Question

multiply and divide rational numbers: application of dividing rational numbers quick check.

Answer 1

Dividing rational numbers involves a few simple steps, similar to multiplying them. Here’s a quick guide on how to do it, along with an example application:

### Steps to Divide Rational Numbers:

1. **Identify the Rational Numbers**: A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

2. **Convert Division to Multiplication**: To divide by a fraction, multiply by its reciprocal. For example, to divide \( \frac{a}{b} \) by \( \frac{c}{d} \), this is the same as multiplying \( \frac{a}{b} \) by \( \frac{d}{c} \).

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

3. **Multiply**: Multiply the numerators together and the denominators together.

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

4. **Simplify**: If possible, simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common factor (GCF).

### Example Application:

Let's consider dividing \( \frac{3}{4} \) by \( \frac{2}{5} \).

1. **Identify the Rational Numbers**: \( \frac{3}{4} \) and \( \frac{2}{5} \).

2. **Convert Division to Multiplication**:
\[
\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}
\]

3. **Multiply**:
- Numerator: \( 3 \times 5 = 15 \)
- Denominator: \( 4 \times 2 = 8 \)
- Result: \( \frac{15}{8} \)

4. **Simplify**: In this case, \( \frac{15}{8} \) cannot be simplified further, so the final answer is \( \frac{15}{8} \) or \( 1 \frac{7}{8} \) in mixed number form.

### Quick Check:

- **Convert back to check**:
- Multiply \( \frac{3}{4} \) by \( \frac{5}{2} \):
\( \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} \).
- It matches the result from our division, confirming the calculation is correct.

This process will help you effectively divide rational numbers in a variety of applications!