Question
Multiply & Divide Rational Numbers Sample Work
Answers
GPT-4o mini
Sure! Let's go through how to multiply and divide rational numbers, step by step.
### Rational Numbers
Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
### Multiplying Rational Numbers
To multiply two rational numbers, follow these steps:
1. **Multiply the numerators**: Multiply the top numbers of the fractions together.
2. **Multiply the denominators**: Multiply the bottom numbers of the fractions together.
3. **Simplify the result if possible**: Reduce the fraction to its simplest form.
#### Example:
Multiply \( \frac{2}{3} \) and \( \frac{4}{5} \).
1. Multiply the numerators: \( 2 \times 4 = 8 \)
2. Multiply the denominators: \( 3 \times 5 = 15 \)
3. Combine: \( \frac{8}{15} \)
4. **Conclusion**: \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)
### Dividing Rational Numbers
To divide two rational numbers, follow these steps:
1. **Multiply by the reciprocal**: Change the division into multiplication by flipping the second fraction (taking its reciprocal).
2. **Proceed with multiplication**: Follow the steps for multiplying rational numbers as outlined above.
3. **Simplify the result if possible**: Reduce the fraction to its simplest form.
#### Example:
Divide \( \frac{3}{4} \) by \( \frac{2}{3} \).
1. Find the reciprocal of the second fraction: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
2. Multiply \( \frac{3}{4} \) by \( \frac{3}{2} \):
- Multiply the numerators: \( 3 \times 3 = 9 \)
- Multiply the denominators: \( 4 \times 2 = 8 \)
- Combine: \( \frac{9}{8} \)
3. **Conclusion**: \( \frac{3}{4} \div \frac{2}{3} = \frac{9}{8} \)
### Additional Examples
1. **Multiply**: \( \frac{5}{6} \times \frac{3}{7} \)
- Numerators: \( 5 \times 3 = 15 \)
- Denominators: \( 6 \times 7 = 42 \)
- Result: \( \frac{15}{42} \) (simplifies to \( \frac{5}{14} \))
2. **Divide**: \( \frac{7}{8} \div \frac{1}{4} \)
- Reciprocal: \( \frac{1}{4} \) becomes \( \frac{4}{1} \)
- Multiply: \( \frac{7}{8} \times \frac{4}{1} \)
- Numerators: \( 7 \times 4 = 28 \)
- Denominators: \( 8 \times 1 = 8 \)
- Result: \( \frac{28}{8} \) (simplifies to \( \frac{7}{2} \))
### Summary
- To **multiply**, multiply the numerators and denominators.
- To **divide**, multiply by the reciprocal of the second fraction.
Feel free to reach out if you need more examples or have any specific questions!
### Rational Numbers
Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
### Multiplying Rational Numbers
To multiply two rational numbers, follow these steps:
1. **Multiply the numerators**: Multiply the top numbers of the fractions together.
2. **Multiply the denominators**: Multiply the bottom numbers of the fractions together.
3. **Simplify the result if possible**: Reduce the fraction to its simplest form.
#### Example:
Multiply \( \frac{2}{3} \) and \( \frac{4}{5} \).
1. Multiply the numerators: \( 2 \times 4 = 8 \)
2. Multiply the denominators: \( 3 \times 5 = 15 \)
3. Combine: \( \frac{8}{15} \)
4. **Conclusion**: \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)
### Dividing Rational Numbers
To divide two rational numbers, follow these steps:
1. **Multiply by the reciprocal**: Change the division into multiplication by flipping the second fraction (taking its reciprocal).
2. **Proceed with multiplication**: Follow the steps for multiplying rational numbers as outlined above.
3. **Simplify the result if possible**: Reduce the fraction to its simplest form.
#### Example:
Divide \( \frac{3}{4} \) by \( \frac{2}{3} \).
1. Find the reciprocal of the second fraction: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
2. Multiply \( \frac{3}{4} \) by \( \frac{3}{2} \):
- Multiply the numerators: \( 3 \times 3 = 9 \)
- Multiply the denominators: \( 4 \times 2 = 8 \)
- Combine: \( \frac{9}{8} \)
3. **Conclusion**: \( \frac{3}{4} \div \frac{2}{3} = \frac{9}{8} \)
### Additional Examples
1. **Multiply**: \( \frac{5}{6} \times \frac{3}{7} \)
- Numerators: \( 5 \times 3 = 15 \)
- Denominators: \( 6 \times 7 = 42 \)
- Result: \( \frac{15}{42} \) (simplifies to \( \frac{5}{14} \))
2. **Divide**: \( \frac{7}{8} \div \frac{1}{4} \)
- Reciprocal: \( \frac{1}{4} \) becomes \( \frac{4}{1} \)
- Multiply: \( \frac{7}{8} \times \frac{4}{1} \)
- Numerators: \( 7 \times 4 = 28 \)
- Denominators: \( 8 \times 1 = 8 \)
- Result: \( \frac{28}{8} \) (simplifies to \( \frac{7}{2} \))
### Summary
- To **multiply**, multiply the numerators and denominators.
- To **divide**, multiply by the reciprocal of the second fraction.
Feel free to reach out if you need more examples or have any specific questions!