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

• 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