## The operation of sorting fractions in ascending order:

^{- 21}/_{141}, ^{- 36}/_{195}, ^{- 33}/_{162}

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

#### negative proper fractions: - ^{21}/_{141}, - ^{36}/_{195}, - ^{33}/_{162};

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

#### - ^{21}/_{141} = - ^{(3 × 7)}/_{(3 × 47)} = - ^{((3 × 7) ÷ 3)}/_{((3 × 47) ÷ 3)} = - ^{7}/_{47}

#### - ^{36}/_{195} = - ^{(22 × 32)}/_{(3 × 5 × 13)} = - ^{((22 × 32) ÷ 3)}/_{((3 × 5 × 13) ÷ 3)} = - ^{12}/_{65}

#### - ^{33}/_{162} = - ^{(3 × 11)}/_{(2 × 34)} = - ^{((3 × 11) ÷ 3)}/_{((2 × 34) ÷ 3)} = - ^{11}/_{54}

## 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:

#### 7 is a prime number

#### 12 = 2^{2} × 3

#### 11 is a prime number

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

#### LCM (7, 12, 11) = 2^{2} × 3 × 7 × 11 = 924

### Calculate the expanding number of each fraction

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

#### For fraction: - ^{7}/_{47} is 924 ÷ 7 = (2^{2} × 3 × 7 × 11) ÷ 7 = 132

#### For fraction: - ^{12}/_{65} is 924 ÷ 12 = (2^{2} × 3 × 7 × 11) ÷ (2^{2} × 3) = 77

#### For fraction: - ^{11}/_{54} is 924 ÷ 11 = (2^{2} × 3 × 7 × 11) ÷ 11 = 84

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

#### - ^{7}/_{47} = - ^{(132 × 7)}/_{(132 × 47)} = - ^{924}/_{6,204}

#### - ^{12}/_{65} = - ^{(77 × 12)}/_{(77 × 65)} = - ^{924}/_{5,005}

#### - ^{11}/_{54} = - ^{(84 × 11)}/_{(84 × 54)} = - ^{924}/_{4,536}

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

#### The larger the denominator the larger the negative fraction.

## ::: Comparing operation :::

The final answer: