiplication**: Total decimal places in factors = decimal places in product.
• **0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4, 0.2 = 1/5** — memorize these for speed.
• **Simplify fractions by dividing both terms by their HCF.**