## The operation of comparing fractions:

^{13}/_{51} and ^{19}/_{53}

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

^{13}/_{51} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

13 is a prime number;

51 = 3 × 17;

^{19}/_{53} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

19 is a prime number;

53 is a prime number;

## 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:

#### 13 is a prime number

#### 19 is a prime number

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

#### LCM (13, 19) = 13 × 19 = 247

### Calculate the expanding number of each fraction

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

#### For fraction: ^{13}/_{51} is 247 ÷ 13 = (13 × 19) ÷ 13 = 19

#### For fraction: ^{19}/_{53} is 247 ÷ 19 = (13 × 19) ÷ 19 = 13

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{13}/_{51} = ^{(19 × 13)}/_{(19 × 51)} = ^{247}/_{969}

^{19}/_{53} = ^{(13 × 19)}/_{(13 × 53)} = ^{247}/_{689}

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

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

## ::: Comparing operation :::

The final answer: