# Working with Fractions

## Overview

This chapter explores how to multiply and divide fractions through real-life contexts, area models, and formal mathematical rules. It explains both conceptual and procedural methods for operations on fractions, including simplification techniques and historical contributions from Indian mathematicians like Brahmagupta and Bhāskarāchārya.

---

## Key Topics Covered

### 1. **Multiplication of Fractions**

#### A. Whole Number × Fraction

- Multiply the whole number by the numerator; keep the denominator the same.
- Examples:
- \( 3 \times \frac{1}{4} = \frac{3}{4} \)
- \( 5 \times \frac{2}{3} = \frac{10}{3} \)

#### B. Fraction × Whole Number

- Multiply the numerator with the whole number.
- Convert mixed numbers into improper fractions before multiplying.

#### C. Fraction × Fraction

- Multiply the numerators.
- Multiply the denominators.
- Simplify the result if possible.
- Example:

$$
\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}
$$

- Visual interpretation: Imagine a rectangle with fractional length and breadth; the product gives the area.

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### 2. **Division of Fractions**

- Division of fractions is the same as multiplying by the reciprocal.

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

Examples:

- \( \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2 \)
- \( \frac{2}{3} \div \frac{3}{5} = \frac{2}{3} \times \frac{5}{3} = \frac{10}{9} \)

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### 3. **Simplifying Fractions During Multiplication**

- Cancel common factors before multiplying to simplify.
- Example:

$$
\frac{12}{7} \times \frac{5}{24} = \frac{12 \times 5}{7 \times 24} = \frac{60}{168} = \frac{5}{14}
$$

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### 4. **Using Area Models**

- Use unit squares or rectangles to visualize multiplication of fractions.
- Area of the rectangle = product of its fractional length and width.

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### 5. **When is the Product Bigger or Smaller?**

- The product:
- Is greater than both factors if both > 1.
- Is less than both if both < 1.
- Lies between the two if one is > 1 and the other < 1.

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### 6. **Order of Multiplication**

- Multiplication is commutative:

$$
a \times b = b \times a
$$

- This rule applies to both whole numbers and fractions.

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### 7. **Word Problems Involving Fractions**

- Apply multiplication/division of fractions in real-world problems:
- Amount of milk per cup
- Bricks needed to cover an area
- Time to fill a tank
- Sharing and comparison tasks

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### 8. **History and Contributions**

- **Brahmagupta** (628 CE): Formalized rules for multiplication and division of fractions.
- **Bhāskarāchārya (Bhāskara II)**: Clearly explained the reciprocal method in his book *Līlāvatī* (1150 CE).
- Ancient Indian texts like *Śulba Sūtra* (~800 BCE) used fractions, and Bhāskara I (629 CE) introduced visual models for them.

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## New Terms and Definitions

| Term | Definition |
|------------------|----------------------------------------------------------------------------|
| Fraction | A number that represents a part of a whole. Written as one number over another. |
| Numerator | The top part of a fraction; shows how many parts are taken. |
| Denominator | The bottom part of a fraction; shows the total number of parts. |
| Reciprocal | When two numbers multiply to make 1. The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). |
| Simplify | To reduce a fraction to its lowest terms. |
| Area model | A method using shapes to visualize multiplication of fractions. |
| Mixed number | A number made of a whole and a fraction part, like \( 1 \frac{1}{2} \). |
| Improper fraction| A fraction where the numerator is larger than the denominator. |
| Product | The result of multiplication. |
| Quotient | The result of division. |

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## Practice Problems

### Easy (3)

1. $\dfrac{2}{3} \times 3 = \dfrac{6}{3} = 2$

2. $\dfrac{1}{4} \div 2 = \dfrac{1}{4} \times \dfrac{1}{2} = \dfrac{1}{8}$

3. $5 \times \dfrac{1}{5} = \dfrac{5}{5} = 1$

### Medium (2)

4. $\dfrac{2}{3} \times \dfrac{3}{5} = \dfrac{6}{15} = \dfrac{2}{5}$

5. $\dfrac{5}{8} \div \dfrac{1}{4} = \dfrac{5}{8} \times \dfrac{4}{1} = \dfrac{20}{8} = \dfrac{5}{2}$

### Difficult (3)

6. $\dfrac{4}{7} \times \dfrac{14}{15} = \dfrac{56}{105} = \dfrac{8}{15}$

7. $\dfrac{9}{10} \div \dfrac{3}{5} = \dfrac{9}{10} \times \dfrac{5}{3} = \dfrac{45}{30} = \dfrac{3}{2}$

8. 5 cups of tea use $\dfrac{1}{4}$ L of milk. Milk per cup:

$$
\dfrac{1}{4} \div 5 = \dfrac{1}{4} \times \dfrac{1}{5} = \dfrac{1}{20} \, \text{L}
$$

### Very Difficult (2)

9. Shaded region = $\dfrac{3}{4}$ of a triangle which is $\dfrac{1}{2}$ of a square:

$$
\dfrac{3}{4} \times \dfrac{1}{2} = \dfrac{3}{8}
$$

10. A baker needs $\dfrac{1}{6}$ kg flour per loaf. For 5 kg:

$$
5 \div \dfrac{1}{6} = 5 \times 6 = 30 \, \text{loaves}