Mathematical Tutorial

How to Simplify a Ratio to Lowest Terms

Simplifying a ratio reduces its terms to the smallest possible positive whole numbers while preserving the identical proportional value.

Method 1: Factoring by Greatest Common Divisor (GCD)

The most reliable method for simplifying any whole-number ratio is dividing both terms by their Greatest Common Divisor (also called the Greatest Common Factor or GCF).

Example: Simplify 84 : 144

  1. Find factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
  2. Find factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
  3. The largest common number is 12 (GCD = 12).
  4. Divide both terms: 84 ÷ 12 = 7; and 144 ÷ 12 = 12.
  5. The simplified ratio is 7 : 12.

Method 2: Simplifying Ratios with Decimals

If one or both terms in a ratio contain decimals (such as 1.75 : 2.5), you cannot calculate GCD immediately. First, eliminate the decimals by multiplying both numbers by a power of 10:

  1. Count the highest number of decimal places: 1.75 has two decimals (hundredths), 2.5 has one decimal.
  2. Multiply both terms by 100: 1.75 × 100 = 175, and 2.5 × 100 = 250.
  3. Now simplify whole-number ratio 175 : 250.
  4. GCD(175, 250) = 25.
  5. 175 ÷ 25 = 7; 250 ÷ 25 = 10.
  6. Final simplified ratio: 7 : 10.

Method 3: Simplifying Ratios with Fractions

When ratios are written with fractions (e.g. 2/3 : 4/5), multiply both sides by the Least Common Multiple (LCM) of the denominators:

  • Denominators are 3 and 5 → LCM(3, 5) = 15.
  • Multiply: (2/3 × 15) : (4/5 × 15) → 10 : 12.
  • Divide by GCD(10, 12) = 2 → 5 : 6.

Interactive Tools & Next Guides:

Instant Ratio Simplifier → How to Solve Proportions → Practice Simplification Problems →
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Reviewed for accuracy: October 2026 by Ratio Calculator Mathematical Review Team • Topic: Ratio Simplification Algorithms