Quick Revision Notes: Class 6 Mathematics

Chapter: Fractions

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1. What is a Fraction? (भिन्न क्या है?)

A fraction represents a part of a whole. When a whole thing is divided into equal parts, each part is called a fraction.

• Notation: Written as a/b, where a and b are whole numbers and b ≠ 0.
• Numerator (अंश): The top number a, which shows how many parts we have taken.
• Denominator (हर): The bottom number b, which shows the total number of equal parts into which the whole is divided.

Example: In the fraction 3/5,

• 3 is the numerator.
• 5 is the denominator.

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2. Types of Fractions (भिन्न के प्रकार)

• Proper Fraction (उचित भिन्न): A fraction where the numerator is less than the denominator (Numerator < Denominator).

  • Value is always less than 1.
  • Example: 2/3, 5/7, 11/15

• Improper Fraction (विषम भिन्न): A fraction where the numerator is greater than or equal to the denominator (Numerator ≥ Denominator).

  • Value is 1 or greater than 1.
  • Example: 5/3, 7/4, 9/9

• Mixed Fraction (मिश्रित भिन्न): A combination of a whole number and a proper fraction.

  • Example: 1 (2/3), 2 (1/4)

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3. Converting Between Mixed and Improper Fractions

• Converting Mixed Fraction to Improper Fraction: $$\text{Improper Fraction} = \frac{(\text{Whole} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$$

  • Example: Convert 2 (3/4) into an improper fraction. $$(2 \times 4) + 3 = 8 + 3 = 11 \implies \frac{11}{4}$$

• Converting Improper Fraction to Mixed Fraction:

  • Divide the numerator by the denominator.
  • Quotient becomes the whole number, Remainder becomes the new numerator, and the Denominator remains the same.
  • Formula: $\text{Quotient } \frac{\text{Remainder}}{\text{Denominator}}$
  • Example: Convert 11/4 into a mixed fraction.
  • $11 \div 4 \rightarrow \text{Quotient } = 2, \text{ Remainder } = 3 \implies 2 \frac{3}{4}$

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4. Equivalent Fractions (तुल्य भिन्न)

Fractions that represent the same part or value of a whole are called equivalent fractions.

• How to find: Multiply or divide both the numerator and the denominator by the same non-zero number.
• Example: Find two equivalent fractions of 2/3.
  • Multiply by 2: (2 × 2) / (3 × 2) = 4/6
  • Multiply by 3: (2 × 3) / (3 × 3) = 6/9
  • So, 2/3, 4/6, and 6/9 are equivalent fractions.

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5. Simplest Form of a Fraction (सरलतम रूप / न्यूनतम रूप)

A fraction is said to be in the simplest (or lowest) form if its numerator and denominator have no common factor other than 1 (i.e., their HCF is 1).

• Method: Divide the numerator and denominator by their HCF.
• Example: Reduce 15/45 to its simplest form.
  • HCF of 15 and 45 is 15.
  • (15 ÷ 15) / (45 ÷ 15) = 1/3

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6. Like and Unlike Fractions (समान और असमान भिन्न)

• Like Fractions (समान भिन्न): Fractions with the same denominators.
  • Example: 1/7, 3/7, 5/7, 6/7
• Unlike Fractions (असमान भिन्न): Fractions with different denominators.
  • Example: 1/3, 2/5, 4/7

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7. Comparing Fractions (भिन्न की तुलना)

• Comparing Like Fractions: The fraction with the greater numerator is greater.

  • Example: 5/9 > 2/9 (since $5 > 2$)

• Comparing Unlike Fractions:

  1. Make the denominators same by finding the LCM of the denominators.
  2. Convert them into like fractions.
  3. Compare their numerators.

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8. Addition and Subtraction of Fractions (भिन्न का जोड़ और घटाना)

• Case 1: Addition/Subtraction of Like Fractions $$\text{Result} = \frac{\text{Sum or Difference of Numerators}}{\text{Common Denominator}}$$

  • Example: 2/7 + 3/7 = (2 + 3) / 7 = 5/7

• Case 2: Addition/Subtraction of Unlike Fractions

  1. Find the LCM of the denominators.
  2. Convert each fraction into an equivalent fraction with the LCM as the denominator.
  3. Add or subtract the like fractions obtained.
  • Example: 1/2 + 1/3
    • LCM of 2 and 3 is 6.
    • 1/2 = 3/6 and 1/3 = 2/6
    • 3/6 + 2/6 = 5/6