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When Adding Fractions How Do You Find a Common Denominator?

Published in Fractions 2 mins read

To find a common denominator when adding fractions, you need to find a common multiple of the denominators of the fractions you are adding. The least common multiple (LCM) is often the easiest choice, but any common multiple will work.

Methods for Finding a Common Denominator

Here are a few common methods for finding a common denominator:

1. Listing Multiples

  • List the multiples of each denominator.

  • Identify the smallest multiple that appears in both lists. This is the Least Common Multiple (LCM) and can be used as the common denominator.

    Example:

    Adding 1/4 and 1/6.

    Multiples of 4: 4, 8, 12, 16, 20, 24,...
    Multiples of 6: 6, 12, 18, 24, 30,...

    The LCM of 4 and 6 is 12.

2. Prime Factorization

  • Find the prime factorization of each denominator.

  • Identify all the unique prime factors and their highest powers present in any of the factorizations.

  • Multiply these prime factors raised to their highest powers to find the LCM.

    Example:

    Adding 1/4 and 1/6.

    Prime factorization of 4: 2 x 2 = 22
    Prime factorization of 6: 2 x 3

    The LCM is 22 x 3 = 4 x 3 = 12.

3. Multiplying Denominators

  • Multiply all the denominators together. This will always give you a common denominator, although it may not be the least common denominator. This method is particularly useful when the denominators have no common factors.

  • After finding the common denominator, you'll need to adjust the numerators accordingly so that the value of each fraction remains the same.

    Example:

    Adding 1/4 and 1/6.

    Common denominator: 4 x 6 = 24

    Then, convert the fractions:

    1/4 = (1 x 6)/(4 x 6) = 6/24
    1/6 = (1 x 4)/(6 x 4) = 4/24

Adjusting the Numerators

Once you have a common denominator, you must adjust the numerators of each fraction. To do this, determine what factor you multiplied the original denominator by to get the common denominator. Then, multiply the original numerator by the same factor.

Example (Using LCM of 12):

Adding 1/4 and 1/6 using a common denominator of 12:

  • For 1/4, we multiplied the denominator 4 by 3 to get 12. So, we multiply the numerator 1 by 3, resulting in 3/12.
  • For 1/6, we multiplied the denominator 6 by 2 to get 12. So, we multiply the numerator 1 by 2, resulting in 2/12.

Now we can add the fractions: 3/12 + 2/12 = 5/12.

By finding a common denominator, you can express the fractions with the same sized "pieces," making addition (and subtraction) straightforward.

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