# Fractions, theory: how to sort (in ascending order) many fractions with unlike (different) numerators and denominators?

## How to sort multiple fractions?

### 1. EQUAL DENOMINATORS but unlike numerators fractions

• a) To sort multiple positive fractions that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), simply sort the numerators: the larger the numerator the larger the fraction, ie: 24/25 > 19/25 > 16/25
• b) To sort multiple negative fractions that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), simply sort the numerators: the larger the numerator the smaller the fraction, ie: -19/25 < -17/25 < -12/25
• c) To compare multiple fractions of different signs (positive and negative) that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), the rule is that any positive fraction is larger than any negative fraction; after that use the a) and b) points above, ie: 4/25 > 2/25 > -1/25 > -11/25

### 2. EQUAL NUMERATORS but unlike denominators fractions

• a) To sort multiple positive fractions that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), simply sort the denominators: the larger the denominator the smaller the fraction, ie: 24/25 > 24/26 > 24/29
• b) To sort multiple negative fractions that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), simply sort the denominators: the larger the denominator the larger the fraction, ie: -17/25 < -17/29 < -17/35
• c) To sort multiple fractions of different signs (positive and negative) that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), the rule is that any negative fraction is smaller than any positive fraction; after that use the a) and b) points above, ie: -1/19 < -1/25 < 1/200 < 1/20

### 3. Different denominators and numerators (unlike denominators and unlike numerators) fractions

• a) To sort multiple fractions of the same sign (all positive or all negative) that have different denominators and numerators (unlike denominators and unlike numerators), fractions should be brought to the same denominator (or if it's easier, to the same numerators). Please see the next paragraph, 3.a)
• b) To sort multiple fractions of different signs (positive and negative) that have different denominators and numerators (unlike denominators and unlike numerators), the rule is that any negative fraction is smaller than any positive fraction; after that separately sort the positive fractions and separately the negative ones.

### 3.a) How to sort multiple fractions of the same sign (all positive or all negative) that have unlike denominators and unlike numerators? Below, we will bring the fractions to the same denominator, but you can sort by bringing them to the same numerator.

#### Calculate the lowest common multiple, LCM, of fractions' denominators

• Factor the fractions' denominators down to their prime factors
• The fractions' denominators lowest common multiple, LCM, will contain all their unique prime factors, by the highest powers.
• .
• If you don't know how, or if you'd like to practice the process, go to this page on numere-prime.ro website: two numbers' lowest common multiple, LCM.

#### Bring the fractions to the same denominator.

• Calculate each fraction's expanding number: that is a non-zero number obtained by dividing the lowest common multiple LCM calculated above by each fraction's denominator
• Expand each fraction: multiply each fraction's both numerator and denominator by the corresponding expanding number calculated above.
• At this point, fractions are brought to the same denominator, so it's now only a simple task of sorting fractions' numerators.
• If the fractions are positive, the larger the numerator the larger the fraction. If they are negative, the larger the numerator the smaller the fraction.

### An example of sorting three fractions of the same sign that have different denominators and numerators (unlike denominators and numerators), with explanations: 1/2 vs. 16/24 vs. 45/75

#### Reduce (simplify) each fraction to lowest terms:

• Factor both the numerator and denominator of each fraction down to prime factors
• Divide each fraction's both numerator and denominator by their greatest common factor GCF (greatest common divisor GCD)
• If you don't know how to calculate the greatest common divisor, go to numere-prime.ro: greatest common factor (GCF).
• Reduce (simplify) the fraction 1/2 ... the fraction's numerator and 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 (simplify) the fraction 16/24 = 24 / (23 * 3) = (24 ÷ 23) / ((23 * 3) ÷ 23) = 2/3
• Reduce (simplify) the 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 și 45/75 = 3/5

#### Calculate the fractions' denominators lowest common multiple, LCM:

• Factor fractions' denominators down to prime factors and then take all the unique prime factors, by the highest powers
• First fraction's denominator factored down to prime factors: denominator is 2, it's already a prime number, it cannot be prime factorized
• Second fraction's denominator factored down to prime factors: denominator is 3, it's already a prime number, it cannot be prime factorized
• Third fraction's denominator factored down to prime factors: denominator is 5, it's a prime number, it cannot be prime factorized
• The fractions' denominators lowest common multiple LCM must contain all their unique prime factors, by the highest powers: LCM (2; 3; 5) = 2 * 3 * 5 = 30.

#### Bring fractions to the same denominator:

• Each fraction's expanding number is calculated for each fraction by dividing the lowest common multiple LCM by each fraction's denominator:
• first fraction's expanding number is: 30 ÷ 2 = 15
• second fraction's expanding number is: 30 ÷ 3 = 10
• third fraction's expanding number ÷ 30 ÷ 5 = 6
• To bring fractions to the same denominator, expand fractions: multiply each fraction's both numerator and denominator by their corresponding expanding number:
• the first fraction is expanding as 1/2 = (15 * 1) / (15 * 2) = 15/30
• the second fraction is expanding as 2/3 = (10 * 2) / (10 * 3) = 20/30
• the third fraction is expanded as 3/5 = (6 * 3) / (6 * 5) = 18/30
• Sorted fractions are: 20/30 > 18/30 > 15/30, which it means that the initial fractions are sorted as: 16/24 > 45/75 > 1/2.