### 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:

#### 19 is a prime number

#### 40 = 2^{3} × 5

#### 50 = 2 × 5^{2}

#### 41 is a prime number

#### 13 is a prime number

#### 3 is a prime number

#### 5 is a prime number

#### 51 = 3 × 17

#### 52 = 2^{2} × 13

#### 53 is a prime number

#### 22 = 2 × 11

#### 49 = 7^{2}

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

#### LCM (19, 40, 50, 41, 13, 3, 5, 51, 52, 53, 22, 49) = 2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53 = 2,950,839,691,800

### Calculate the expanding number of each fraction

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

#### For fraction: ^{27}/_{19} is 2,950,839,691,800 ÷ 19 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 19 = 155,307,352,200

#### For fraction: ^{89}/_{40} is 2,950,839,691,800 ÷ 40 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ (2^{3} × 5) = 73,770,992,295

#### For fraction: ^{111}/_{50} is 2,950,839,691,800 ÷ 50 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ (2 × 5^{2}) = 59,016,793,836

#### For fraction: ^{119}/_{41} is 2,950,839,691,800 ÷ 41 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 41 = 71,971,699,800

#### For fraction: ^{27}/_{13} is 2,950,839,691,800 ÷ 13 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 13 = 226,987,668,600

#### For fraction: ^{13}/_{3} is 2,950,839,691,800 ÷ 3 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 3 = 983,613,230,600

#### For fraction: ^{26}/_{5} is 2,950,839,691,800 ÷ 5 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 5 = 590,167,938,360

#### For fraction: ^{311}/_{51} is 2,950,839,691,800 ÷ 51 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ (3 × 17) = 57,859,601,800

#### For fraction: ^{323}/_{52} is 2,950,839,691,800 ÷ 52 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ (2^{2} × 13) = 56,746,917,150

#### For fraction: ^{526}/_{53} is 2,950,839,691,800 ÷ 53 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 53 = 55,676,220,600

#### For fraction: ^{221}/_{22} is 2,950,839,691,800 ÷ 22 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ (2 × 11) = 134,129,076,900

#### For fraction: ^{502}/_{49} is 2,950,839,691,800 ÷ 49 = (2^{3} × 3 × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 × 41 × 53) ÷ 7^{2} = 60,221,218,200

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{27}/_{19} = ^{(155,307,352,200 × 27)}/_{(155,307,352,200 × 19)} = ^{4,193,298,509,400}/_{2,950,839,691,800}

^{89}/_{40} = ^{(73,770,992,295 × 89)}/_{(73,770,992,295 × 40)} = ^{6,565,618,314,255}/_{2,950,839,691,800}

^{111}/_{50} = ^{(59,016,793,836 × 111)}/_{(59,016,793,836 × 50)} = ^{6,550,864,115,796}/_{2,950,839,691,800}

^{119}/_{41} = ^{(71,971,699,800 × 119)}/_{(71,971,699,800 × 41)} = ^{8,564,632,276,200}/_{2,950,839,691,800}

^{27}/_{13} = ^{(226,987,668,600 × 27)}/_{(226,987,668,600 × 13)} = ^{6,128,667,052,200}/_{2,950,839,691,800}

^{13}/_{3} = ^{(983,613,230,600 × 13)}/_{(983,613,230,600 × 3)} = ^{12,786,971,997,800}/_{2,950,839,691,800}

^{26}/_{5} = ^{(590,167,938,360 × 26)}/_{(590,167,938,360 × 5)} = ^{15,344,366,397,360}/_{2,950,839,691,800}

^{311}/_{51} = ^{(57,859,601,800 × 311)}/_{(57,859,601,800 × 51)} = ^{17,994,336,159,800}/_{2,950,839,691,800}

^{323}/_{52} = ^{(56,746,917,150 × 323)}/_{(56,746,917,150 × 52)} = ^{18,329,254,239,450}/_{2,950,839,691,800}

^{526}/_{53} = ^{(55,676,220,600 × 526)}/_{(55,676,220,600 × 53)} = ^{29,285,692,035,600}/_{2,950,839,691,800}

^{221}/_{22} = ^{(134,129,076,900 × 221)}/_{(134,129,076,900 × 22)} = ^{29,642,525,994,900}/_{2,950,839,691,800}

^{502}/_{49} = ^{(60,221,218,200 × 502)}/_{(60,221,218,200 × 49)} = ^{30,231,051,536,400}/_{2,950,839,691,800}

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

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

## The fractions sorted in ascending order:

^{4,193,298,509,400}/_{2,950,839,691,800} < ^{6,128,667,052,200}/_{2,950,839,691,800} < ^{6,550,864,115,796}/_{2,950,839,691,800} < ^{6,565,618,314,255}/_{2,950,839,691,800} < ^{8,564,632,276,200}/_{2,950,839,691,800} < ^{12,786,971,997,800}/_{2,950,839,691,800} < ^{15,344,366,397,360}/_{2,950,839,691,800} < ^{17,994,336,159,800}/_{2,950,839,691,800} < ^{18,329,254,239,450}/_{2,950,839,691,800} < ^{29,285,692,035,600}/_{2,950,839,691,800} < ^{29,642,525,994,900}/_{2,950,839,691,800} < ^{30,231,051,536,400}/_{2,950,839,691,800}

The initial fractions in ascending order:

^{54}/_{38} < ^{27}/_{13} < ^{111}/_{50} < ^{89}/_{40} < ^{119}/_{41} < ^{260}/_{60} < ^{260}/_{50} < ^{311}/_{51} < ^{323}/_{52} < ^{526}/_{53} < ^{442}/_{44} < ^{502}/_{49}

## ::: Comparing operation :::

The final answer: