## The operation of comparing fractions:

^{- 99}/_{1,000} and ^{- 101}/_{1,002}

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

#### - ^{99}/_{1,000} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

99 = 3^{2} × 11;

1,000 = 2^{3} × 5^{3};

#### - ^{101}/_{1,002} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

101 is a prime number;

1,002 = 2 × 3 × 167;

## 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:

#### 99 = 3^{2} × 11

#### 101 is a prime number

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

#### LCM (99, 101) = 3^{2} × 11 × 101 = 9,999

### Calculate the expanding number of each fraction

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

#### For fraction: - ^{99}/_{1,000} is 9,999 ÷ 99 = (3^{2} × 11 × 101) ÷ (3^{2} × 11) = 101

#### For fraction: - ^{101}/_{1,002} is 9,999 ÷ 101 = (3^{2} × 11 × 101) ÷ 101 = 99

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

#### - ^{99}/_{1,000} = - ^{(101 × 99)}/_{(101 × 1,000)} = - ^{9,999}/_{101,000}

#### - ^{101}/_{1,002} = - ^{(99 × 101)}/_{(99 × 1,002)} = - ^{9,999}/_{99,198}

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

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

## ::: Comparing operation :::

The final answer: