Adding Fractions

• Adding fractions involves combining two or more fractions into a single fraction. The procedure differs slightly depending on whether the fractions have the same denominator (common denominator) or different denominators (uncommon denominators).

Fractions with Common Denominators

• When fractions have the same denominator, they are called like fractions.

• To add like fractions, you add the numerators and keep the denominator the same

• Formula: a/c + b/c = (a+b)/c, where a, b, and c are numbers and c ≠ 0.

• Example: 2/7 + 3/7 = (2+3)/7 = 5/7

Fractions with Uncommon Denominators

• When fractions have different denominators, they are called unlike fractions.

• To add unlike fractions, you must first find a common denominator. The most common choice is the Least Common Denominator (LCD).

• The LCD is the smallest multiple that the denominators have in common.

• Steps to add fractions with unlike denominators:

  • Find the LCD of the denominators.
  • Convert each fraction to an equivalent fraction with the LCD as the denominator.
  • Add the new like fractions by adding the numerators and keeping the LCD as the denominator.
  • Simplify the resulting fraction, if possible.

• Example: 1/4 + 2/5

  • Find the LCD of 4 and 5. The multiples of 4 are 4, 8, 12, 16, 20, ... and the multiples of 5 are 5, 10, 15, 20, .... The LCD is 20.
  • Convert each fraction to an equivalent fraction with a denominator of 20.
    • 1/4 = (1 * 5)/(4 * 5) = 5/20
    • 2/5 = (2 * 4)/(5 * 4) = 8/20
  • Add the like fractions: 5/20 + 8/20 = (5+8)/20 = 13/20
  • Simplify the fraction. In this case, 13/20 is already in its simplest form.

Finding the Least Common Denominator (LCD)

• Listing Multiples, list the multiples of each denominator until you find the smallest multiple that is common to all denominators.

• Prime Factorization, find the prime factorization of each denominator, then take the highest power of each prime factor that appears in any of the factorizations, and multiply them together.

• Example using Listing Multiples: Find the LCD of 3, 4, and 6.

  • Multiples of 3: 3, 6, 9, 12, 15, 18, ...
  • Multiples of 4: 4, 8, 12, 16, 20, ...
  • Multiples of 6: 6, 12, 18, 24, ...
  • The LCD is 12

• Example using Prime Factorization: Find the LCD of 8, 12, and 18

  • Prime factorization of 8: 2^3
  • Prime factorization of 12: 2^2 * 3
  • Prime factorization of 18: 2 * 3^2
  • Take the highest power of each prime factor: 2^3 and 3^2
  • Multiply them together: 2^3 * 3^2 = 8 * 9 = 72. The LCD is 72.

Adding Mixed Numbers

• A mixed number is a number consisting of an integer and a proper fraction

• When adding mixed numbers, there are two common approaches: converting to improper fractions or adding whole numbers and fractions separately.

• Method 1: Converting to Improper Fractions

  • Convert each mixed number to an improper fraction.

  • Find a common denominator (if necessary) and add the fractions.

  • Convert the resulting improper fraction back to a mixed number, if desired.

  • Example: 2 1/3 + 1 1/2

    • Convert to improper fractions: 2 1/3 = (23 + 1)/3 = 7/3 and 1 1/2 = (12 + 1)/2 = 3/2
    • Find the LCD of 3 and 2: The LCD is 6.
    • Convert each fraction to an equivalent fraction with a denominator of 6:
      • 7/3 = (72)/(32) = 14/6
      • 3/2 = (33)/(23) = 9/6
    • Add the fractions: 14/6 + 9/6 = 23/6
    • Convert back to a mixed number: 23/6 = 3 5/6

• Method 2: Adding Whole Numbers and Fractions Separately

  • Add the whole numbers.

  • Add the fractions, finding a common denominator if necessary.

  • If the sum of the fractions is an improper fraction, convert it to a mixed number and add the whole number part to the sum of the whole numbers.

  • Example: 2 1/3 + 1 1/2

    • Add the whole numbers: 2 + 1 = 3
    • Add the fractions: 1/3 + 1/2. The LCD of 3 and 2 is 6.
      • 1/3 = 2/6 and 1/2 = 3/6
      • 2/6 + 3/6 = 5/6
    • Combine the sum of the whole numbers and the sum of the fractions: 3 + 5/6 = 3 5/6

Simplifying Fractions

• Simplifying fractions means reducing the fraction to its lowest terms. This is done by dividing both the numerator and the denominator by their greatest common factor (GCF).

• Steps to simplify a fraction:

  • Find the GCF of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCF.

• Example: Simplify 12/18

  • Find the GCF of 12 and 18: The factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6.
  • Divide both the numerator and the denominator by 6:
    • 12/6 = 2
    • 18/6 = 3
  • Simplified fraction: 2/3

Properties of Addition

• Commutative Property: The order in which fractions are added does not affect the sum.

  • a/b + c/d = c/d + a/b

• Associative Property: The grouping of fractions being added does not affect the sum.

  • (a/b + c/d) + e/f = a/b + (c/d + e/f)

• Identity Property: The sum of any fraction and zero is the original fraction. Zero is the additive identity.

  • a/b + 0 = a/b

Tips and Tricks

• Always look for opportunities to simplify fractions before adding. This can make the numbers smaller and easier to work with.
• Double-check that the final answer is simplified.
• When adding multiple fractions, find the LCD of all the denominators before converting the fractions.
• Practice regularly to become more comfortable with adding fractions.