Mixed numbers combine whole numbers and fractions, making them tricky to work with. Adding and subtracting mixed numbers requires careful handling of both parts. Visual models can help us understand the process better.

When adding mixed numbers, we combine whole numbers and fractions separately. For subtraction, we might need to borrow from the whole number. Different denominators? No problem! We'll find a common denominator first.

Adding and Subtracting Mixed Numbers

Addition of mixed numbers

• Represent each mixed number using a visual model
  • Whole numbers depicted as fully shaded regions (3 whole squares)
  • Fractions depicted as partially shaded regions (2/3 of a square shaded)
• Combine the whole number parts by adding them together (3 + 2 = 5 whole squares)
• Add the fractional parts of the mixed numbers
  • If the sum of fractions is less than 1, result is a proper fraction (1/4 + 1/4 = 2/4 or 1/2)
  • If the sum of fractions is greater than or equal to 1, convert improper fraction to a mixed number (3/4 + 5/4 = 8/4 = 2)

Mixed number addition calculations

• Add the whole number parts of the mixed numbers (2 + 3 = 5)
• Add the fractional parts of the mixed numbers (1/5 + 2/5 = 3/5)
  • If the sum of fractions is less than 1, write result as a proper fraction (1/6 + 1/6 = 2/6 or 1/3)
  • If the sum of fractions is greater than or equal to 1, convert improper fraction to a mixed number and add to whole number part (3 1/4 + 1 3/4 = 3 + 1 + 4/4 = 5)

Visual models for mixed number subtraction

• Represent each mixed number using a visual model
  • Whole numbers depicted as fully shaded regions (5 whole squares)
  • Fractions depicted as partially shaded regions (3/4 of a square shaded)
• If fractional part of subtrahend is larger than fractional part of minuend, borrow 1 from whole number part of minuend and add to fractional part (4 2/3 - 1 5/6 becomes 3 + 1 2/3 - 1 5/6 = 3 8/6 - 1 5/6)
• Subtract the fractional parts of the mixed numbers (8/6 - 5/6 = 3/6 or 1/2)
• Subtract the whole number parts of the mixed numbers (3 - 1 = 2)

Mixed number subtraction calculations

• If fractional part of subtrahend is larger than fractional part of minuend, borrow 1 from whole number part of minuend and add to fractional part
  • Convert 1 whole to an equivalent fraction with same denominator as fractional parts (1 = 4/4 in the case of 3 1/4 - 1 3/4)
• Subtract the fractional parts of the mixed numbers (5/4 - 3/4 = 2/4 or 1/2)
• Subtract the whole number parts of the mixed numbers (3 - 1 = 2)

Mixed number operations with different denominators

• Find the least common denominator (LCD) of the fractional parts (LCD of 1/2 and 1/3 is 6)
• Convert each mixed number to an equivalent mixed number with the LCD
  • Multiply both numerator and denominator of fractional part by same factor to obtain LCD (2 1/2 = 2 3/6 and 1 1/3 = 1 2/6)
  • Simplify resulting fraction if possible (2 4/6 = 2 2/3)
• Add or subtract the mixed numbers with the common denominator
  • Follow steps for adding or subtracting mixed numbers with common denominators (2 3/6 + 1 2/6 = 3 5/6)
• Simplify the result, if necessary
  • Convert any improper fraction to a mixed number (11/4 = 2 3/4)
  • Reduce fractional part to lowest terms (10/12 = 5/6)

Additional Concepts for Mixed Number Operations

• Equivalent fractions: Fractions that represent the same value but have different numerators and denominators
• Cross multiplication: A method to compare fractions by multiplying the numerator of each fraction by the denominator of the other
• Cancellation: Simplifying fractions by dividing both the numerator and denominator by their common factors
• Reciprocal: The fraction obtained by flipping the numerator and denominator (used in division of fractions)
• Fraction bar: The horizontal line separating the numerator and denominator in a fraction