Lecture 5: Multiplication and Division of Rational Numbers

Introduction

• Review of fractions, mixed numbers, rational numbers, and decimals from the previous lecture.
• Focus on multiplication and division of rational numbers (Section 1.3 of the textbook).

Multiplication of Rational Numbers

Multiplication of Fractions

• Three interpretations:
  1. Multiplication by a positive integer as repeated addition.
  2. Positive integer times a fraction (e.g., 2/5) interpreted as parts of an object.
  3. Fraction times a fraction (e.g., 2/3 of 4/5) as part of another part.

Example Calculations

• Example 1: Compute ( \frac{2}{5} \times 3 )
  • Interpreted as repeated addition: ( \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{6}{5} )
• Example 2: Compute ( 3 \times \frac{2}{5} )
  • Interpreted as 2/5 of 3 objects, resulting in ( \frac{6}{5} ).
• Observation: Multiplying the numerators and denominators directly gives the result.

Multiplication of Fraction by Fraction

• Example 3: ( \frac{2}{3} \times \frac{4}{5} ) interpreted as 2/3 of 4/5 of an object.
  • Result: ( \frac{8}{15} )
• Example 4: ( \frac{4}{5} \times \frac{2}{3} ) interpreted as 4/5 of 2/3 of an object.
  • Result: ( \frac{8}{15} )
• Observation: Multiplication involves multiplying numerators and denominators.

Extending to Rational Numbers

• Definition: ( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} )
• Consistency in arithmetic operations.

Simplifying Rational Numbers

• Example 5: Multiplication of fractions and simplifying:
  • Part A: Simplify ( \frac{10}{21} \times \frac{12}{55} = \frac{8}{77} )
  • Part B: ( -\frac{15}{16} \times \frac{24}{66} = -\frac{15}{44} )
  • Part C: Simplifying negative fractions.
  • Part D: Multiplying decimals by converting to fractions.

Division of Rational Numbers

Division of Fractions

• Interpretation: "How many ( \frac{C}{D} ) units fit in ( \frac{A}{B} ) units?"
• Example 6: Division through steps:
  • Step 1: ( 1 \div \frac{1}{7} = 7 )
  • Step 2: ( 1 \div \frac{4}{7} = \frac{7}{4} )
  • Step 3: ( \frac{9}{11} \div \frac{4}{7} = \frac{63}{44} )

Rational Number Division Definition

• Division by a fraction means multiplying by its reciprocal:
  • ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} )

Example Calculations

• Example A: ( \frac{3}{10} \div -\frac{15}{14} = -\frac{7}{25} )
• Example B: Dividing decimals by converting to fractions and using a calculator for practical purposes.

Conclusion

• Understanding multiplication and division of rational numbers through interpretation and computation.
• Homework assigned for practice.