4.2 Multiply and Divide Fractions

Fractions are like slices of pizza. You can multiply them to make bigger portions or divide them to share. Simplifying fractions makes them easier to handle, just like cutting a pizza into fewer, larger slices.

Multiplying fractions is straightforward: multiply the tops, multiply the bottoms. Dividing is trickier, but using reciprocals makes it a breeze. Remember, flipping the second fraction turns division into multiplication.

Multiplying and Dividing Fractions

Simplification of fractions

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• Find the greatest common factor (GCF) shared by the numerator and denominator (12 and 18 have a GCF of 6)
• Reduce the fraction by dividing both the numerator and denominator by the GCF $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
• The resulting reduced fraction is in its simplest form with the lowest possible terms ($\frac{2}{3}$ cannot be further simplified)
• Simplifying fractions makes them easier to work with in calculations and comparisons ($\frac{2}{3}$ vs $\frac{12}{18}$)
• The fraction bar (horizontal line) separates the numerator from the denominator and represents division

Multiplication of fractions

• Multiply fractions by multiplying their numerators together and denominators together $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ ($\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12}$)
• Simplify the resulting fraction to its lowest terms by finding the GCF of the numerator and denominator and dividing by it ($\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}$)
• Convert mixed numbers to improper fractions before multiplying ($2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$)
  • Multiply the whole number by the denominator
  • Add the result to the numerator
  • Place the sum over the original denominator
• Multiply the resulting improper fractions together and simplify the product ($\frac{7}{3} \times \frac{4}{5} = \frac{28}{15}$)
• Use cross cancellation to simplify fractions before multiplying, reducing common factors between numerators and denominators

Reciprocals in fraction operations

• A fraction's reciprocal (multiplicative inverse) is found by flipping its numerator and denominator ($\frac{3}{4}$ has a reciprocal of $\frac{4}{3}$)
• Multiplying a fraction by its reciprocal always equals 1 $\frac{a}{b} \times \frac{b}{a} = 1$ ($\frac{2}{5} \times \frac{5}{2} = 1$)
• Reciprocals play a key role in dividing fractions by allowing division to be rewritten as multiplication ($\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1}$)
• Understanding reciprocals simplifies the process of dividing fractions ($\frac{2}{3} \div \frac{5}{6} = \frac{2}{3} \times \frac{6}{5}$)

Division using reciprocals

• To divide fractions, multiply the first fraction by the reciprocal of the second $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ ($\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4}$)
• Simplify the result to its lowest terms by dividing the numerator and denominator by their GCF ($\frac{6}{4} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2}$)
• When dividing mixed numbers:
  1. Convert them to improper fractions ($1\frac{1}{3} = \frac{4}{3}$ and $2\frac{1}{4} = \frac{9}{4}$)
  2. Multiply the first improper fraction by the reciprocal of the second ($\frac{4}{3} \div \frac{9}{4} = \frac{4}{3} \times \frac{4}{9}$)
  3. Simplify the resulting fraction ($\frac{16}{27}$)
• Dividing fractions using reciprocals avoids the more complex common denominator method

Special Types of Fractions

• A complex fraction is a fraction that contains fractions in its numerator, denominator, or both, and can be simplified by division
• A unit fraction has a numerator of 1 and is the reciprocal of a whole number (e.g., $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$)
• Understanding these special types of fractions helps in various mathematical operations and problem-solving