Class 6
Mathematics

Chapter 7: Fractions

Master the world of fractions! Learn how whole quantities are split into equal shares, explore all 7 primary types of fractions with visual models, master conversions, and practice addition/subtraction using historical and modern methods[cite: 5].

1. Introduction & Concept of Equal Sharing

A fraction represents a part of a whole quantity or any number of equal parts[cite: 5]. When an item or group of items is divided into equal portions, fractions tell us exactly how much each share represents[cite: 5].

Sharing Principle: Sharing 1 roti equally among 2 children gives each child $\frac{1}{2}$ roti[cite: 5]. Sharing 1 roti among 4 children gives each child $\frac{1}{4}$ roti[cite: 5]. As the number of equal shares increases, each individual share becomes smaller ($\frac{1}{2} > \frac{1}{4}$)[cite: 5].

Visual Model of $\frac{3}{4}$ (3 out of 4 Equal Parts Shaded):

1/4
1/4
1/4
1/4

Numerator = 3 (Parts selected) | Denominator = 4 (Total equal parts)[cite: 5]

2. Detailed Classification: Types of Fractions

Fractions are classified based on the relationships between their numerators, denominators, and whole components.

Fraction Type Definition / Condition Examples
1. Proper Fraction Numerator is strictly less than denominator ($Numerator < Denominator$)[cite: 5]. Value is less than 1. $\frac{1}{2}, \frac{3}{5}, \frac{7}{10}$[cite: 5]
2. Improper Fraction Numerator is greater than or equal to denominator ($Numerator \ge Denominator$)[cite: 5]. Value is $\ge 1$. $\frac{5}{3}, \frac{7}{4}, \frac{9}{2}, \frac{4}{4}$[cite: 5]
3. Mixed Fraction Combination of a non-zero whole number and a proper fraction[cite: 5]. $1\frac{2}{3}, 2\frac{3}{4}, 5\frac{1}{2}$[cite: 5]
4. Unit Fraction Numerator is always equal to 1[cite: 5]. Represents a single fractional unit. $\frac{1}{2}, \frac{1}{3}, \frac{1}{6}, \frac{1}{100}$[cite: 5]
5. Equivalent Fractions Fractions having different numerators & denominators that represent the same value[cite: 5]. $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$[cite: 5]
6. Like Fractions Fractions that share the exact same denominator[cite: 5]. Easy to compare directly. $\frac{1}{7}, \frac{3}{7}, \frac{5}{7}$[cite: 5]
7. Unlike Fractions Fractions that have different denominators[cite: 5]. Must be converted before adding/subtracting. $\frac{2}{3}, \frac{4}{5}, \frac{1}{6}$[cite: 5]

Proper Fractions

Represents a portion strictly smaller than one complete whole unit[cite: 5]. E.g., $\frac{2}{5}$ of a pizza.

Improper Fractions

Represents an amount equal to or greater than one full whole unit[cite: 5]. E.g., $\frac{7}{4}$ chikkis.

Mixed Numbers

Expresses improper fractions as whole units + remaining parts[cite: 5]. E.g., $1\frac{3}{4} = \frac{7}{4}$.

3. Conversions Between Fraction Types

Type A: Converting Improper Fraction $\longrightarrow$ Mixed Fraction

Rule: Divide Numerator by Denominator.

$$\text{Mixed Fraction} = \text{Quotient} \frac{\text{Remainder}}{\text{Denominator}}$$

Example 1: Convert $\frac{11}{4}$ into a Mixed Fraction

  • Divide $11 \div 4 \implies \text{Quotient} = 2$, $\text{Remainder} = 3$[cite: 5].
  • Therefore, $\frac{11}{4} = \mathbf{2\frac{3}{4}}$[cite: 5].

Type B: Converting Mixed Fraction $\longrightarrow$ Improper Fraction

Rule: Multiply whole number by denominator, add numerator, keep denominator intact.

$$\text{Improper Fraction} = \frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$$

Example 2: Convert $3\frac{2}{5}$ into an Improper Fraction

  • Calculation: $\frac{(3 \times 5) + 2}{5} = \frac{15 + 2}{5} = \mathbf{\frac{17}{5}}$[cite: 5].

4. Fractions on the Number Line

Just like whole numbers, every fraction has a unique position on the number line[cite: 5]. The space between whole numbers is partitioned into equal segments equal to the denominator[cite: 5].

Example: Locating $\frac{3}{5}$ on the Number Line

Divide the line segment from 0 to 1 into 5 equal parts[cite: 5]:

|
0
|
$\frac{1}{5}$
|
$\frac{2}{5}$
|
$\frac{3}{5}$
|
$\frac{4}{5}$
|
1 ($\frac{5}{5}$)

5. Equivalent Fractions & Simplification

Equivalent fractions represent equal proportions of the whole even though they use different numerators and denominators[cite: 5].

Fraction Wall Diagram

1 Whole Unit
1/2
1/2
1/4
1/4
1/4
1/4
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8

Observing the wall: $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$[cite: 5]

Simplifying Fractions to Simplest / Lowest Form

A fraction is in its lowest terms when numerator and denominator share no common factors other than 1[cite: 5].

Example: Simplify $\frac{24}{36}$

  • Find HCF of 24 and 36, which is 12[cite: 5].
  • Divide top and bottom by 12: $$\frac{24 \div 12}{36 \div 12} = \mathbf{\frac{2}{3}}$$[cite: 5]

6. Comparing Fractions

Case 1: Comparing Like Fractions (Same Denominators)

Simply compare numerators. Larger numerator means greater fraction[cite: 5].

$$\frac{7}{11} > \frac{4}{11} \quad \text{because } 7 > 4$$[cite: 5]

Case 2: Comparing Unlike Fractions (Different Denominators)

Convert unlike fractions into equivalent like fractions using the LCM of their denominators[cite: 5].

Example: Compare $\frac{3}{4}$ and $\frac{5}{6}$

  1. LCM of 4 and 6 = 12[cite: 5].
  2. Convert $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$[cite: 5].
  3. Convert $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$[cite: 5].
  4. Since $\frac{10}{12} > \frac{9}{12}$, we conclude $\mathbf{\frac{5}{6} > \frac{3}{4}}$[cite: 5].

7. Operations: Addition & Subtraction (Brahmagupta's Method)

First codified systematically by the Indian mathematician Brahmagupta in 628 CE, operations on fractions require expressing them with a common denominator first[cite: 5].

Example A: Addition of Unlike Fractions ($\frac{2}{5} + \frac{3}{4}$)

1. LCM of 5 and 4 = 20[cite: 5].

2. Convert to equivalent fractions: $$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}, \quad \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$$[cite: 5]

3. Add numerators: $$\frac{8}{20} + \frac{15}{20} = \frac{8 + 15}{20} = \frac{23}{20} = \mathbf{1\frac{3}{20}}$$[cite: 5]

Example B: Subtraction of Mixed Numbers ($4\frac{1}{2} - 2\frac{2}{3}$)

1. Convert mixed to improper fractions: $$4\frac{1}{2} = \frac{9}{2}, \quad 2\frac{2}{3} = \frac{8}{3}$$[cite: 5]

2. LCM of 2 and 3 = 6[cite: 5].

3. Express with common denominator: $$\frac{9}{2} = \frac{27}{6}, \quad \frac{8}{3} = \frac{16}{6}$$[cite: 5]

4. Subtract numerators: $$\frac{27}{6} - \frac{16}{6} = \mathbf{\frac{11}{6} = 1\frac{5}{6}}$$[cite: 5]

8. History & Unit Fraction Puzzles

In ancient India, fractions were called bhinna ('broken') or bhaga ('part')[cite: 5]. Historical texts like the Bakhshali manuscript (~300 CE) wrote numerators directly above denominators[cite: 5]. Later, the horizontal fraction bar was introduced by Al-Hassar in the 12th century[cite: 5].

Classic Egyptian Unit Fraction Puzzle

Can you find three different unit fractions (numerator = 1) that add up exactly to 1?[cite: 5]

Solution: $$\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1$$[cite: 5]

Key Takeaways

  • A Fraction represents an equal portion of a whole quantity[cite: 5].
  • Proper fractions are $< 1$; improper fractions are $\ge 1$[cite: 5].
  • Mixed numbers contain a whole number and a proper fraction[cite: 5].
  • Equivalent fractions share the same value when reduced to lowest terms[cite: 5].
  • Addition/subtraction of unlike fractions requires finding a common denominator (LCM)[cite: 5].

Multiple Choice Practice Quiz

1. Which of the following is an improper fraction?

A. $\frac{3}{7}$
B. $\frac{8}{5}$
C. $\frac{1}{4}$
D. $\frac{2}{3}$

Answer: B ($\frac{8}{5}$ has numerator > denominator)[cite: 5]

2. What is the lowest term of $\frac{18}{24}$?

A. $\frac{9}{12}$
B. $\frac{6}{8}$
C. $\frac{3}{4}$
D. $\frac{3}{8}$

Answer: C ($\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$)[cite: 5]

3. Calculate: $\frac{5}{6} - \frac{1}{3}$

A. $\frac{1}{2}$
B. $\frac{4}{3}$
C. $\frac{4}{6}$
D. $\frac{1}{3}$

Answer: A ($\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$)[cite: 5]

Textbook Exercise Solutions

Step-by-Step Solved Problems

  1. Solve: $\frac{2}{3} + \frac{4}{5}$
    LCM of 3 and 5 = 15[cite: 5].
    $\frac{2 \times 5}{15} + \frac{4 \times 3}{15} = \frac{10 + 12}{15} = \mathbf{\frac{22}{15} = 1\frac{7}{15}}$[cite: 5]
  2. Word Problem: Rahim mixed $\frac{2}{3}$ L of yellow paint and $\frac{3}{4}$ L of blue paint. Total volume of mixed paint?
    Total = $\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \mathbf{\frac{17}{12} = 1\frac{5}{12}\text{ Litres}}$[cite: 5].

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