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