## The operation of comparing fractions:

^{60}/_{137} and ^{65}/_{144}

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

^{60}/_{137} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

60 = 2^{2} × 3 × 5;

137 is a prime number;

^{65}/_{144} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

65 = 5 × 13;

144 = 2^{4} × 3^{2};

## 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:

#### 60 = 2^{2} × 3 × 5

#### 65 = 5 × 13

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

#### LCM (60, 65) = 2^{2} × 3 × 5 × 13 = 780

### Calculate the expanding number of each fraction

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

#### For fraction: ^{60}/_{137} is 780 ÷ 60 = (2^{2} × 3 × 5 × 13) ÷ (2^{2} × 3 × 5) = 13

#### For fraction: ^{65}/_{144} is 780 ÷ 65 = (2^{2} × 3 × 5 × 13) ÷ (5 × 13) = 12

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{60}/_{137} = ^{(13 × 60)}/_{(13 × 137)} = ^{780}/_{1,781}

^{65}/_{144} = ^{(12 × 65)}/_{(12 × 144)} = ^{780}/_{1,728}

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

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

## ::: Comparing operation :::

The final answer: