## The operation of comparing fractions:

^{41}/_{54} and ^{50}/_{56}

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

^{41}/_{54} already reduced to the lowest terms;

the numerator and denominator have no common prime factors:

41 is a prime number;

54 = 2 × 3^{3};

^{50}/_{56} = ^{(2 × 52)}/_{(23 × 7)} = ^{((2 × 52) ÷ 2)}/_{((23 × 7) ÷ 2)} = ^{25}/_{28}

## 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:

#### 54 = 2 × 3^{3}

#### 28 = 2^{2} × 7

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

#### LCM (54, 28) = 2^{2} × 3^{3} × 7 = 756

### Calculate the expanding number of each fraction

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

#### For fraction: ^{41}/_{54} is 756 ÷ 54 = (2^{2} × 3^{3} × 7) ÷ (2 × 3^{3}) = 14

#### For fraction: ^{25}/_{28} is 756 ÷ 28 = (2^{2} × 3^{3} × 7) ÷ (2^{2} × 7) = 27

### Expand the fractions

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

Multiply the numerators and denominators by their expanding number:

^{41}/_{54} = ^{(14 × 41)}/_{(14 × 54)} = ^{574}/_{756}

^{25}/_{28} = ^{(27 × 25)}/_{(27 × 28)} = ^{675}/_{756}

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

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

## ::: Comparing operation :::

The final answer: