## The operation of comparing fractions:

^{56}/_{65} and ^{65}/_{73}

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

^{56}/_{65} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

56 = 2^{3} × 7;

65 = 5 × 13;

^{65}/_{73} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

65 = 5 × 13;

73 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:

#### 56 = 2^{3} × 7

#### 65 = 5 × 13

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

#### LCM (56, 65) = 2^{3} × 5 × 7 × 13 = 3,640

### Calculate the expanding number of each fraction

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

#### For fraction: ^{56}/_{65} is 3,640 ÷ 56 = (2^{3} × 5 × 7 × 13) ÷ (2^{3} × 7) = 65

#### For fraction: ^{65}/_{73} is 3,640 ÷ 65 = (2^{3} × 5 × 7 × 13) ÷ (5 × 13) = 56

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{56}/_{65} = ^{(65 × 56)}/_{(65 × 65)} = ^{3,640}/_{4,225}

^{65}/_{73} = ^{(56 × 65)}/_{(56 × 73)} = ^{3,640}/_{4,088}

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

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

## ::: Comparing operation :::

The final answer: