# 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.

### 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