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

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

### Simply put, (a / b) * (c / d) = (a * c) / (b * d) - true if b and d are coprime numbers (no common prime factors).

- Where appropriate, you should start by reducing fractions to lowest terms (simplifying). If you don't know how, or if you'd like to practice the whole process, with examples, go to www.fractii.ro: reduce ordinary math fractions to lower terms, online, with explanations.
- Factorize into prime factors the numerators and denominators of the reduced fractions.
- The end fraction will have as the numerator the product of all the numerators' prime factors and as the denominator the product of the denominators' prime factors.
- Reduce all the common prime factors that appear in the end fraction's both factorized nominator and denominator.
- Perform effectively the multiplying operations of the remaining prime factors above the fraction line, for the numerator, and under the line, for the denominator.
- The end fraction cannot be reduced anymore, since all common prime factors were reduced previously.
- If the case, if the fraction numerator is higher than the denominator (improper fraction), the end fraction could be written as a mixed fraction, consisting of an integer and a proper fraction.

### An example of multiplying of ordinary fractions, with explanations

- Let's multiply three ordinary fractions: 6/90 * 16/24 * 30/75.
**Reduce each fraction to lower terms (simplify):**- Factorize each fraction's both numerator and denominator into prime factors
- Divide both the numerator and denominator of each fracton by the number containing only their common factors, at lowest powers - this is the greatest common factor GCF (greatest common divisor GCD) of each fraction's numerator and denominator
- If you don't know how to calculate the greatest common divisor, access www.numere-prime.ro: greatest common factor (GCF) of numbers.
- Reduce the first fraction to lowest terms (simplify): 6/90 = (2 * 3) / (2 * 3
^{2}* 5) = [ (2 * 3) : (2 * 3) ] / [ (2 * 3^{2}* 5) : (2 * 3) ] = 1/(3 * 5) = 1/15 - Reduce the second fraction to lowest terms (simplify): 16/24 = (2
^{4}) / (2^{3}* 3) = [ (2^{4}) : (2^{3}) ] / [ (2^{3}* 3) : (2^{3}) ] = 2/3 - Reduce the third fraction to lowest terms (simplify): 30/75 = (2 * 3 * 5) / (3 * 5
^{2}) = [ (2 * 3 * 5) : (3 * 5) ] / [ (3 * 2^{5}) : (3 * 5) ] = 2/5

- At this point, fractions are reduced (simplified): 6/90 * 16/24 * 30/75 = 1/15 * 2/3 * 2/5.
- 1/15 * 2/3 * 2/5 = (1 * 2 * 2) / (3 * 5 * 3 * 5) = 4 / 225