🔵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:
- Find the LCD (smallest number both denominators divide into)
- Convert each fraction to equivalent fraction with LCD
- Add or subtract the numerators
- Keep the denominator the same
- 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!👀 See a worked example
Add ⅔ + ⅕
- 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
Subtract ⅞ - ¼
- LCD of 8 and 4 = 8
- ⅞ = (7)/(8) (stays)
- ¼ = (1×2)/(4×2) = 2/8
- (7)/(8) - 2/8 = (5)/(8)
Add 2⅓ + 1½
- 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
🤖 Vidi's 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
🟢Multiplying Fractions
Multiplying Fractions is EASY!
Unlike addition, you DON'T need common denominators. Just multiply straight across!
Rule: (a/b) × (c/d) = (a×c)/(b×d)
Steps:
- Multiply the numerators
- Multiply the denominators
- Simplify if needed
Example: ⅔ × ⅘ = (2×4)/(3×5) = 8/15
Multiplying Mixed Numbers:
- Convert mixed numbers to improper fractions
- Multiply straight across
- Convert back to mixed number
VISUAL REPRESENTATION: ⅔ × ¾
Step 1: Start with ⅔ of a whole
┌───┬───┐
│░░░│░░░│░░░ (2 of 3 columns)
└───┴───┘
Step 2: Take ¾ of that shaded part
┌───┬───┐
│░░░│░░░│░░░
│░░░│░░░│░░░
└───┴───┘
Result: 6 out of 12 squares = 6/12 = ½
AREA MODEL: ⅔ × ¾
Total rectangle = 3 × 4 = 12 equal parts
┌─────┬─────┬─────┬─────┐
│ │ │ │ │
│░░░░░│░░░░░│░░░░░│ │ Row 1 (¾)
├─────┼─────┼─────┼─────┤
│░░░░░│░░░░░│░░░░░│ │ Row 2 (¾)
├─────┼─────┼─────┼─────┤
│ │ │ │ │ Row 3
└─────┴─────┴─────┴─────┘
Col1 Col2 Col3 Col4
(⅔ of columns)
Overlap = 6 squares = 6/12 = ½👀 See a worked example
Multiply ⅖ × ⅗
- Multiply numerators: 2 × 3 = 6
- Multiply denominators: 5 × 7 = 35
- Answer: 6/35 (already simplified)
Multiply 1½ × 2⅓
- Convert: 1½ = (3)/(2), 2⅓ = (7)/(3)
- Multiply: (3)/(2) × (7)/(3) = 21/6
- Simplify: 21/6 = (7)/(2) = 3½
Multiply ⅚ × 9
- Write 9 as 9/1
- Multiply: (5×9)/(6×1) = 45/6
- Simplify: 45/6 = 15/2 = 7½
🤖 Vidi's Key Points
- Multiply numerators, multiply denominators
- No common denominator needed!
- Simplify before multiplying (cross-cancel) to make easier
- Convert mixed numbers to improper fractions first
- Any whole number = that number over 1
🟣Dividing Fractions
The "Keep-Change-Flip" Rule
Dividing fractions is easy with this trick:
Keep the first fraction
Change ÷ to ×
Flip the second fraction (reciprocal)
Rule: (a)/(b) ÷ (c)/(d) = (a)/(b) × (d)/(c) = (a × d)/(b × c)
Example: ¾ ÷ 2 = ¾ ÷ 2/1 = ¾ × ½ = (3)/(8)
Dividing Mixed Numbers:
- Convert mixed numbers to improper fractions
- Use Keep-Change-Flip
- Simplify
VISUAL EXPLANATION: ½ ÷ ¼
"How many ¼ are in ½?"
½ of a whole: ┌─────────────┐
│░░░░░░░░░░░░░│
└─────────────┘
¼ of a whole: ┌───────┐
│░░░░░░░│
└───────┘
Count how many ¼ fit into ½:
┌───────┬───────┐
│░░░░░░░│░░░░░░░│
└───────┴───────┘
¼ ¼
Answer: 2 (½ ÷ ¼ = 2)
KEEP-CHANGE-FLIP STEPS:
¾ ÷ ⅚
KEEP CHANGE FLIP
¾ → × → ⁶⁄₅
= ¾ × ⁶⁄₅ = ¹⁸⁄₂₀ = ⁹⁄₁₀👀 See a worked example
Divide ⅔ ÷ ⅘
- Keep: ⅔
- Change: ×
- Flip: ⅘ → 5/4
- Multiply: ⅔ × 5/4 = (2×5)/(3×4) = 10/12 = (5)/(6)
Divide ⅚ ÷ 3
- Write 3 as 3/1
- Keep: ⅚
- Change: ×
- Flip: 3/1 → (1)/(3)
- Multiply: ⅚ × ⅓ = 5/18
Divide 2⅓ ÷ 1½
- Convert: 2⅓ = (7)/(3), 1½ = (3)/(2)
- Keep: (7)/(3)
- Change: ×
- Flip: (3)/(2) → (2)/(3)
- Multiply: (7)/(3) × (2)/(3) = 14/9 = 1⁵⁄₉
🤖 Vidi's Key Points
- Keep-Change-Flip: Keep 1st, Change ÷ to ×, Flip 2nd
- Reciprocal of a fraction = flip numerator and denominator
- Whole number = number over 1
- Convert mixed numbers before starting
- Simplify your answer
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