## The operation of comparing fractions:

^{39}/_{90} and ^{45}/_{98}

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

^{39}/_{90} = ^{(3 × 13)}/_{(2 × 32 × 5)} = ^{((3 × 13) ÷ 3)}/_{((2 × 32 × 5) ÷ 3)} = ^{13}/_{30}

^{45}/_{98} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

45 = 3^{2} × 5;

98 = 2 × 7^{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:

#### 13 is a prime number

#### 45 = 3^{2} × 5

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

#### LCM (13, 45) = 3^{2} × 5 × 13 = 585

### Calculate the expanding number of each fraction

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

#### For fraction: ^{13}/_{30} is 585 ÷ 13 = (3^{2} × 5 × 13) ÷ 13 = 45

#### For fraction: ^{45}/_{98} is 585 ÷ 45 = (3^{2} × 5 × 13) ÷ (3^{2} × 5) = 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}/_{30} = ^{(45 × 13)}/_{(45 × 30)} = ^{585}/_{1,350}

^{45}/_{98} = ^{(13 × 45)}/_{(13 × 98)} = ^{585}/_{1,274}

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

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

## ::: Comparing operation :::

The final answer: