Operations with Fractions (Unlike Denominators) • Topic 1 of 3

Adding & Subtracting Fractions with Unlike Denominators

The Problem with Unlike Denominators

You cannot add fractions that have different denominators directly:

  • ½ + ⅓ = ❌ (can't add yet)

The Solution: Find a Common Denominator

The common denominator is a number that both denominators divide into evenly. The easiest is the Least Common Denominator (LCD).

Steps:

  1. Find the LCD (smallest number both denominators divide into)
  2. Convert each fraction to equivalent fraction with LCD
  3. Add or subtract the numerators
  4. Keep the denominator the same
  5. Simplify if needed

Example: ½ + ⅓

  • LCD of 2 and 3 = 6
  • ½ = 3/6
  • ⅓ = 2/6
  • 3/6 + 2/6 = (5)/(6)
ADDING ½ + ⅓ VISUALLY:

    ½                     ⅓
    ┌─────────────┐       ┌─────────┐
    │░░░░░░░░░░░░░│       │░░░░░░░░░│
    │░░░░░░░░░░░░░│       │░░░░░░░░░│
    └─────────────┘       └─────────┘
    
    CONVERT TO SIXTHS:
    
    ½ = 3/6               ⅓ = 2/6
    
    ┌───┬───┬───┐        ┌───┬───┐
    │░░░│░░░│░░░│        │░░░│░░░│
    └───┴───┴───┘        └───┴───┘
    
    ADD: 3 sixths + 2 sixths = 5 sixths (5/6)


FINDING LCD (Number Line Method):

    Multiples of 2: 2, 4, 6, 8, 10...
    Multiples of 3: 3, 6, 9, 12, 15...
    
    First common multiple: 6 ← LCD!
Solved Examples
1
Worked Example

Add ⅔ + ⅕

Solution
  • LCD of 3 and 5 = 15
  • ⅔ = (2×5)/(3×5) = 10/15
  • ⅕ = (1×3)/(5×3) = 3/15
  • 10/15 + 3/15 = 13/15
2
Worked Example

Subtract ⅞ - ¼

Solution
  • LCD of 8 and 4 = 8
  • ⅞ = (7)/(8) (stays)
  • ¼ = (1×2)/(4×2) = 2/8
  • (7)/(8) - 2/8 = (5)/(8)
3
Worked Example

Add 2⅓ + 1½

Solution
  • Convert to improper fractions: 2⅓ = (7)/(3), 1½ = (3)/(2)
  • LCD of 3 and 2 = 6
  • (7)/(3) = 14/6
  • (3)/(2) = 9/6
  • 14/6 + 9/6 = 23/6 = 3⅚

Key Points

  • Find common denominator before adding/subtracting
  • LCD is the smallest number both denominators divide into
  • Multiply numerator and denominator by same number
  • Never add denominators, only numerators
  • Convert mixed numbers to improper fractions first
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.To add unlike fractions, first find a:
Explanation: Common denominator.
Q2.$\tfrac12+\tfrac13=$
Explanation: $\tfrac36+\tfrac26$.
Q3.$\tfrac34-\tfrac14=$
Explanation: $\tfrac24=\tfrac12$.
Q4.The common denominator is usually the ___ of the denominators.
Explanation: LCM.
Practice more MCQs → 📝 Assignment · 35 marks