## The operation of sorting fractions in ascending order:

^{97}/_{72}, ^{56}/_{77}, ^{59}/_{85}, ^{50}/_{103}

### Analyze the fractions to be compared and ordered, by category:

#### positive proper fractions: ^{56}/_{77}, ^{59}/_{85}, ^{50}/_{103};

#### 1 positive improper fraction: ^{97}/_{72};

### How to sort and order fractions by categories:

#### Any positive proper fraction is smaller than

#### any positive improper fraction

## Sort the positive proper fractions:

^{56}/_{77}, ^{59}/_{85}, ^{50}/_{103}

### Reduce (simplify) fractions to their lowest terms equivalents:

^{56}/_{77} = ^{(23 × 7)}/_{(7 × 11)} = ^{((23 × 7) ÷ 7)}/_{((7 × 11) ÷ 7)} = ^{8}/_{11}

^{59}/_{85} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

59 is a prime number;

85 = 5 × 17;

^{50}/_{103} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

50 = 2 × 5^{2};

103 is a prime number;

## To sort fractions, build them up to the same numerator.

### Calculate LCM, the least common multiple of the fractions' numerators

#### LCM will be the common numerator of the compared fractions.

#### The prime factorization of the numerators:

#### 8 = 2^{3}

#### 59 is a prime number

#### 50 = 2 × 5^{2}

#### Multiply all the unique prime factors, by the largest exponents:

#### LCM (8, 59, 50) = 2^{3} × 5^{2} × 59 = 11,800

### Calculate the expanding number of each fraction

#### Divide LCM by the numerator of each fraction:

#### For fraction: ^{8}/_{11} is 11,800 ÷ 8 = (2^{3} × 5^{2} × 59) ÷ 2^{3} = 1,475

#### For fraction: ^{59}/_{85} is 11,800 ÷ 59 = (2^{3} × 5^{2} × 59) ÷ 59 = 200

#### For fraction: ^{50}/_{103} is 11,800 ÷ 50 = (2^{3} × 5^{2} × 59) ÷ (2 × 5^{2}) = 236

### Expand the fractions

#### Build up all the fractions to the same numerator (which is LCM).

Multiply the numerators and denominators by their expanding number:

^{8}/_{11} = ^{(1,475 × 8)}/_{(1,475 × 11)} = ^{11,800}/_{16,225}

^{59}/_{85} = ^{(200 × 59)}/_{(200 × 85)} = ^{11,800}/_{17,000}

^{50}/_{103} = ^{(236 × 50)}/_{(236 × 103)} = ^{11,800}/_{24,308}

### The fractions have the same numerator, compare their denominators.

#### The larger the denominator the smaller the positive fraction.

## The fractions sorted in ascending order:

^{11,800}/_{24,308} < ^{11,800}/_{17,000} < ^{11,800}/_{16,225}

The initial fractions in ascending order:

^{50}/_{103} < ^{59}/_{85} < ^{56}/_{77}

## ::: Comparing operation :::

The final answer: