# Fractions

## Overview

This chapter introduces the concept of fractions through real-life sharing examples, visual models, number lines, and ancient Indian mathematics. Students learn to understand, compare, and compute with fractions, including equivalent fractions, mixed numbers, and operations of addition and subtraction. The chapter also highlights Brahmagupta's historical contribution to the arithmetic of fractions.

## Key Topics Covered

### 1. Understanding Fractions

* **Sharing as Fractions**: When a whole is shared equally, each part is a fraction. For example, 1 roti shared by 2 children gives each ½ roti.
* **Comparing Fractions**: More parts mean smaller shares. So, ¼ is less than ½.
* **Fractional Units**: When a whole is divided into equal parts like ½, ⅓, ¼, etc., each part is a fractional unit.

### 2. Fractions as Parts of a Whole

* **Equal Pieces**: Different shapes can represent the same fractional value if they are equal in size.
* **Visual Representation**: Food items like chikki are used to show fractional parts (e.g., ¾, ⅓).
* **Activity**: Students are encouraged to identify and compare fractional parts of shapes.

### 3. Measuring with Fractions

* **Paper Strip Activity**: Folding strips to create equal parts and understand unit fractions like ½, ¼, ⅛.
* **Addition Using Unit Fractions**: Repeating a fractional unit shows the total (e.g., 3 × ¼ = ¾).
* **Number Names in Fractions**: Top number = numerator, bottom number = denominator.

### 4. Fractions on the Number Line

* **Marking Fractions**: Fractions like ½, ⅓, ⅗ are marked on a number line between whole numbers.
* **Understanding Size**: Visualising fractions helps in understanding their magnitude.

### 5. Mixed Fractions

* **Definition**: A mixed fraction has a whole number and a proper fraction (e.g., 2 ⅔).
* **Conversion**: Improper fractions like 7/3 can be written as 2 1/3.
* **Reverse Conversion**: Mixed numbers can be written as improper fractions (e.g., 3 ¾ = 15/4).

### 6. Equivalent Fractions

* **Same Value, Different Forms**: ½ = 2/4 = 4/8.
* **Fraction Wall Activity**: Comparing different fractions visually to find equivalent pairs.
* **Simplest Form**: A fraction in its lowest terms has no common factor between numerator and denominator except 1.

### 7. Comparing Fractions

* **Same Denominator**: Compare numerators directly.
* **Different Denominators**: Convert to equivalent fractions with same denominator (LCM method).
* **Visual Comparison**: Use number lines or real-world sharing examples.

### 8. Addition and Subtraction of Fractions

* **Same Denominator**: Add or subtract numerators directly.
* **Different Denominators**: Use Brahmagupta’s method—find a common denominator, convert, then compute.
* **Mixed Number Operations**: Add or subtract whole and fractional parts separately.

### 9. History and Puzzle

* **Ancient Indian Mathematicians**: Brahmagupta formalised rules for fractions in the 7th century.
* **Egyptian Fractions**: Writing a number as a sum of different unit fractions (e.g., 1 = ½ + ⅓ + 1/6).
* **Puzzle Activity**: Find combinations of different fractional units that sum to 1.

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## New Terms and Definitions

| Term | Definition |
| -------------------- | ------------------------------------------------------------------------------ |
| Fraction | A part of a whole |
| Numerator | The top number in a fraction showing parts taken |
| Denominator | The bottom number in a fraction showing total equal parts |
| Unit Fraction | A fraction with 1 as the numerator |
| Mixed Fraction | A number made of a whole number and a fraction |
| Improper Fraction | A fraction where numerator > denominator |
| Equivalent Fractions | Different fractions showing the same amount |
| Simplest Form | A fraction with numerator and denominator having no common factor other than 1 |
| Number Line | A line where numbers and fractions are placed at equal distances |
| Fraction Wall | A visual tool for comparing sizes of fractions |

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## Practice Questions

### 🟢 Easy (3 Questions)

**1. What is the fraction of a pizza if it is cut into 4 equal parts and you eat 1 piece?**
**Answer:** ¼
**Explanation:** 1 piece out of 4 is ¼.

**2. Which is more: ½ or ⅓?**
**Answer:** ½
**Explanation:** When the denominator is smaller, the fraction is larger (if numerator is 1).

**3. Express 2 ½ as an improper fraction.**
**Answer:** 5/2
**Explanation:** (2 × 2) + 1 = 4 + 1 = 5, so 5/2.

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### 🟡 Medium (2 Questions)

**4. Find the sum: 2/5 + 1/5**
**Answer:** 3/5
**Explanation:** Same denominator, add numerators: 2 + 1 = 3.

**5. What is ¾ – ¼ ?**
**Answer:** ½
**Explanation:** Same denominator → 3 – 1 = 2 ⇒ 2/4 = ½ in simplest form.

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### 🔴 Difficult (3 Questions)

**6. Add: 1/3 + 1/4**
**Answer:** 7/12
**Explanation:** LCM of 3 and 4 = 12.
1/3 = 4/12, 1/4 = 3/12 → 4/12 + 3/12 = 7/12.

**7. Subtract: 5/6 – 1/3**
**Answer:** ½
**Explanation:** Convert 1/3 = 2/6 → 5/6 – 2/6 = 3/6 = ½.

**8. Arrange in ascending order: 2/3, 3/4, 5/6**
**Answer:** 2/3 < 3/4 < 5/6
**Explanation:** Convert to common denominator or decimal:
2/3 = 0.666..., 3/4 = 0.75, 5/6 = 0.833...

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### ⚫ Very Difficult (2 Questions)

**9. Add 3/4 + 1/3 + 1/5**
**Answer:** 47/60
**Explanation:**
LCM of 4, 3, 5 = 60
→ 3/4 = 45/60, 1/3 = 20/60, 1/5 = 12/60
→ Sum = 45 + 20 + 12 = 77/60 = 1 17/60

**10. Subtract 13/4 from 23/3**
**Answer:** 35/12
**Explanation:**
Convert both to same denominator:
13/4 = 39/12, 23/3 = 92/12
→ 92 – 39 = 53 → 53/12 = 4 5/12

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