- To sort fractions easier, first, we must place them into categories: positive and negative fractions, improper and proper fractions.
- As a general rule:
- any positive improper fraction is larger...
- ... than any positive proper fraction, which is larger...
- ... than zero, which is larger...
- ... than any negative proper fraction, which is larger...
- ... than any negative improper fraction.

- If the fractions to be compared and sorted are all from different categories, then our work is very easy, just follow the rule above.
- If we have more than one fraction in each category, first, we must compare the fractions in each category, then sort them by following the rule above.
- Below we will sort in ascending order three positive proper fractions.

^{1}/_{2}vs.^{16}/_{24}vs.^{45}/_{75}#### Reduce (simplify) each fraction to the lowest terms:

- Factor the numerator and the denominator of each fraction, break them down to prime factors.
- Factor numbers, break them down to prime factors, online.
- Divide the numerator and denominator of each fraction by the GCF, the greatest common factor. GCF is the product of all the common prime factors of the numerator and the denominator, multiplied by the smallest exponents - this is the greatest common factor, GCF;
- Calculate the greatest common factor (GCF), online.
- Reduce fraction
^{1}/_{2}- the numerator and the denominator are coprime numbers, they don't have any common prime factors and as a result the fraction cannot be reduced (simplified), it's irreducible. - Reduce fraction
^{16}/_{24}=^{24}/_{(23 × 3)}=^{(24 ÷ 23)}/_{((23 × 3) ÷ 23)}=^{2}/_{3} - Reduce fraction
^{45}/_{75}=^{(32 × 5)}/_{(3 × 52)}=^{((32 × 5) ÷ (3 × 5))}/_{((3 × 52) ÷ (3 × 5))}=^{3}/_{5} - At this point, the three fractions are reduced to the lowest terms (simplified):
^{1}/_{2},^{16}/_{24}=^{2}/_{3},^{45}/_{75}=^{3}/_{5}- Reduce (simplify) ordinary fractions, online.

#### Calculate the least common multiple, LCM, of the new denominators of the reduced fractions:

- LCM will be the common denominator of the sorted fractions, it is also called the
**least common denominator**. - Factor the denominators of the fractions and multiply all the unique prime factors, by the largest exponents, if there are any.
- 2 it's already a prime number, it cannot be factored anymore;
- 3 it's already a prime number, it cannot be factored anymore;
- 5 it's a prime number, it cannot be factored anymore.
- LCM (2; 3; 5) = 2 × 3 × 5 = 30.
- Calculate the least common multiple, LCM, online.

- LCM will be the common denominator of the sorted fractions, it is also called the
#### Build up the fractions to the same denominator, by expanding them:

**Each fraction's expanding number**is calculated by dividing LCM by the denominator:- first fraction: 30 ÷ 2 = 15;
- second fraction: 30 ÷ 3 = 10;
- third fraction: ÷ 30 ÷ 5 = 6.
- To
**build fractions up to the same denominator**, expand them: multiply the numerator and the denominator by their corresponding expanding number: - the first fraction:
^{1}/_{2}=^{(15 × 1)}/_{(15 × 2)}=^{15}/_{30} - the second fraction:
^{2}/_{3}=^{(10 × 2)}/_{(10 × 3)}=^{20}/_{30} - the third fraction:
^{3}/_{5}=^{(6 × 3)}/_{(6 × 5)}=^{18}/_{30}

#### The sorted fractions are:

^{15}/_{30}<^{18}/_{30}<^{20}/_{30}=>^{1}/_{2}<^{3}/_{5}<^{2}/_{3}=>^{1}/_{2}<^{45}/_{75}<^{16}/_{24}