Class 6 Notes

Class 6 Mathematics Notes Chapter 7 (Fractions) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Detailed Notes with MCQs of Chapter 7, 'Fractions'. This is a fundamental chapter, and understanding it well is crucial not just for your Class 6 exams but also forms the base for many concepts you'll encounter in higher classes and competitive government exams. Pay close attention!

Chapter 7: Fractions - Detailed Notes for Exam Preparation

1. What is a Fraction?

  • A fraction represents a part of a whole. The whole can be a single object (like a cake, a pizza) or a collection of objects (like 10 marbles).
  • It is written in the form Numerator / Denominator.
    • Numerator: The top number. It tells us how many parts we have or are considering.
    • Denominator: The bottom number. It tells us the total number of equal parts the whole is divided into. The denominator can never be zero.
  • Example: If a pizza is cut into 8 equal slices and you eat 3 slices, the fraction of the pizza you ate is 3/8. Here, 3 is the numerator, and 8 is the denominator.

2. Representing Fractions on a Number Line

  • Fractions can be shown on a number line.
  • To represent a fraction like 3/4 on a number line:
    • Consider the segment between 0 and 1 (since 3/4 is less than 1).
    • Divide this segment into 4 equal parts (because the denominator is 4).
    • Mark the 3rd point from 0 (because the numerator is 3). This point represents 3/4.
    • Similarly, 7/5 would be located between 1 and 2. Divide the segment between 1 (which is 5/5) and 2 (which is 10/5) into 5 equal parts and mark the 2nd point after 1 (representing 5/5 + 2/5 = 7/5).

3. Types of Fractions

  • (a) Proper Fractions:
    • The numerator is smaller than the denominator (Numerator < Denominator).
    • Represents a quantity less than one whole.
    • Examples: 1/2, 3/4, 5/8, 9/10.
  • (b) Improper Fractions:
    • The numerator is greater than or equal to the denominator (Numerator ≥ Denominator).
    • Represents a quantity equal to or greater than one whole.
    • Examples: 5/4 (greater than 1), 8/8 (equal to 1), 11/6 (greater than 1), 7/7 (equal to 1).
  • (c) Mixed Fractions (or Mixed Numbers):
    • A combination of a whole number and a proper fraction.
    • Represents a quantity greater than one whole.
    • Examples: 1 ¾ (means 1 whole and 3/4), 2 ½ (means 2 wholes and 1/2), 5 ⅓.

4. Conversion between Improper and Mixed Fractions

  • (a) Improper to Mixed:
    • Divide the numerator by the denominator.
    • The quotient becomes the whole number part.
    • The remainder becomes the new numerator.
    • The denominator stays the same.
    • Example: Convert 11/4 to a mixed fraction.
      • 11 ÷ 4 = 2 with a remainder of 3.
      • So, 11/4 = 2 ¾.
  • (b) Mixed to Improper:
    • Multiply the whole number by the denominator.
    • Add the result to the numerator.
    • This sum becomes the new numerator.
    • The denominator stays the same.
    • Example: Convert 3 ½ to an improper fraction.
      • (3 × 2) + 1 = 6 + 1 = 7.
      • The denominator is 2.
      • So, 3 ½ = 7/2.

5. Equivalent Fractions

  • Fractions that represent the same value or the same part of a whole, even though they have different numerators and denominators.
  • How to find equivalent fractions: Multiply or divide both the numerator and the denominator of a fraction by the same non-zero number.
  • Examples:
    • 1/2 is equivalent to 2/4 (multiplied by 2/2), 3/6 (multiplied by 3/3), 5/10 (multiplied by 5/5).
    • 6/9 is equivalent to 2/3 (divided by 3/3).
  • Checking for equivalence: Two fractions a/b and c/d are equivalent if a × d = b × c (cross-multiplication).

6. Simplest Form (or Lowest Term) of a Fraction

  • A fraction is in its simplest form when its numerator and denominator have no common factor other than 1. In other words, their Highest Common Factor (HCF) is 1.
  • How to reduce to simplest form:
    • Find the HCF of the numerator and the denominator.
    • Divide both the numerator and the denominator by their HCF.
    • Alternatively, keep dividing the numerator and denominator by common factors until no more common factors (other than 1) exist.
  • Example: Reduce 12/18 to its simplest form.
    • Factors of 12: 1, 2, 3, 4, 6, 12.
    • Factors of 18: 1, 2, 3, 6, 9, 18.
    • Common factors: 1, 2, 3, 6.
    • HCF = 6.
    • Divide numerator and denominator by 6: 12 ÷ 6 = 2; 18 ÷ 6 = 3.
    • Simplest form = 2/3.

7. Like and Unlike Fractions

  • (a) Like Fractions: Fractions that have the same denominator.
    • Examples: 1/7, 3/7, 5/7, 6/7.
    • Easy to compare, add, and subtract.
  • (b) Unlike Fractions: Fractions that have different denominators.
    • Examples: 1/3, 2/5, 3/4, 5/8.
    • Need to be converted to like fractions (using LCM) before comparing, adding, or subtracting.

