Fractions, theory: adding ordinary 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. Wheever the case, you should start by reducing all the fractions to lowest terms (simplifying).

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

    • All fractions' denominators must be factored down to constituent prime factors (prime factorization of all denominators).
    • Lowest common multiple LCM will contain all of the denominators factors, at the highest powers.
    • Whether you don't know how or you'd like to practice the process, go to the address on the numere-prime.ro website: LCM lowest common multiple of numbers.
  • 3. Calculate each fraction's expanding number, the number that will be used to multiply fraction's both numerator and denominator 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, getting the expanding number; that number will be used to multiply both the numerator and denominator for 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 result fraction will have as a denominator the lowest common multiple calculated above.
  • 5. Whenever the case, reduce the result fraction to the lowest terms (simplify).

An example of adding fractions with different denominators (unlike denominators)

  • Let' add these three fractions:

    6/90 + 16/24 + 30/75

  • Reduce to lower terms (simplify) each fraction:
    • Factor each fraction's both numerator and denominator down to constituent prime factors (numerator and denominator prime factorization), 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); GCF contains only their common factors, by the lowest powers.
    • Whether you don't know how to calculate the greatest common divisor of two numbers or you just need some refreshment, access this page on numere-prime.ro website: greatest common factor or divisor (GCF, GCD).
    • Reduce to lowest terms (simplify) the fraction:

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

    • Reduce to lowest terms (simplify) the fraction:

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

    • Reduce to lowest terms (simplify) the fraction:

      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 three fractions' denominators. For that, factor each fraction's denominator down to constituent prime factors (each fraction's denominator prime factorization); then take ALL the denominators' 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 fractions' denominators must contain all of the denominators' factors by the highest powers:
      LCM (15, 3, 5) = LCM (3 * 5, 3, 5) = 3 * 5 = 15
  • Calculate each fraction's expanding number - the 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 expanding number:
      15 : 15 = 1
    • second fraction expanding fraction:
      15 : 3 = 5
    • third fraction expanding fraction:
      15 : 5 = 3
  • To bring fractions to the same denominator, multiply each fraction's both numerator and denominator by its corresponding expanding number calculated above:
    • the first fraction stays unchanged:

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

    • the second fraction becomes:

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

    • the third fraction becomes:

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

  • The final result of adding the fractions:

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

  • In this 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 improper, or also called a top-heavy fraction, in other words it is greater than 1, 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

More on ordinary math fractions theory:

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