## The operation of comparing fractions:

^{3}/_{4} and ^{14}/_{15}

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

^{3}/_{4} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

3 is a prime number;

4 = 2^{2};

^{14}/_{15} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

14 = 2 × 7;

15 = 3 × 5;

## 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:

#### 3 is a prime number

#### 14 = 2 × 7

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

#### LCM (3, 14) = 2 × 3 × 7 = 42

### Calculate the expanding number of each fraction

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

#### For fraction: ^{3}/_{4} is 42 ÷ 3 = (2 × 3 × 7) ÷ 3 = 14

#### For fraction: ^{14}/_{15} is 42 ÷ 14 = (2 × 3 × 7) ÷ (2 × 7) = 3

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{3}/_{4} = ^{(14 × 3)}/_{(14 × 4)} = ^{42}/_{56}

^{14}/_{15} = ^{(3 × 14)}/_{(3 × 15)} = ^{42}/_{45}

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

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

## ::: Comparing operation :::

The final answer: