## The operation of sorting fractions in ascending order:

^{- 31}/_{16}, ^{- 32}/_{15}, ^{- 473}/_{18}, ^{- 515}/_{14}, ^{- 505}/_{15}, ^{- 469}/_{12}

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

#### negative improper fractions: - ^{31}/_{16}, - ^{32}/_{15}, - ^{473}/_{18}, - ^{515}/_{14}, - ^{505}/_{15}, - ^{469}/_{12};

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

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

the numerator and the denominator have no common prime factors:

31 is a prime number;

16 = 2^{4};

#### - ^{32}/_{15} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

32 = 2^{5};

15 = 3 × 5;

#### - ^{473}/_{18} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

473 = 11 × 43;

18 = 2 × 3^{2};

#### - ^{515}/_{14} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

515 = 5 × 103;

14 = 2 × 7;

#### - ^{505}/_{15} = - ^{(5 × 101)}/_{(3 × 5)} = - ^{((5 × 101) ÷ 5)}/_{((3 × 5) ÷ 5)} = - ^{101}/_{3};

#### - ^{469}/_{12} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

469 = 7 × 67;

12 = 2^{2} × 3;

## To sort fractions in ascending order, build up their denominators the same.

### Calculate LCM, the least common multiple of the denominators of the fractions.

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

In this case, LCM is also called LCD, the least common denominator.

#### The prime factorization of the denominators:

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

#### 15 = 3 × 5;

#### 18 = 2 × 3^{2};

#### 14 = 2 × 7;

#### 3 is a prime number;

#### 12 = 2^{2} × 3;

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

#### LCM (16, 15, 18, 14, 3, 12) = 2^{4} × 3^{2} × 5 × 7 = 5,040

### Calculate the expanding number of each fraction

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

#### For fraction: - ^{31}/_{16} is 5,040 ÷ 16 = (2^{4} × 3^{2} × 5 × 7) ÷ 2^{4} = 315;

#### For fraction: - ^{32}/_{15} is 5,040 ÷ 15 = (2^{4} × 3^{2} × 5 × 7) ÷ (3 × 5) = 336;

#### For fraction: - ^{473}/_{18} is 5,040 ÷ 18 = (2^{4} × 3^{2} × 5 × 7) ÷ (2 × 3^{2}) = 280;

#### For fraction: - ^{515}/_{14} is 5,040 ÷ 14 = (2^{4} × 3^{2} × 5 × 7) ÷ (2 × 7) = 360;

#### For fraction: - ^{101}/_{3} is 5,040 ÷ 3 = (2^{4} × 3^{2} × 5 × 7) ÷ 3 = 1,680;

#### For fraction: - ^{469}/_{12} is 5,040 ÷ 12 = (2^{4} × 3^{2} × 5 × 7) ÷ (2^{2} × 3) = 420;

### Expand the fractions

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

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

#### - ^{31}/_{16} = - ^{(315 × 31)}/_{(315 × 16)} = - ^{9,765}/_{5,040};

#### - ^{32}/_{15} = - ^{(336 × 32)}/_{(336 × 15)} = - ^{10,752}/_{5,040};

#### - ^{473}/_{18} = - ^{(280 × 473)}/_{(280 × 18)} = - ^{132,440}/_{5,040};

#### - ^{515}/_{14} = - ^{(360 × 515)}/_{(360 × 14)} = - ^{185,400}/_{5,040};

#### - ^{101}/_{3} = - ^{(1,680 × 101)}/_{(1,680 × 3)} = - ^{169,680}/_{5,040};

#### - ^{469}/_{12} = - ^{(420 × 469)}/_{(420 × 12)} = - ^{196,980}/_{5,040};

### The fractions have the same denominator, compare their numerators.

#### The larger the numerator the smaller the negative fraction.

## ::: Comparing operation :::

The final answer: