## The operation of sorting fractions in ascending order:

^{34}/_{10}, ^{24}/_{13}, ^{20}/_{14}

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

#### positive improper fractions: ^{34}/_{10}, ^{24}/_{13}, ^{20}/_{14};

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

^{34}/_{10} = ^{(2 × 17)}/_{(2 × 5)} = ^{((2 × 17) ÷ 2)}/_{((2 × 5) ÷ 2)} = ^{17}/_{5}

^{24}/_{13} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

24 = 2^{3} × 3;

13 is a prime number;

^{20}/_{14} = ^{(22 × 5)}/_{(2 × 7)} = ^{((22 × 5) ÷ 2)}/_{((2 × 7) ÷ 2)} = ^{10}/_{7}

## 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:

#### 5 is a prime number

#### 13 is a prime number

#### 7 is a prime number

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

#### LCM (5, 13, 7) = 5 × 7 × 13 = 455

### Calculate the expanding number of each fraction

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

#### For fraction: ^{17}/_{5} is 455 ÷ 5 = (5 × 7 × 13) ÷ 5 = 91

#### For fraction: ^{24}/_{13} is 455 ÷ 13 = (5 × 7 × 13) ÷ 13 = 35

#### For fraction: ^{10}/_{7} is 455 ÷ 7 = (5 × 7 × 13) ÷ 7 = 65

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{17}/_{5} = ^{(91 × 17)}/_{(91 × 5)} = ^{1,547}/_{455}

^{24}/_{13} = ^{(35 × 24)}/_{(35 × 13)} = ^{840}/_{455}

^{10}/_{7} = ^{(65 × 10)}/_{(65 × 7)} = ^{650}/_{455}

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

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

## ::: Comparing operation :::

The final answer: