## The operation of comparing fractions:

^{9}/_{4} vs. ^{12}/_{11}

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

^{9}/_{4} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

9 = 3^{2};

4 = 2^{2};

^{12}/_{11} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

12 = 2^{2} × 3;

11 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:

#### 9 = 3^{2};

#### 12 = 2^{2} × 3;

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

#### LCM (9, 12) = 2^{2} × 3^{2} = 36

### Calculate the expanding number of each fraction

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

#### For fraction: ^{9}/_{4} is 36 ÷ 9 = (2^{2} × 3^{2}) ÷ 3^{2} = 4;

#### For fraction: ^{12}/_{11} is 36 ÷ 12 = (2^{2} × 3^{2}) ÷ (2^{2} × 3) = 3;

### 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:

^{9}/_{4} = ^{(4 × 9)}/_{(4 × 4)} = ^{36}/_{16};

^{12}/_{11} = ^{(3 × 12)}/_{(3 × 11)} = ^{36}/_{33};

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

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

## ::: Comparing operation :::

The final answer: