## The operation of sorting fractions in ascending order:

^{2}/_{3}, ^{7}/_{9}, ^{5}/_{6}, ^{11}/_{18}

### Analyze the fractions to be compared and ordered, by category:

#### positive proper fractions: ^{2}/_{3}, ^{7}/_{9}, ^{5}/_{6}, ^{11}/_{18};

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

^{2}/_{3} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

2 is a prime number;

3 is a prime number;

^{7}/_{9} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

7 is a prime number;

9 = 3^{2};

^{5}/_{6} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

5 is a prime number;

6 = 2 × 3;

^{11}/_{18} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

11 is a prime number;

18 = 2 × 3^{2};

## To sort fractions in ascending order, build up their denominators the same.

### Calculate LCM, the least common multiple of the denominators of the fractions.

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

In this case, LCM is also called LCD, the least common denominator.

#### The prime factorization of the denominators:

#### 3 is a prime number

#### 9 = 3^{2}

#### 6 = 2 × 3

#### 18 = 2 × 3^{2}

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

#### LCM (3, 9, 6, 18) = 2 × 3^{2} = 18

### Calculate the expanding number of each fraction

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

#### For fraction: ^{2}/_{3} is 18 ÷ 3 = (2 × 3^{2}) ÷ 3 = 6

#### For fraction: ^{7}/_{9} is 18 ÷ 9 = (2 × 3^{2}) ÷ 3^{2} = 2

#### For fraction: ^{5}/_{6} is 18 ÷ 6 = (2 × 3^{2}) ÷ (2 × 3) = 3

#### For fraction: ^{11}/_{18} is 18 ÷ 18 = (2 × 3^{2}) ÷ (2 × 3^{2}) = 1

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{2}/_{3} = ^{(6 × 2)}/_{(6 × 3)} = ^{12}/_{18}

^{7}/_{9} = ^{(2 × 7)}/_{(2 × 9)} = ^{14}/_{18}

^{5}/_{6} = ^{(3 × 5)}/_{(3 × 6)} = ^{15}/_{18}

^{11}/_{18} = ^{(1 × 11)}/_{(1 × 18)} = ^{11}/_{18}

### The fractions have the same denominator, compare their numerators.

#### The larger the numerator the larger the positive fraction.

## ::: Comparing operation :::

The final answer: