## 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}