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CC.3 Simplify fractions to lowest terms

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What does it mean to simplify a fraction?

Simplifying a fraction means making it as small as possible while keeping the same value. A simplified fraction uses the smallest possible whole numbers for the numerator and denominator.

Examples:
  • 24 simplifies to 12
  • 36 simplifies to 12
  • 48 simplifies to 12
Note

Think of simplifying like reducing a recipe. You use fewer ingredients, but the taste stays the same. The fraction's value doesn't change when you simplify it.

How to find the greatest common factor

The Greatest Common Factor (GCF) is the largest number that divides evenly into both the numerator and the denominator.

Steps to find the GCF:
  • List the factors of the numerator (numbers that divide into it evenly)
  • List the factors of the denominator
  • Find the largest number that appears in both lists
Example for 812:
  • Factors of 8: 1, 2, 4, 8
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Common factors: 1, 2, 4
  • Greatest Common Factor: 4
Note

The number 1 is always a common factor, but it won't help simplify the fraction. Look for factors greater than 1.

The steps to simplify any fraction

Follow these three steps to simplify any fraction correctly.

Simplify 69:
  1. Find the GCF of 6 and 9
    • Factors of 6: 1, 2, 3, 6
    • Factors of 9: 1, 3, 9
    • GCF = 3
  2. Divide both numerator and denominator by the GCF
    • 6 ÷ 3 = 2
    • 9 ÷ 3 = 3
  3. Write the simplified fraction: 23
Note

Always check your work! Multiply the simplified fraction's numerator and denominator by the GCF. You should get back to your original fraction.

Special cases to remember

Some fractions need special attention when simplifying.

Examples:
  • Fractions that are already simplest: 35 (GCF = 1, so it's already simplified)
  • Fractions that equal 1: 77 simplifies to 1
  • Fractions with even numbers: 1015 (GCF = 5, simplifies to 23)
  • Improper fractions: 96 (GCF = 3, simplifies to 32)
Note

When the numerator and denominator are the same number, the fraction always equals 1. When the numerator is larger, you have an improper fraction, which can be simplified or converted to a mixed number.

Why simplifying fractions matters

Simplified fractions are easier to compare, add, subtract, and use in real-life situations.

Real-world examples:
  • Recipes: A recipe calls for 24 cup of sugar, which is easier to measure as 12 cup
  • Comparing distances: Which is shorter: 36 mile or 24 mile? When simplified (12 and 12), you can see they're equal
  • Sharing fairly: If 4 out of 8 slices of pizza remain, that's 12 of the pizza left
Note

Mathematicians always write fractions in simplest form. It's like cleaning up your work so others can understand it easily.