Fractions, theory: multiplying ordinary math fractions, how to

Tutoring: multiplying fractions - how to multiply ordinary math fractions?

The end fraction will have as a numerator the result of multiplying all the fractions' numerators and as a denominator the result of multiplying all the fractions' denominators.

Simply put, a/b * c/d = (a * c) / (b * d) - true if the numbers are coprime (no common prime factors).

How to multiply ordinary fractions? Steps.

  • Where appropriate, you should start by reducing fractions to lower terms (simplifying). If you don't know how, or if you'd like to practice the whole process, with examples, go to www.fractii.ro: reduce ordinary math fractions to lower terms, online, with explanations.
  • Factor the reduced fractions' numerators and denominators down to their prime factors.
  • The end fraction will have as a numerator the product of all the numerators' prime factors and as a denominator the product of all the denominators' prime factors.
  • Reduce all the common prime factors of the prime factorized nominator and denominator.
  • Perform effectively the multiplication operations of the remaining prime factors above the fraction line, for the numerator, and under the line, for the denominator.
  • The end fraction cannot be reduced anymore, since all the common prime factors were already reduced.
  • Whenever the case, if the fraction is an improper one, in other words if the absolute value of the fraction's numerator is larger than the absolute value of the fraction's denominator (also called a top-heavy fraction), the end fraction could be written as a mixed number (also called a mixed fraction), consisting of an integer and a proper fraction.

An example of multiplying three ordinary fractions, with explanations: 6/90 * 16/24 * 30/75.

  • to reduce each fraction to lower terms (simplify), factor each fraction's both numerator and denominator down to their prime factors.
  • Divide both the numerator and denominator of each fracton by their greatest common factor GCF (greatest common divisor GCD).
  • If you don't know how to calculate the greatest common divisor, access www.numere-prime.ro: greatest common factor (GCF) of numbers.
  • Reduce the first 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 second fraction to lowest terms (simplify): 16/24 = (24) / (23 * 3) = [(24) ÷ (23)] / [(23 * 3) ÷ (23)] = 2/3
  • Reduce the third fraction to lowest terms (simplify): 30/75 = (2 * 3 * 5) / (3 * 52) = [(2 * 3 * 5) ÷ (3 * 5)] / [(3 * 52) ÷ (3 * 5)] = 2/5
  • At this point, fractions are reduced (simplified): 6/90 * 16/24 * 30/75 = 1/15 * 2/3 * 2/5
  • Effectively multiply all the fractions's numerators and denominators:

    1/15 * 2/3 * 2/5 = (1 * 2 * 2) / (3 * 5 * 3 * 5) = 4/225

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