Operations with Fractions: Add, Subtract, Multiply, Divide

1. Adding Fractions

To add fractions, the denominators must be the same.

Same Denominator

  • Add the numerators and keep the denominator
  • Example: 2/5 + 1/5 = (2+1)/5 = 3/5

Different Denominators

  1. Find a common denominator (usually the LCM)
  2. Convert each fraction
  3. Add the numerators

Example: 1/3 + 1/4

  • LCM of 3 and 4 is 12
  • 1/3 = 4/12 (multiply by 4)
  • 1/4 = 3/12 (multiply by 3)
  • 4/12 + 3/12 = 7/12

2. Subtracting Fractions

Subtraction follows the same steps as addition.

Same Denominator

  • Subtract the numerators and keep the denominator
  • Example: 4/7 - 2/7 = (4-2)/7 = 2/7

Different Denominators

  1. Find the LCM
  2. Convert each fraction
  3. Subtract the numerators

Example: 3/4 - 1/6

  • LCM of 4 and 6 is 12
  • 3/4 = 9/12
  • 1/6 = 2/12
  • 9/12 - 2/12 = 7/12

3. Multiplying Fractions

Multiplying is simpler—no common denominator needed.

Steps:

  1. Multiply the numerators together
  2. Multiply the denominators together
  3. Simplify if possible

Example: 2/3 × 3/5

  • (2×3)/(3×5) = 6/15
  • Simplify: 6/15 ÷ 3/3 = 2/5

4. Dividing Fractions

To divide, multiply by the reciprocal (flip the second fraction).

Steps:

  1. Find the reciprocal of the second fraction
  2. Multiply the first fraction by the reciprocal
  3. Simplify if possible

Example: 3/4 ÷ 2/5

  • Reciprocal of 2/5 is 5/2
  • 3/4 × 5/2 = (3×5)/(4×2) = 15/8
  • Convert to mixed number: 15/8 = 1 7/8

Tips for Success

  • Simplify Early: Reduce fractions before calculating to make numbers smaller
  • Check Denominators: For addition and subtraction, always align denominators first
  • Practice Mixed Numbers: Convert to improper fractions for calculations, then back if needed

Example with Mixed Numbers: 1 1/2 + 2 1/3

  • Convert to improper: 3/2 + 7/3
  • Find LCM (6): 9/6 + 14/6
  • Add: 23/6
  • Convert back: 3 5/6

Practice Problem

Solve these problems:

  1. 2/3 + 1/6
  2. 5/8 - 1/4
  3. 3/7 × 2/5
  4. 4/9 ÷ 2/3

Answers:

  1. 5/6
  2. 3/8
  3. 6/35
  4. 2/3

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