Addition and Subtraction of Fractions

Addition and Subtraction of Fractional Numbers

Improper Fractions

• Definition: A fraction where the numerator is greater than or equal to the denominator (e.g., 7/4, 5/5).

• Conversion to Mixed Numbers:

  • Divide the numerator by the denominator.
  • The quotient is the whole number part, and the remainder over the denominator is the fractional part.
  • Example: 9/4 → 2 (whole) and 1/4 (fraction) → Mixed number: 2 1/4.

• Addition/Subtraction with Improper Fractions:

  • Ensure fractions have a common denominator.
  • Add or subtract the numerators, keeping the denominator the same.
  • Simplify if necessary.
  • Example: 7/4 + 5/4 = (7 + 5)/4 = 12/4 = 3 (improper fraction simplified).

Mixed Numbers

• Definition: A combination of a whole number and a fraction (e.g., 2 3/5).

• Conversion to Improper Fractions:

  • Multiply the whole number by the denominator and add the numerator.
  • Place the result over the original denominator.
  • Example: 2 3/5 → (2×5 + 3)/5 = 13/5.

• Addition/Subtraction with Mixed Numbers:

  • Convert mixed numbers to improper fractions.
  • Find a common denominator for the fractions.
  • Add or subtract the numerators and keep the denominator the same.
  • Convert the result back to a mixed number if necessary.
  • Example:
    • 1 1/2 + 2 1/3
    • Convert: 3/2 + 7/3.
    • Common denominator: 6 → (3×3 + 7×2)/6 = (9 + 14)/6 = 23/6.
    • Convert back: 23/6 = 3 5/6.

Key Points

• Always simplify fractions when possible.
• Keep track of proper and improper fractions during calculations.
• Converting between mixed numbers and improper fractions is crucial for addition and subtraction.

Improper Fractions

• Improper fractions have numerators equal to or greater than their denominators (e.g., 7/4, 5/5).
• To convert an improper fraction to a mixed number, divide the numerator by the denominator to obtain the whole number, while the remainder becomes the fractional part.
• Example of conversion: 9/4 results in 2 (whole) and 1/4 (fraction), yielding the mixed number 2 1/4.
• For addition or subtraction of improper fractions, ensure a common denominator is established first.
• Adjust the numerators and keep the denominator constant. Simplify the final result if needed.
• Example calculation: For 7/4 + 5/4, add numerators: (7 + 5)/4 = 12/4 = 3 after simplification.

Mixed Numbers

• Mixed numbers consist of a whole number combined with a fraction (e.g., 2 3/5).
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place this result over the original denominator.
• Example conversion: 2 3/5 converts to (2×5 + 3)/5 = 13/5.
• When adding or subtracting mixed numbers, first convert them to improper fractions.
• Find a common denominator, adjust the numerators, and maintain the common denominator through calculations.
• After obtaining the result, convert back to a mixed number when necessary.
• Example operation: Adding 1 1/2 and 2 1/3 converts to 3/2 + 7/3, with a common denominator of 6 leading to (3×3 + 7×2)/6 = (9 + 14)/6 = 23/6, which converts back to 3 5/6.

Key Points

• Always simplify fractions to their lowest form when possible for clarity.
• Differentiate between proper and improper fractions during calculations for accuracy.
• Conversions between mixed numbers and improper fractions play a crucial role in adding and subtracting fractional numbers.