8. Comparing Fractions

  • (a) Comparing Like Fractions: The fraction with the larger numerator is the larger fraction.
    • Example: Compare 5/8 and 3/8. Since 5 > 3, 5/8 > 3/8.
  • (b) Comparing Unlike Fractions:
    • Method 1: Convert to Like Fractions:
      • Find the Least Common Multiple (LCM) of the denominators.
      • Convert each fraction into an equivalent fraction with the LCM as the new denominator.
      • Compare the numerators of the resulting like fractions.
      • Example: Compare 2/3 and 3/4.
        • LCM of 3 and 4 is 12.
        • 2/3 = (2×4)/(3×4) = 8/12.
        • 3/4 = (3×3)/(4×3) = 9/12.
        • Since 9 > 8, 9/12 > 8/12. Therefore, 3/4 > 2/3.
    • Method 2: Cross-Multiplication (for two fractions a/b and c/d):
      • Calculate a × d and b × c.
      • If a × d > b × c, then a/b > c/d.
      • If a × d < b × c, then a/b < c/d.
      • If a × d = b × c, then a/b = c/d.
      • Example: Compare 2/3 and 3/4.
        • 2 × 4 = 8.
        • 3 × 3 = 9.
        • Since 8 < 9, 2/3 < 3/4.

9. Addition and Subtraction of Fractions

  • (a) Adding/Subtracting Like Fractions:
    • Add or subtract the numerators.
    • Keep the same denominator.
    • Simplify the result if possible.
    • Example: 2/7 + 3/7 = (2+3)/7 = 5/7.
    • Example: 5/8 - 1/8 = (5-1)/8 = 4/8 = 1/2 (simplified).
  • (b) Adding/Subtracting Unlike Fractions:
    • Find the LCM of the denominators.
    • Convert each fraction into an equivalent fraction with the LCM as the denominator.
    • Add or subtract the numerators of the resulting like fractions.
    • Keep the LCM as the denominator.
    • Simplify the result if possible.
    • Example: 1/4 + 2/3
      • LCM of 4 and 3 is 12.
      • 1/4 = (1×3)/(4×3) = 3/12.
      • 2/3 = (2×4)/(3×4) = 8/12.
      • 1/4 + 2/3 = 3/12 + 8/12 = (3+8)/12 = 11/12.
    • Example: 3/5 - 1/2
      • LCM of 5 and 2 is 10.
      • 3/5 = (3×2)/(5×2) = 6/10.
      • 1/2 = (1×5)/(2×5) = 5/10.
      • 3/5 - 1/2 = 6/10 - 5/10 = (6-5)/10 = 1/10.
  • Adding/Subtracting Mixed Fractions:
    • Option 1: Convert mixed fractions to improper fractions first, then add/subtract as usual.
    • Option 2: Add/subtract the whole number parts and the fractional parts separately. Be careful if subtraction of fractions requires borrowing from the whole number. (Option 1 is often safer).

Key Takeaways for Exams:

  • Know the definitions: Numerator, Denominator, Proper, Improper, Mixed, Like, Unlike, Equivalent, Simplest Form.
  • Master conversions: Improper <-> Mixed.
  • Master operations: Finding Equivalent Fractions, Reducing to Simplest Form, Comparing, Adding, Subtracting (both like and unlike).
  • Understand LCM and HCF and their application in fractions.
  • Practice word problems involving fractions.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on this chapter. Try to solve them yourself first!

  1. Which of the following is a proper fraction?
    (A) 7/5
    (B) 8/8
    (C) 9/10
    (D) 3 ½

  2. The improper fraction representation of 2 ¾ is:
    (A) 9/4
    (B) 11/4
    (C) 6/4
    (D) 10/4

  3. Which fraction is equivalent to 3/5?
    (A) 6/9
    (B) 9/12
    (C) 12/20
    (D) 15/20

  4. The simplest form of the fraction 24/36 is:
    (A) 12/18
    (B) 6/9
    (C) 4/6
    (D) 2/3

  5. Which set contains only like fractions?
    (A) 1/5, 2/10, 3/15
    (B) 2/7, 3/7, 9/7
    (C) 1/4, 2/4, 3/5
    (D) 5/8, 5/9, 5/10

  6. Which symbol should be placed in the box: 5/8 ☐ 3/4?
    (A) >
    (B) <
    (C) =
    (D) None of these

  7. What is the sum of 2/9 + 5/9?
    (A) 7/18
    (B) 7/9
    (C) 3/9
    (D) 10/81

  8. What is the result of 3/4 - 1/3?
    (A) 2/1
    (B) 2/12
    (C) 5/12
    (D) 1/12

  9. A ribbon of length 5 ¼ metres is cut into two pieces. If one piece is 2 ½ metres long, what is the length of the other piece?
    (A) 3 ¼ m
    (B) 2 ¾ m
    (C) 3 ¾ m
    (D) 7 ¾ m

  10. On a number line, the fraction 11/5 lies between which two consecutive whole numbers?
    (A) 0 and 1
    (B) 1 and 2
    (C) 2 and 3
    (D) 10 and 11

Answer Key:

  1. (C)
  2. (B) [(2*4)+3 / 4 = 11/4]
  3. (C) [3x4=12, 5x4=20]
  4. (D) [HCF of 24 and 36 is 12. 24/12=2, 36/12=3]
  5. (B) [Same denominator 7]
  6. (B) [Convert to like fractions: 5/8 and 6/8. Since 5 < 6, 5/8 < 6/8 or 3/4]
  7. (B) [(2+5)/9 = 7/9]
  8. (C) [LCM is 12. (9/12) - (4/12) = 5/12]
  9. (B) [5 ¼ - 2 ½ = 21/4 - 5/2 = 21/4 - 10/4 = 11/4 = 2 ¾ m]
  10. (C) [11/5 = 2 1/5, which is between 2 and 3]

Revise these concepts thoroughly. Remember, practice is key to mastering fractions! Let me know if any part needs further clarification. Good luck with your preparation!

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