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How Do You Find the Common Denominator When Ordering Fractions?

Published in Fractions 2 mins read

To find a common denominator when ordering fractions, you need to find a common multiple of the denominators of all the fractions. The most efficient approach is to find the least common multiple (LCM), which then becomes the least common denominator (LCD).

Here's a breakdown of the process:

  1. Identify the Denominators: List all the denominators of the fractions you want to order.

    • Example: If you're ordering 1/2, 2/3, and 3/4, your denominators are 2, 3, and 4.

  2. Find the Least Common Multiple (LCM): Determine the LCM of the denominators. There are a couple of ways to do this:

    • Listing Multiples: List multiples of each denominator until you find a common one.

      • Multiples of 2: 2, 4, 6, 8, 10, 12...
      • Multiples of 3: 3, 6, 9, 12...
      • Multiples of 4: 4, 8, 12...
      • In this case, the LCM is 12.

    • Prime Factorization: Find the prime factorization of each denominator. Then, take the highest power of each prime factor that appears in any of the factorizations and multiply them together.

      • 2 = 2
      • 3 = 3
      • 4 = 2 x 2 = 22
      • LCM = 22 x 3 = 4 x 3 = 12

  3. The LCM is the Common Denominator: The LCM you found will be your common denominator (LCD). In our example, the LCD is 12.

  4. Convert Fractions to Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator. To do this, divide the common denominator by the original denominator, and then multiply both the numerator and the denominator of the original fraction by the result.

    • 1/2 = (1 x 6) / (2 x 6) = 6/12 (Because 12 / 2 = 6)
    • 2/3 = (2 x 4) / (3 x 4) = 8/12 (Because 12 / 3 = 4)
    • 3/4 = (3 x 3) / (4 x 3) = 9/12 (Because 12 / 4 = 3)

  5. Order the Fractions: Now that all the fractions have the same denominator, you can easily order them by comparing their numerators.

    • 6/12 < 8/12 < 9/12, which means 1/2 < 2/3 < 3/4.

Example with Larger Numbers:

Let's order 5/8 and 7/12.

  1. Denominators: 8 and 12
  2. Find the LCM:
    • Multiples of 8: 8, 16, 24, 32...
    • Multiples of 12: 12, 24, 36...
    • LCM = 24
  3. LCD = 24
  4. Convert:
    • 5/8 = (5 x 3) / (8 x 3) = 15/24
    • 7/12 = (7 x 2) / (12 x 2) = 14/24
  5. Order: 14/24 < 15/24, so 7/12 < 5/8

By converting fractions to equivalent fractions with a common denominator, you can easily compare and order them.

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