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How Do You Order Fractions with Large Denominators?

Published in Fraction Comparison 3 mins read

To order fractions with large denominators, the most common and reliable method is to find a common denominator and then compare the numerators. Here's a breakdown of the process:

1. Finding a Common Denominator

The key to comparing fractions is to have them all expressed with the same denominator. Here's how to find one:

  • Least Common Multiple (LCM): The most efficient approach is to find the least common multiple (LCM) of all the denominators. The LCM is the smallest number that all the denominators divide into evenly.

    • Prime Factorization Method: Break down each denominator into its prime factors. Then, take the highest power of each prime factor that appears in any of the denominators and multiply them together.
    • Example: If the denominators are 24, 36, and 40:
      • 24 = 23 x 3
      • 36 = 22 x 32
      • 40 = 23 x 5
      • LCM = 23 x 32 x 5 = 360

  • Common Multiple (if LCM is too difficult): If finding the LCM is too complex, you can simply multiply all the denominators together. This will always result in a common denominator, but it might be a very large number, making the subsequent calculations more difficult.

2. Converting the Fractions

Once you have a common denominator, convert each fraction so that it has that denominator. To do this, multiply both the numerator and the denominator of each fraction by the factor that will make the original denominator equal to the common denominator.

  • Example: If the fractions are 5/24, 7/36, and 9/40, and the LCM is 360:
    • 5/24 = (5 x 15) / (24 x 15) = 75/360
    • 7/36 = (7 x 10) / (36 x 10) = 70/360
    • 9/40 = (9 x 9) / (40 x 9) = 81/360

3. Comparing the Numerators

Now that all the fractions have the same denominator, you can directly compare their numerators. The fraction with the smallest numerator is the smallest fraction, and the fraction with the largest numerator is the largest fraction.

  • Example: Using the fractions 75/360, 70/360, and 81/360:
    • 70/360 < 75/360 < 81/360
    • Therefore, 7/36 < 5/24 < 9/40

4. Alternative Methods (for approximations or specific cases)

  • Convert to Decimals: Divide the numerator by the denominator for each fraction to obtain its decimal equivalent. Then, compare the decimal values. This method can be useful for large denominators, as calculators can easily handle the division.
  • Cross-Multiplication (for comparing two fractions): If you only need to compare two fractions (a/b and c/d), cross-multiply: a*d and b*c. If a*d < b*c, then a/b < c/d.
  • Benchmarking: Compare each fraction to a common benchmark, such as 1/2 or 1. This is useful if some fractions are clearly larger or smaller than the benchmark.

Example Summarized

  1. Fractions: 3/50, 7/100, 1/20
  2. LCM of Denominators (50, 100, 20): 100
  3. Convert Fractions:
    • 3/50 = 6/100
    • 7/100 = 7/100
    • 1/20 = 5/100
  4. Compare Numerators: 5 < 6 < 7
  5. Order: 1/20 < 3/50 < 7/100

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