## The operation of sorting fractions in ascending order:

^{9}/_{20}, ^{6}/_{10}, ^{10}/_{17}, ^{11}/_{15}, ^{6}/_{7}, ^{10}/_{8}, ^{14}/_{5}

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

#### positive proper fractions: ^{9}/_{20}, ^{6}/_{10}, ^{10}/_{17}, ^{11}/_{15}, ^{6}/_{7};

#### positive improper fractions: ^{10}/_{8}, ^{14}/_{5};

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

#### Any positive proper fraction is smaller than

#### any positive improper fraction

### We will sort the fractions of each of the above categories separately.

## Sort the positive proper fractions:

^{9}/_{20}, ^{6}/_{10}, ^{10}/_{17}, ^{11}/_{15}, ^{6}/_{7}

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

^{9}/_{20} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

9 = 3^{2};

20 = 2^{2} × 5;

^{6}/_{10} = ^{(2 × 3)}/_{(2 × 5)} = ^{((2 × 3) ÷ 2)}/_{((2 × 5) ÷ 2)} = ^{3}/_{5};

^{10}/_{17} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

10 = 2 × 5;

17 is a prime number;

^{11}/_{15} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

11 is a prime number;

15 = 3 × 5;

^{6}/_{7} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

6 = 2 × 3;

7 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};

#### 3 is a prime number;

#### 10 = 2 × 5;

#### 11 is a prime number;

#### 6 = 2 × 3;

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

#### LCM (9, 3, 10, 11, 6) = 2 × 3^{2} × 5 × 11 = 990

### Calculate the expanding number of each fraction

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

#### For fraction: ^{9}/_{20} is 990 ÷ 9 = (2 × 3^{2} × 5 × 11) ÷ 3^{2} = 110;

#### For fraction: ^{3}/_{5} is 990 ÷ 3 = (2 × 3^{2} × 5 × 11) ÷ 3 = 330;

#### For fraction: ^{10}/_{17} is 990 ÷ 10 = (2 × 3^{2} × 5 × 11) ÷ (2 × 5) = 99;

#### For fraction: ^{11}/_{15} is 990 ÷ 11 = (2 × 3^{2} × 5 × 11) ÷ 11 = 90;

#### For fraction: ^{6}/_{7} is 990 ÷ 6 = (2 × 3^{2} × 5 × 11) ÷ (2 × 3) = 165;

### 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}/_{20} = ^{(110 × 9)}/_{(110 × 20)} = ^{990}/_{2,200};

^{3}/_{5} = ^{(330 × 3)}/_{(330 × 5)} = ^{990}/_{1,650};

^{10}/_{17} = ^{(99 × 10)}/_{(99 × 17)} = ^{990}/_{1,683};

^{11}/_{15} = ^{(90 × 11)}/_{(90 × 15)} = ^{990}/_{1,350};

^{6}/_{7} = ^{(165 × 6)}/_{(165 × 7)} = ^{990}/_{1,155};

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

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

## The fractions sorted in ascending order:

^{990}/_{2,200} < ^{990}/_{1,683} < ^{990}/_{1,650} < ^{990}/_{1,350} < ^{990}/_{1,155}

The initial fractions in ascending order:

^{9}/_{20} < ^{10}/_{17} < ^{6}/_{10} < ^{11}/_{15} < ^{6}/_{7}

## Sort the positive improper fractions:

^{10}/_{8} vs. ^{14}/_{5}

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

^{10}/_{8} = ^{(2 × 5)}/_{23} = ^{((2 × 5) ÷ 2)}/_{(23 ÷ 2)} = ^{5}/_{4};

^{14}/_{5} already reduced to the lowest terms;

the numerator and the denominator have no common prime factors:

14 = 2 × 7;

5 is a prime number;

## 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:

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

#### 5 is a prime number;

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

#### LCM (4, 5) = 2^{2} × 5 = 20

### Calculate the expanding number of each fraction

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

#### For fraction: ^{5}/_{4} is 20 ÷ 4 = (2^{2} × 5) ÷ 2^{2} = 5;

#### For fraction: ^{14}/_{5} is 20 ÷ 5 = (2^{2} × 5) ÷ 5 = 4;

### Expand the fractions

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

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

^{5}/_{4} = ^{(5 × 5)}/_{(5 × 4)} = ^{25}/_{20};

^{14}/_{5} = ^{(4 × 14)}/_{(4 × 5)} = ^{56}/_{20};

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

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

## The fractions sorted in ascending order:

^{25}/_{20} < ^{56}/_{20}

The initial fractions in ascending order:

^{10}/_{8} < ^{14}/_{5}

## ::: Comparing operation :::

The final answer: