Fractions, theory: adding ordinary (common, simple) math fractions - how to add fractions with unlike (different) denominators?

Tutoring and a practical example: Adding ordinary (simple, common) math fractions

To add fractions with different denominators (unlike denominators), fractions should be first brought to the same denominator. How is it done?

  • 1. Whenever the case, you should start by reducing all the fractions to the lowest terms (simplifying).

  • 2. To bring all the fractions to the same denominator, you must calculate all the fractions' denominators lowest common multiple, LCM:

    • All fractions' denominators must be factored down to their constituent prime factors (all denominators prime factorizations).
    • The lowest common multiple, LCM, will contain all the denominators unique prime factors, by the highest powers.
    • Whether you don't know how or you'd like to practice the process, go to the address at numere-prime.ro: calculate LCM, the lowest common multiple of two numbers.
  • 3. Calculate each fraction's expanding number, the non-zero number that will be used to multiply both the numerator and denominator of each fraction, in order to bring all the fractions to the same common denominator:

    • Divide the lowest common multiple LCM calculated above by each fraction's denominator, in order to get to the expanding number; that non-zero number will be used to multiply both the numerator and denominator of each fraction.
  • 4. Expand fractions - multiply each fraction's both numerator and denominator by its expanding number.

    • At this point, fractions are brought to the same denominator. In order to add all the fractions simply add all the fractions' numerators. The end fraction will have as a denominator the lowest common multiple calculated above.
  • 5. Whenever the case, reduce the end fraction to the lowest terms (simplify it).

A practical example of adding fractions with different denominators (unlike denominators)

  • Let's add these 3 fractions:

    6/90 + 16/24 + 30/75

  • Reduce each fraction to lower terms (simplify):
    • Factor each fraction's both numerator and denominator down to their constituent prime factors (numerator and denominator's prime factorizations), then divide each fraction's both numerator and denominator by their greatest common factor, GCF (also called the greatest common divisor, GCD, or the highest common factor, HCF); each GCF has only the unique common factors of each pair composed of a numerator and a denominator, by the lowest powers.
    • Whether you don't know how to calculate the greatest common divisor of two numbers or just need some refreshment, access this page on numere-prime.ro: greatest common factor or divisor (GCF, GCD).
    • Reduce the 6/90 fraction to lowest terms (simplify):

      6/90 = (2 * 3)/(2 * 32 * 5) = ((2 * 3) ÷ (2 * 3))/((2 * 32 * 5) ÷ (2 * 3)) = 1/(3 * 5) = 1/15

    • Reduce the 16/24 fraction to lowest terms (simplify):

      16/24 = (24)/(23 * 3) = ((24) ÷ (23))/((23 * 3) ÷ (23)) = 2/3

    • Reduce the 30/75 fraction to lowest terms (simplify):

      30/75 = (2 * 3 * 5)/(3 * 52) = ((2 * 3 * 5) ÷ (3 * 5))/((3 * 25) ÷ (3 * 5)) = 2/5

    • At this point all the fractions are reduced (simplified):

      6/90 + 16/24 + 30/75 = 1/15 + 2/3 + 2/5

  • Next, we calculate the lowest common multiple, LCM, of all the three fractions' denominators. For that, factor each fraction's denominator down to its constituent prime factors (each fraction's denominator prime factorization); then take ALL the three denominators' unique prime factors, by the highest powers.
    • First fraction's denominator prime factorization:
      15 = 3 * 5
    • Second fraction's denominator prime factorization:
      3 is already a prime number, it cannot be prime factorized
    • Third fraction's denominator prime factorization:
      5 is a prime number, it cannot be prime factorized
    • The lowest common multiple LCM of all the fractions' denominators must contain all the denominators' unique prime factors by the highest powers:
      LCM (15, 3, 5) = LCM (3 * 5, 3, 5) = 3 * 5 = 15
  • Calculate each fraction's expanding number - the non-zero number that will be used to multiply each fraction's both numerator and denominator by. This number is calculated for each fraction by dividing the lowest common multiple LCM by each fraction's denominator:
    • first fraction's expanding number:
      15 ÷ 15 = 1
    • second fraction's expanding fraction:
      15 ÷ 3 = 5
    • third fraction's expanding fraction:
      15 ÷ 5 = 3
  • To bring the three fractions to the same denominator, expand each fraction by its corresponding expanding number calculated above:
    • the first fraction stays unchanged:

      1/15 = (1 * 1)/(1 * 15) = 1/15

    • the second fraction expands to:

      2/3 = (5 * 2)/(5 * 3) = 10/15

    • the third fraction expands to:

      2/5 = (3 * 2)/(3 * 5) = 6/15

  • The final result of adding the three fractions:

    6/90 + 16/24 + 30/75 = 1/15 + 2/3 + 2/5 = 1/15 + 10/15 + 6/15 = 17/15

  • In this particular case it was no longer needed to reduce the fraction, as the numerator and denominators are coprime numbers (prime to each other, no other common factors than 1).
  • Since the final fraction is an improper one (also called a top-heavy fraction), in other words the absolute value of the numerator is larger than the absolute value of the denominator, it can be written as a mixed number (also called a mixed fraction):

    17/15 = (15 + 2)/15 = 15/15 + 2/15 = 1 + 2/15 = 1 2/15.

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