## The operation of comparing fractions:

^{27}/_{41} and ^{37}/_{44}

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

^{27}/_{41} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

27 = 3^{3};

41 is a prime number;

^{37}/_{44} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

37 is a prime number;

44 = 2^{2} × 11;

## 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:

#### 27 = 3^{3}

#### 37 is a prime number

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

#### LCM (27, 37) = 3^{3} × 37 = 999

### Calculate the expanding number of each fraction

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

#### For fraction: ^{27}/_{41} is 999 ÷ 27 = (3^{3} × 37) ÷ 3^{3} = 37

#### For fraction: ^{37}/_{44} is 999 ÷ 37 = (3^{3} × 37) ÷ 37 = 27

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{27}/_{41} = ^{(37 × 27)}/_{(37 × 41)} = ^{999}/_{1,517}

^{37}/_{44} = ^{(27 × 37)}/_{(27 × 44)} = ^{999}/_{1,188}

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

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

## ::: Comparing operation :::

The final answer: