## The operation of comparing fractions:

^{132}/_{171} and ^{135}/_{179}

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

^{132}/_{171} = ^{(22 × 3 × 11)}/_{(32 × 19)} = ^{((22 × 3 × 11) ÷ 3)}/_{((32 × 19) ÷ 3)} = ^{44}/_{57}

^{135}/_{179} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

135 = 3^{3} × 5;

179 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:

#### 44 = 2^{2} × 11

#### 135 = 3^{3} × 5

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

#### LCM (44, 135) = 2^{2} × 3^{3} × 5 × 11 = 5,940

### Calculate the expanding number of each fraction

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

#### For fraction: ^{44}/_{57} is 5,940 ÷ 44 = (2^{2} × 3^{3} × 5 × 11) ÷ (2^{2} × 11) = 135

#### For fraction: ^{135}/_{179} is 5,940 ÷ 135 = (2^{2} × 3^{3} × 5 × 11) ÷ (3^{3} × 5) = 44

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{44}/_{57} = ^{(135 × 44)}/_{(135 × 57)} = ^{5,940}/_{7,695}

^{135}/_{179} = ^{(44 × 135)}/_{(44 × 179)} = ^{5,940}/_{7,876}

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

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

## ::: Comparing operation :::

The final answer: