## The operation of sorting fractions in ascending order:

^{10}/_{10}, ^{11}/_{13}, ^{11}/_{9}, ^{15}/_{4}, ^{16}/_{6}

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

#### 1 positive proper fraction: ^{11}/_{13};

#### positive improper fractions: ^{10}/_{10}, ^{11}/_{9}, ^{15}/_{4}, ^{16}/_{6};

### How to sort and order fractions by categories:

#### Any positive proper fraction is smaller than

#### any positive improper fraction

## Sort the positive improper fractions:

^{10}/_{10}, ^{11}/_{9}, ^{15}/_{4}, ^{16}/_{6}

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

^{10}/_{10} = ^{(10 ÷ 10)}/_{(10 ÷ 10)} = ^{1}/_{1} = 1;

^{11}/_{9} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

11 is a prime number;

9 = 3^{2};

^{15}/_{4} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

15 = 3 × 5;

4 = 2^{2};

^{16}/_{6} = ^{24}/_{(2 × 3)} = ^{(24 ÷ 2)}/_{((2 × 3) ÷ 2)} = ^{8}/_{3};

## 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:

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

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

#### 3 is a prime number;

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

#### LCM (9, 4, 3) = 2^{2} × 3^{2} = 36

### Calculate the expanding number of each fraction

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

#### For fraction: 1 is 36;

#### For fraction: ^{11}/_{9} is 36 ÷ 9 = (2^{2} × 3^{2}) ÷ 3^{2} = 4;

#### For fraction: ^{15}/_{4} is 36 ÷ 4 = (2^{2} × 3^{2}) ÷ 2^{2} = 9;

#### For fraction: ^{8}/_{3} is 36 ÷ 3 = (2^{2} × 3^{2}) ÷ 3 = 12;