## The operation of sorting fractions in ascending order:

^{22}/_{28}, ^{17}/_{25}, ^{25}/_{32}, ^{13}/_{31}

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

#### positive proper fractions: ^{22}/_{28}, ^{17}/_{25}, ^{25}/_{32}, ^{13}/_{31};

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

^{22}/_{28} = ^{(2 × 11)}/_{(22 × 7)} = ^{((2 × 11) ÷ 2)}/_{((22 × 7) ÷ 2)} = ^{11}/_{14};

^{17}/_{25} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

17 is a prime number;

25 = 5^{2};

^{25}/_{32} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

25 = 5^{2};

32 = 2^{5};

^{13}/_{31} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

13 is a prime number;

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:

#### 11 is a prime number;

#### 17 is a prime number;

#### 25 = 5^{2};

#### 13 is a prime number;

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

#### LCM (11, 17, 25, 13) = 5^{2} × 11 × 13 × 17 = 60,775

### Calculate the expanding number of each fraction

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

#### For fraction: ^{11}/_{14} is 60,775 ÷ 11 = (5^{2} × 11 × 13 × 17) ÷ 11 = 5,525;

#### For fraction: ^{17}/_{25} is 60,775 ÷ 17 = (5^{2} × 11 × 13 × 17) ÷ 17 = 3,575;

#### For fraction: ^{25}/_{32} is 60,775 ÷ 25 = (5^{2} × 11 × 13 × 17) ÷ 5^{2} = 2,431;

#### For fraction: ^{13}/_{31} is 60,775 ÷ 13 = (5^{2} × 11 × 13 × 17) ÷ 13 = 4,675;

### 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:

^{11}/_{14} = ^{(5,525 × 11)}/_{(5,525 × 14)} = ^{60,775}/_{77,350};

^{17}/_{25} = ^{(3,575 × 17)}/_{(3,575 × 25)} = ^{60,775}/_{89,375};

^{25}/_{32} = ^{(2,431 × 25)}/_{(2,431 × 32)} = ^{60,775}/_{77,792};

^{13}/_{31} = ^{(4,675 × 13)}/_{(4,675 × 31)} = ^{60,775}/_{144,925};

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

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

## ::: Comparing operation :::

The final answer: