1.6 Add and Subtract Fractions

Adding and subtracting fractions is a key skill in algebra. You'll learn to work with common and different denominators, simplify complex fractions, and solve equations involving fractions.

This knowledge builds on basic fraction concepts and prepares you for more advanced algebraic operations. Mastering these techniques will help you tackle more complex math problems with confidence.

Adding and Subtracting Fractions

Adding fractions with common denominators

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• Add the numerators of the fractions together while keeping the common denominator the same
• Simplify the resulting fraction by dividing the numerator and denominator by their greatest common factor (GCF) if possible
• Examples:
  • $\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$
  • $\frac{5}{12} + \frac{7}{12} = \frac{12}{12} = 1$

Adding fractions with different denominators

• Find the least common multiple (LCM) of the denominators to determine the common denominator
  • Multiply the numerator and denominator of each fraction by the factor needed to obtain the common denominator
• Add the resulting numerators together while keeping the common denominator
• Simplify the resulting fraction by dividing the numerator and denominator by their GCF if possible
• Examples:
  • $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$
  • $\frac{2}{3} + \frac{3}{5} = \frac{10}{15} + \frac{9}{15} = \frac{19}{15}$ (This is an example of an improper fraction)

Simplifying complex fractions

• Simplify the numerator and denominator of the complex fraction separately
  • Apply the order of operations (PEMDAS) to simplify each part
• Divide the simplified numerator by the simplified denominator
• Simplify the resulting fraction by dividing the numerator and denominator by their GCF if possible
• Examples:
  • $\frac{\frac{1}{2} + \frac{1}{3}}{\frac{2}{5} - \frac{1}{6}} = \frac{\frac{5}{6}}{\frac{7}{30}} = \frac{5}{6} \div \frac{7}{30} = \frac{5}{6} \cdot \frac{30}{7} = \frac{25}{7}$
  • $\frac{\frac{3}{4} - \frac{1}{6}}{\frac{2}{3} + \frac{1}{2}} = \frac{\frac{7}{12}}{\frac{7}{6}} = \frac{7}{12} \div \frac{7}{6} = \frac{7}{12} \cdot \frac{6}{7} = \frac{1}{2}$

Solving expressions with fractions

• Simplify the expression by combining like terms and performing any necessary operations
• Multiply both sides of the equation by the common denominator to eliminate fractions
• Solve the resulting equation using algebra techniques
  1. Isolate the variable on one side of the equation
  2. Perform the same operation on both sides of the equation to maintain equality
• Examples:
  • $\frac{2x}{3} + \frac{1}{4} = \frac{5}{6}$
    • Multiply both sides by 12: $8x + 3 = 10$
    • Subtract 3 from both sides: $8x = 7$
    • Divide both sides by 8: $x = \frac{7}{8}$
  • $\frac{3}{4}x - \frac{1}{2} = \frac{1}{3}$
    • Multiply both sides by 12: $9x - 6 = 4$
    • Add 6 to both sides: $9x = 10$
    • Divide both sides by 9: $x = \frac{10}{9}$

Working with Mixed Numbers and Equivalent Fractions

• Mixed numbers are a combination of a whole number and a proper fraction (e.g., 3½)
• To add or subtract mixed numbers, convert them to improper fractions first
• Equivalent fractions are fractions that represent the same value (e.g., ½ and 2/4)
• Use cross-multiplication to determine if two fractions are equivalent
• Reciprocals are fractions where the numerator and denominator are swapped (e.g., 2/3 and 3/2 are reciprocals)