Learn how to multiply fractions

Multiplying fractions. How to multiply ordinary math fractions? Steps. Example.

How to multiply two fractions?

When we multiply ordinary fractions, the end fraction will have:

  • as a numerator, the result of multiplying all the numerators of the fractions,
  • as a denominator, the result of multiplying all the denominators of the fractions.
  • a/b × c/d = (a × c) / (b × d)
  • a, b, c, d are integer numbers;
  • if the pairs (a × c) and (b × d) are not coprime (they have common prime factors) the end fraction should be reduced (simplified) to lower terms.

How to multiply ordinary fractions? Steps.

  • Start by reducing fractions to lower terms (simplifying).
  • Reduce math fractions to lower terms, online, with explanations.
  • Factor the numerators and the denominators of the reduced fractions: break them down to their prime factors.
  • Calculate the prime factors of numbers, online calculator
  • Above the fraction bar we write the product of all the prime factors of the fractions' numerators, without doing any calculations.
  • Below the fraction bar we write the product of all the prime factors of the fractions' denominators, without doing any calculations.
  • Cross out all the common prime factors that appear both above and below the fraction bar.
  • Multiply the remaining prime factors above the fraction bar - this will be the numerator of the resulted fraction.
  • Multiply the remaining prime factors below the fraction bar - this will be the denominator of the resulted fraction.
  • There is no need to reduce (simplify) the resulting fraction, since we have already crossed out all the common prime factors.
  • If the resulted fraction is an improper one (without considering the sign, the numerator is larger than the denominator), it could be written as a mixed number, consisting of an integer and a proper fraction of the same sign.
  • Write improper fractions as mixed numbers, online.
  • Multiply ordinary fractions, online, with explanations.

An example of multiplying three ordinary fractions, with explanations:

  • 6/90 × 80/24 × 30/75 = ?
  • Factor the numerators and the denominators of the fractions (break them down as products of prime factors) and reduce the original fractions.
    • 6/90 = (2 × 3) / (2 × 32 × 5) = ((2 × 3) ÷ (2 × 3)) / ((2 × 32 × 5) ÷ (2 × 3)) = 1/(3 × 5) = 1/15
    • 80/24 = (24 × 5) / (23 × 3) = ((24 × 5) ÷ (23)) / ((23 × 3) ÷ (23)) = (2 × 5)/3 = 10/3
    • 30/75 = (2 × 3 × 5) / (3 × 52) = ((2 × 3 × 5) ÷ (3 × 5)) / ((3 × 52) ÷ (3 × 5)) = 2/5
  • At this point, the fractions are reduced (simplified) and their numerators and denominators are factored:
    • 6/90 × 80/24 × 30/75 = 1/(3 × 5) × (2 × 5)/3 × 2/5
  • Multiply all the prime factors above and respectively below the fraction bar, crossing out the common factors:
    • 1/(3 × 5) × (2 × 5)/3 × 2/5
    • = (1 × 2 × 5 × 2) / (3 × 5 × 3 × 5)
    • = (1 × 2 × 2 × 5) / (3 × 3 × 5 × 5)
    • = (1 × 2 × 2 × 5) / (3 × 3 × 5 × 5)
    • = (2 × 2) / (3 × 3 × 5)
    • = 4/45

More on ordinary (common) math fractions theory: