Operations with fractions involve several fundamental actions: addition, subtraction, multiplication, and division. Below is a brief overview of how each operation works with fractions.

1. Addition of Fractions

To add fractions, follow these steps:

• Same Denominator: If the fractions have the same denominator, simply add the numerators and keep the denominator the same. $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$
• Different Denominators: If the fractions have different denominators, first find a common denominator, usually the least common denominator (LCD). Then, adjust the fractions accordingly and add the numerators. $\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + b \cdot c}{b \cdot d}$

2. Subtraction of Fractions

Subtracting fractions is similar to addition:

• Same Denominator: Subtract the numerators and keep the denominator the same. $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$
• Different Denominators: Find a common denominator, adjust the fractions, and then subtract the numerators. $\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - b \cdot c}{b \cdot d}$

3. Multiplication of Fractions

To multiply fractions:

• Multiply the numerators together and multiply the denominators together. $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$
• Simplify the resulting fraction if possible.

4. Division of Fractions

To divide fractions:

• Multiply the first fraction by the reciprocal of the second fraction. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$
• Simplify the resulting fraction if possible.

Example:

Let’s apply these operations to the fractions $\frac{2}{3}$ and $\frac{4}{5}$:

• Addition: $\frac{2}{3} + \frac{4}{5} = \frac{2 \times 5 + 3 \times 4}{3 \times 5} = \frac{10 + 12}{15} = \frac{22}{15}$
• Subtraction: $\frac{2}{3} - \frac{4}{5} = \frac{2 \times 5 - 3 \times 4}{3 \times 5} = \frac{10 - 12}{15} = \frac{-2}{15}$
• Multiplication: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$
• Division: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6}$

Would you like to delve deeper into any of these operations? Here are some related questions:

1. How can you find the least common denominator (LCD) of two fractions?
2. What is the process for simplifying fractions after performing operations?
3. How do you handle mixed numbers in fraction operations?
4. How do you multiply or divide fractions that include negative numbers?
5. What are the real-life applications of fraction operations?

Tip: When adding or subtracting fractions, always ensure the denominators are the same. If they aren't, convert them to a common denominator before performing the operation.