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How Do You Find the Common Factor of a Ratio?

Published in Ratio Simplification 2 mins read

To find the common factor of a ratio, you need to find the greatest common factor (GCF) of the numbers in the ratio. This process simplifies the ratio to its lowest terms.

Understanding Ratios and Common Factors

A ratio expresses the relationship between two or more quantities. For example, a ratio of 3:4 indicates that for every 3 units of one quantity, there are 4 units of another. Like fractions, ratios can be simplified by dividing both parts by their greatest common factor (GCF).

The GCF is the largest number that divides evenly into both numbers without leaving a remainder. Finding the GCF is crucial for simplifying ratios.

Steps to Find the Common Factor (GCF) of a Ratio:

  1. List the factors: Write down all the factors of each number in the ratio. A factor is a number that divides evenly into another number.

  2. Identify common factors: Compare the lists of factors and identify the factors that appear in both lists.

  3. Determine the greatest common factor: Select the largest number from the list of common factors. This is the greatest common factor (GCF).

  4. Simplify the ratio: Divide both parts of the ratio by the GCF to simplify the ratio to its lowest terms.

Example:

Let's simplify the ratio 12:18.

  1. Factors of 12: 1, 2, 3, 4, 6, 12
  2. Factors of 18: 1, 2, 3, 6, 9, 18
  3. Common factors: 1, 2, 3, 6
  4. GCF: 6
  5. Simplified ratio: 12 ÷ 6 : 18 ÷ 6 = 2:3

Using Prime Factorization (Alternative Method):

For larger numbers, prime factorization can be a more efficient method:

  1. Find the prime factorization: Express each number in the ratio as a product of its prime factors.

  2. Identify common prime factors: Find the prime factors that are common to both numbers.

  3. Calculate the GCF: Multiply the common prime factors together.

  4. Simplify the ratio: Divide both parts of the ratio by the GCF.

Example using Prime Factorization:

Let's simplify the ratio 24:36.

  1. Prime factorization of 24: 2 x 2 x 2 x 3 = 2³ x 3
  2. Prime factorization of 36: 2 x 2 x 3 x 3 = 2² x 3²
  3. Common prime factors: 2 x 2 x 3 = 2² x 3
  4. GCF: 12
  5. Simplified ratio: 24 ÷ 12 : 36 ÷ 12 = 2:3

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