4.5 Add and Subtract Fractions with Different Denominators

Fractions can be tricky, but they're essential in math and everyday life. We'll learn how to find common denominators, add and subtract fractions, and simplify complex fractions. These skills will help you solve problems and work with expressions involving fractions.

We'll also explore problem-solving techniques and how to handle fractions with variables. By mastering these concepts, you'll be better equipped to tackle more advanced math topics and real-world situations involving fractions.

Working with Fractions

Least common denominator (LCD)

• Smallest number divisible by all denominators in a set of fractions
  • Factoring each denominator into prime factors reveals the necessary components (prime factorization)
  • LCD is the product of prime factors using the highest power of each factor (2³ × 5 for denominators 8, 20, and 40)

Equivalent fractions with common denominators

• Multiply numerator and denominator of each fraction by appropriate factor to obtain LCD
  • $\frac{3}{4}$ and $\frac{5}{6}$ with LCD 12: $\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}$, $\frac{5}{6} \times \frac{2}{2} = \frac{10}{12}$
• Ensures fractions have the same denominator for addition and subtraction

Addition and subtraction of unlike fractions

• Convert fractions to equivalent forms with common denominator (LCD) before operating
  • $\frac{2}{3} + \frac{3}{4}$: LCD is 12, so $\frac{2}{3} \times \frac{4}{4} = \frac{8}{12}$ and $\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}$
• Add or subtract numerators, keeping the common denominator ($\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$)
• Simplify result if possible ($\frac{17}{12} = 1\frac{5}{12}$)

Simplification of complex fractions

• Complex fraction has a fraction in numerator, denominator, or both
• Follow order of operations (PEMDAS) to simplify
  1. Simplify fractions within numerator and denominator
  2. Perform multiplication and division in numerator and denominator
  3. Perform addition and subtraction in numerator and denominator
  4. Divide resulting numerator by resulting denominator
• $\frac{\frac{3}{4} - \frac{1}{6}}{\frac{2}{3} + \frac{1}{2}}$: Simplify to $\frac{\frac{7}{12}}{\frac{7}{6}}$, then divide to get $\frac{7}{12} \times \frac{6}{7} = \frac{1}{2}$
  • This process can be simplified using cross multiplication

Problem-solving with fraction operations

• Identify given information and question asked
• Set up equation using given information
• Convert fractions to equivalent forms with common denominator if needed
• Perform required operations (addition or subtraction) to solve equation
• Simplify result and check if answer makes sense in problem context
  • Recipe calls for $1\frac{1}{4}$ cups flour and $\frac{3}{4}$ cup sugar. How much dry ingredients in total? $1\frac{1}{4} + \frac{3}{4} = \frac{5}{4} + \frac{3}{4} = 2$ cups

Expressions with fractions and variables

• Substitute given values for variables in expression
• Simplify fractions within expression
• Perform operations following PEMDAS
• Simplify result and express as fraction in lowest terms
• $\frac{3x}{4} - \frac{y}{6}$ with $x = \frac{2}{3}$ and $y = \frac{1}{2}$:
  1. Substitute: $\frac{3 \times \frac{2}{3}}{4} - \frac{\frac{1}{2}}{6}$
  2. Simplify: $\frac{\frac{1}{2}}{1} - \frac{1}{12} = \frac{6}{12} - \frac{1}{12} = \frac{5}{12}$

Fraction Arithmetic and Simplification Techniques

• Reciprocal: The multiplicative inverse of a fraction, found by flipping the numerator and denominator
• Divisibility rules: Shortcuts for determining if a number is divisible by another without performing division
• Fraction arithmetic: The set of operations (addition, subtraction, multiplication, division) performed on fractions