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Multiply & Divide Rational Numbers Sample Work

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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!