Learn how to multiply common ordinary fractions

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.
• Check whether numbers are prime or not. Break the composite numbers down to prime factors, 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