## The operation of comparing fractions:

^{22}/_{14} and ^{27}/_{19}

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

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

^{27}/_{19} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

27 = 3^{3};

19 is a prime number;

## 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:

#### 7 is a prime number

#### 19 is a prime number

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

#### LCM (7, 19) = 7 × 19 = 133

### Calculate the expanding number of each fraction

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

#### For fraction: ^{11}/_{7} is 133 ÷ 7 = (7 × 19) ÷ 7 = 19

#### For fraction: ^{27}/_{19} is 133 ÷ 19 = (7 × 19) ÷ 19 = 7

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{11}/_{7} = ^{(19 × 11)}/_{(19 × 7)} = ^{209}/_{133}

^{27}/_{19} = ^{(7 × 27)}/_{(7 × 19)} = ^{189}/_{133}

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

#### The larger the numerator the larger the positive fraction.

## ::: Comparing operation :::

The final answer: