Question

Formula for multiplication and division of rational numbers

Answer 1

Rational numbers are numbers that can be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). Here are the formulas for multiplication and division of rational numbers:

### 1. Multiplication of Rational Numbers

To multiply two rational numbers, you multiply their numerators together and their denominators together.

**Formula:**
If you have two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is given by:

\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]

### Example:
\[
\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}
\]

### 2. Division of Rational Numbers

To divide one rational number by another, you multiply the first rational number by the reciprocal of the second rational number.

**Formula:**
If you have two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \), their division is given by:

\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \quad \text{(provided \( c \neq 0 \))}
\]

### Example:
\[
\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6} \quad \text{(after simplification)}
\]

### Summary
- **Multiplication**: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
- **Division**: \(\frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times c} \quad \text{(where \( c \neq 0 \))}\)

These formulas are fundamental for performing operations with rational numbers.