## Tutoring: multiplying fractions - how to multiply ordinary math fractions?

#### The end fraction will have: - - as a numerator, the result of multiplying all the fractions' numerators and
- - as a denominator, the result of multiplying all the fractions' denominators.

Simply put, ^{a}/_{b} × ^{c}/_{d} = ^{(a × c)} / _{(b × d)} - true if the numbers are coprime (no common prime factors); otherwise, the fractions should be reduced.

### How to multiply ordinary fractions? Steps.

- 1) Start by reducing fractions to lower terms (simplifying). Reduce ordinary math fractions to lower terms, online, with explanations.
- 2) Factor the reduced fractions' numerators and denominators down to their prime factors. Calculate the prime factors of numbers, online calculator
- 3) The end fraction will have: 1) as a numerator the product of all the numerators' prime factors and 2) as a denominator the product of all the denominators' prime factors.
- 4) Reduce all the common prime factors of the factored numerators and denominators.
- 5) Perform effectively the multiplication operations of the remaining prime factors above the fraction line, for the numerator, and under the line, for the denominator.
- 6) The end fraction cannot be reduced anymore, since all the common prime factors were already reduced.
- 7) Whenever the case, if the fraction is an improper one (also called a top-heavy fraction - numerator larger than denominator), the end fraction could be written as a mixed number (also called a mixed fraction), consisting of an integer and a proper fraction of the same sign.

### An example of multiplying three ordinary fractions, with explanations: ^{6}/_{90} × ^{80}/_{24} × ^{30}/_{75}.

- Reduce the first fraction to lowest terms (simplify):
^{6}/_{90}=^{(2 × 3)}/_{(2 × 32 × 5)}=^{((2 × 3) ÷ (2 × 3))}/_{((2 × 32 × 5) ÷ (2 × 3))}=^{1}/_{(3 × 5)}=^{1}/_{15} - Reduce the second fraction to lowest terms (simplify):
^{80}/_{24}=^{(24 × 5)}/_{(23 × 3)}=^{((24 × 5) ÷ (23))}/_{((23 × 3) ÷ (23))}=^{(2 × 5)}/_{3}=^{10}/_{3} - Reduce the third fraction to lowest terms (simplify):
^{30}/_{75}=^{(2 × 3 × 5)}/_{(3 × 52)}=^{((2 × 3 × 5) ÷ (3 × 5))}/_{((3 × 52) ÷ (3 × 5))}=^{2}/_{5} - At this point, fractions are reduced (simplified):
^{6}/_{90}×^{80}/_{24}×^{30}/_{75}=^{1}/_{(3 × 5)}×^{(2 × 5)}/_{3}×^{2}/_{5} - Effectively multiply all the fractions's numerators and denominators:

^{1}/_{(3 × 5)}×^{(2 × 5)}/_{3}×^{2}/_{5}=^{(1 × 2 × 2 × 5)}/_{(3 × 3 × 5 × 5)}=^{(2 × 2)}/_{(3 × 3 × 5)}=^{4}/_{45}