## Tutoring: Sorting (ordering) ordinary fractions

## How to sort ascending multiple fractions?

### For ordinary fractions with equal denominators and different numerators (like denominators, unlike numerators), simply sort fractions numerators by value: the smaller fraction is the one with the smaller numerator, and so on, up to the larger (bigger, greater) fraction, which is the one with the larger numerator.

### For ordinary fractions with equal numerators and different denominators (like numerators, unlike denominators), simply sort fractions denominators by value: the smaller fraction is the one with the bigger (larger, greater) denominator, and so on, up to the bigger (larger, greater) fraction, which is the one with the smaller denominator.

### For ordinary fractions with different denominators and numerators (unlike denominators, unlike numerators), fractions should be first brought to the same denominator (or the same numerator, if simpler) and then simply sort fractions numerators by value:

- Where appropriate, you should start by reducing (simplifying) all the fractions to lower terms.

If you don't know how, or if you'd like to practice the whole process, with examples, go to: reduce (simplify) multiple fractions to lower terms, online, with explanations. - To bring all the fractions to the same denominator, you must calculate the lowest common multiple, LCM, of all fractions' denominators:
- All denominators are factorized into prime factors .
- Their lowest common multiple, LCM, will contain ALL the denominators' factors, at the highest powers. .
- If you don't know how, or if you'd like to practice the process, go to this page on www.numere-prime.ro: lowest common multiple, LCM, of two numbers.

- Multiplying number that will multiply each fraction's both numerator and denominator must be calculated for each fraction: divide the lowest common multiple, LCM, calculated above, by each fraction's denominator, getting the corresponding multiplying number
- Multiply each fraction's both the numerator and denominator by the multiplying number calculated above.
- At this point, fractions are brought to the same denominator, so it simply remains to sort (order) fractions' numerators.
- The smaller fraction is the one with the smaller numerator, and so on, up to the bigger (larger, greater) fraction, which is the one with the bigger numerator.

### An example of sorting (ordering) three ordinary fractions with different denominators and numerators (unlike denominators, unlike numerators)

- Let' sort (order) these fractions: 1/2, 16/24 and 45/75.
**Reduce each fraction to lower terms (simplify):**- Factorize each fraction's both the 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 their greatest common factor, GCF, (greatest common divisor, GCD)
- If you don't know how to calculate the greatest common factor (divisor), GCF, go to www.numere-prime.ro: calculate the greatest common factor (GCF).
- Reduce the fraction 1/2 to the lowest terms (simplify): it' already reduced to the lowest terms.
- Reduce the fraction 16/24 to the lowest terms (simplify): 16/24 = 2
^{4}/ (2^{3}* 3) = (2^{4}: 2^{3}) / (2^{3}* 3 : 2^{3}) = 2/3 - Reduce the fraction 45/75 to the lowest terms (simplify): 45/75 = (3
^{2}* 5 : (3 * 5^{2})) = (3^{2}* 5 : (3 * 5)) / (3 * 2^{5}: (3 * 5)) = 3/5

- At this point, fractions are reduced (simplified) to the lowest terms: 1/2, 16/24 = 2/3 and 45/75 = 3/5
- Next,
**calculate the lowest common multiple, LCM**of these fractions' denominators.- Factorize denominators into prime factors and take ALL the factors contained, at the highest powers
- First fraction's denominator factorized into prime factors: denominator is 2, it's a prime number, it cannot be factorized anymore
- Second fraction's denominator factorized into prime factors: denominator is 3, it's a prime number, it cannot be factorized anymore
- Third fraction's denominator factorized into prime factors: denominator is 5, it's a prime number, it cannot be factorized anymore
- The denominators' lowest common multiple, LCM, must contain ALL their factors, at the highest powers: LCM (2, 3, 5) = 2 * 3 * 5 = 30.

**Multiplying number**for each fraction is calculated by dividing the lowest common multiple, LCM, by each fraction's denominator:- multiplying number for the first fraction is: 30 : 2 = 15
- multiplying number for the second fraction is: 30 : 3 = 10
- multiplying number for the third fraction is: 30 : 5 = 6

- To
**bring fractions to the same denominator**, multiply each of them by their own multiplying number:- the first fraction yields: 1/2 = (15 * 1) / (15 * 2) = 15/30
- the second fraction yields: 2/3 = (10 * 2) / (10 * 3) = 20/30
- the third fraction yields: 3/5 = (6 * 3) / (6 * 5) = 18/30

- Obviously, smaller fractions are the ones with smaller numerators, so: 15/30 < 18/30 < 20/30, which it means that the initial fractions are to be sorted (order) as: 1/2 < 45/75 < 16/24.