4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Mixed numbers and complex fractions can be tricky, but they're essential for everyday math. We'll learn how to convert mixed numbers to improper fractions and back, making multiplication and division a breeze. These skills are crucial for cooking, budgeting, and more.

We'll also tackle complex fractions and fraction bar expressions. By breaking them down step-by-step, we'll simplify even the most intimidating fractions. This knowledge will help you solve real-world problems and boost your confidence in math.

Multiplication and division of mixed numbers

Multiplication and division of mixed numbers

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• Convert mixed numbers to improper fractions by multiplying the whole number by the denominator, adding the numerator, and placing the result over the original denominator (e.g., $2\frac{1}{3} = \frac{7}{3}$)
• For multiplication, multiply the numerators and denominators separately, then simplify the result and convert back to a mixed number if necessary (e.g., $2\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = 3\frac{1}{2}$)
• For division, convert mixed numbers to improper fractions, multiply the first fraction by the reciprocal of the second fraction, simplify the result, and convert back to a mixed number if necessary (e.g., $2\frac{1}{3} \div 1\frac{1}{2} = \frac{7}{3} \div \frac{3}{2} = \frac{7}{3} \times \frac{2}{3} = \frac{14}{9} = 1\frac{5}{9}$)

Verbal to fractional expression conversion

• Identify the whole and the parts in the verbal description (e.g., "Two-thirds of a cup" - whole: cup, parts: two-thirds)
• Write the parts as the numerator and the whole as the denominator (e.g., $\frac{2}{3}$ cup)
• Simplify the fraction if possible by dividing the numerator and denominator by their greatest common factor (e.g., "Six out of eight students" = $\frac{6}{8}$ of the students = $\frac{3}{4}$ of the students)

Reduction of complex fractions

Reduction of complex fractions

• Complex fractions contain one or more fractions in the numerator, denominator, or both (e.g., $\frac{\frac{2}{3}}{\frac{1}{4}}$)
• To simplify, multiply the numerator and denominator by the least common denominator (LCD) of all the fractions within the complex fraction (e.g., $\frac{\frac{2}{3}}{\frac{1}{4}} \times \frac{12}{12} = \frac{\frac{2}{3} \times 12}{\frac{1}{4} \times 12} = \frac{8}{3}$)
• Simplify the resulting numerator and denominator by performing the indicated operations and reducing the fraction (e.g., $\frac{8}{3} = 2\frac{2}{3}$)

Simplification of fraction bar expressions

• Fraction bars act as grouping symbols, similar to parentheses (e.g., $\frac{2 + \frac{1}{3}}{4 - \frac{1}{2}}$)
• Simplify the expressions above and below the fraction bar separately by performing the indicated operations and reducing fractions (e.g., numerator: $2 + \frac{1}{3} = \frac{7}{3}$, denominator: $4 - \frac{1}{2} = \frac{7}{2}$)
• Divide the simplified numerator by the simplified denominator to obtain the final result (e.g., $\frac{\frac{7}{3}}{\frac{7}{2}} = \frac{7}{3} \times \frac{2}{7} = \frac{2}{3}$)

Additional concepts for fraction operations

• Simplification: Reduce fractions to their simplest form by dividing both the numerator and denominator by their greatest common factor
• Cancellation: Identify and eliminate common factors in the numerator and denominator before multiplying fractions
• Cross multiplication: A method used to compare fractions or solve equations involving fractions by multiplying the numerator of each fraction by the denominator of the other
• Common denominator: When adding or subtracting fractions, find a common denominator to ensure the fractions have the same base before combining them