## The operation of comparing fractions:

^{13}/_{32} vs. ^{16}/_{37}

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

^{13}/_{32} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

13 is a prime number;

32 = 2^{5};

^{16}/_{37} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

16 = 2^{4};

37 is a prime number;

## To sort fractions, build them up to the same numerator.

### Calculate LCM, the least common multiple of the numerators of the fractions

#### LCM will be the common numerator of the compared fractions.

#### The prime factorization of the numerators:

#### 13 is a prime number;

#### 16 = 2^{4};

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

#### LCM (13, 16) = 2^{4} × 13 = 208

### Calculate the expanding number of each fraction

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

#### For fraction: ^{13}/_{32} is 208 ÷ 13 = (2^{4} × 13) ÷ 13 = 16;

#### For fraction: ^{16}/_{37} is 208 ÷ 16 = (2^{4} × 13) ÷ 2^{4} = 13;

### Expand the fractions

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

#### Multiply the numerators and the denominators by their expanding number:

^{13}/_{32} = ^{(16 × 13)}/_{(16 × 32)} = ^{208}/_{512};

^{16}/_{37} = ^{(13 × 16)}/_{(13 × 37)} = ^{208}/_{481};

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

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

## ::: Comparing operation :::

The final answer: