## The operation of comparing fractions:

^{- 14}/_{27} vs. ^{- 16}/_{31}

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

#### - ^{14}/_{27} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

14 = 2 × 7;

27 = 3^{3};

#### - ^{16}/_{31} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

16 = 2^{4};

31 is a prime number;

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

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

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

#### The prime factorization of the numerators:

#### 14 = 2 × 7;

#### 16 = 2^{4};

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

#### LCM (14, 16) = 2^{4} × 7 = 112

### Calculate the expanding number of each fraction

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

#### For fraction: - ^{14}/_{27} is 112 ÷ 14 = (2^{4} × 7) ÷ (2 × 7) = 8;

#### For fraction: - ^{16}/_{31} is 112 ÷ 16 = (2^{4} × 7) ÷ 2^{4} = 7;

### Expand the fractions

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

#### Multiply the numerators and the denominators by their expanding number:

#### - ^{14}/_{27} = - ^{(8 × 14)}/_{(8 × 27)} = - ^{112}/_{216};

#### - ^{16}/_{31} = - ^{(7 × 16)}/_{(7 × 31)} = - ^{112}/_{217};

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

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

## ::: Comparing operation :::

The final answer: