Class 5 – 10 Curriculum

The Universe of Fractions

A complete journey from basic parts of a whole to complex algebraic rational expressions. Designed for high-retention learning.

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Chapter 1: The Anatomy of a Fraction

📜 Definition & Notation

A fraction represents a part of a whole or, more generally, any number of equal parts. It is written as:

$$ \frac{\text{Numerator}}{\text{Denominator}} = \frac{N}{D} $$

🎯 Numerator ($N$): How many parts we have.
📉 Denominator ($D$): Total parts the whole is divided into ($D \neq 0$).

🖼️ Visualizing Concepts

3 Parts (N)

Representation of $ \frac{3}{4} $

The bar is split into 4 equal sections. 3 are highlighted.

🏷️ Classification of Fractions

1. Proper Fraction

Numerator < Denominator

$\frac{3}{5}, \frac{7}{9}$

Value is always < 1

2. Improper Fraction

Numerator $\ge$ Denominator

$\frac{5}{3}, \frac{12}{7}$

Value is always $\ge 1$

3. Mixed Fraction

Whole Number + Proper Fraction

$2\frac{1}{3}$

Can be converted to Improper

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📌 Key Exam Facts & Points

A fraction with denominator 1 is an integer (e.g., $\frac{5}{1} = 5$).
Value of a Proper fraction is always less than 1.
Value of an Improper fraction is always greater than or equal to 1.

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Chapter 2: Conversions & Equivalency

⚙️ The Mechanics of Conversion

A. Improper to Mixed ($ \frac{17}{3} $)

Divide Numerator by Denominator: $17 \div 3$.
Quotient becomes Whole number ($5$).
Remainder becomes new Numerator ($2$).
Denominator stays same ($3$).

Result: $5 \frac{2}{3}$

B. Mixed to Improper ($ 4 \frac{2}{5} $)

Multiply Whole number by Denominator ($4 \times 5 = 20$).
Add the Numerator ($20 + 2 = 22$).
Place over original Denominator.

Result: $\frac{22}{5}$

⚖️ Equivalent Fractions

Multiplying or dividing both Numerator and Denominator by the same non-zero number does not change the value.

$\frac{1}{2}$ = $\frac{2}{4}$ = $\frac{3}{6}$ = $\frac{50}{100}$

Simplest Form Rule: Divide N and D by their Highest Common Factor (HCF).
Example: $\frac{12}{18} \rightarrow \text{HCF is 6} \rightarrow \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

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Chapter 3: Operations (Add & Subtract)

🧩 The LCM Method (Unlike Fractions)

To add or subtract fractions with different denominators, you must find a common ground: The LCM.

Example: $\frac{2}{3} + \frac{1}{5}$

1
Find LCM: LCM of 3 and 5 is 15.
2
Convert Fractions:
$\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
$\frac{1 \times 3}{5 \times 3} = \frac{3}{15}$
3
Add Numerators: $\frac{10 + 3}{15} = \frac{13}{15}$

🦋 Visual Trick: The Butterfly Method

$$ \frac{a}{b} + \frac{c}{d} = \frac{(a \times d) + (b \times c)}{b \times d} $$

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Chapter 4: Multiplication & Division

💥 Multiplication: Straight Through

$$ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} $$

➗ Division: KCF Rule

KKeep

CChange

FFlip

$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$

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Chapter 5: Advanced (Class 9-10 Level)

🔢 Rational Numbers vs. Fractions

Fraction: $\frac{p}{q}$ where $p, q$ are Whole Numbers, $q \neq 0$. (Always Positive).
Rational Number: $\frac{p}{q}$ where $p, q$ are Integers, $q \neq 0$. (Can be Negative).

All Fractions are Rational Numbers.
Not all Rational Numbers are Fractions.