# Compare and sort in ascending order the two common ordinary fractions, which one is larger: ^{- 4,011}/_{5,005} and ^{- 4,021}/_{5,008}. Common ordinary fractions compared and sorted in ascending order, result explained below

## Compare: ^{- 4,011}/_{5,005} and ^{- 4,021}/_{5,008}

### To compare and sort multiple fractions, they should either have the same denominator or the same numerator.

## The operation of comparing fractions:

^{- 4,011}/_{5,005} and ^{- 4,021}/_{5,008}

### Simplify the operation

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

#### - ^{4,011}/_{5,005} = - ^{(3 × 7 × 191)}/_{(5 × 7 × 11 × 13)} = - ^{((3 × 7 × 191) ÷ 7)}/_{((5 × 7 × 11 × 13) ÷ 7)} = - ^{573}/_{715}

#### - ^{4,021}/_{5,008} is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:

4,021 is a prime number.

5,008 = 2^{4} × 313

## To compare and sort the fractions, build them up to the same numerator.

### To build the fractions up to the same numerator we have to:

#### 1) calculate their common numerator

#### 2) then calculate the expanding number of each fraction

#### 3) then build up their numerators the same by expanding the fractions to equivalent forms, which all have equal numerators

### Calculate the common numerator

#### The common numerator is nothing else than the least common multiple (LCM) of the numerators of the fractions.

#### The LCM will be the common numerator of the compared fractions.

#### To calculate the LCM, we need the prime factorization of the numerators:

#### 573 = 3 × 191

#### 4,021 is a prime number.

#### Multiply all the unique prime factors: if there are repeating prime factors we only take them once, and only the ones having the highest exponent (the highest powers).

#### LCM (573, 4021) = 3 × 191 × 4,021 = 2,304,033

### Calculate the expanding number of each fraction:

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

#### - ^{573}/_{715} : 2,304,033 ÷ 573 = (3 × 191 × 4,021) ÷ (3 × 191) = 4,021

#### - ^{4,021}/_{5,008} : 2,304,033 ÷ 4,021 = (3 × 191 × 4,021) ÷ 4,021 = 573

### Build up the fractions to the same common numerator:

#### Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated at the step 2, above.

#### This way all the fractions will have the same numerator:

#### - ^{573}/_{715} = - ^{(4,021 × 573)}/_{(4,021 × 715)} = - ^{2,304,033}/_{2,875,015}

#### - ^{4,021}/_{5,008} = - ^{(573 × 4,021)}/_{(573 × 5,008)} = - ^{2,304,033}/_{2,869,584}

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

#### The larger the denominator the larger the negative fraction.

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

## ::: The operation of comparing fractions :::

The final answer:

## The fractions sorted in ascending order:

- ^{2,304,033}/_{2,869,584} < - ^{2,304,033}/_{2,875,015}

The initial fractions sorted in ascending order:

^{- 4,021}/_{5,008} < ^{- 4,011}/_{5,005}

#### How are the numbers written: comma ',' used as a thousands separator; point '.' as a decimal separator; numbers rounded off to max. 12 decimals (if the case). Used symbols: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.

## Other similar operations

## Compare and sort common ordinary fractions, online calculator:

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## Tutoring: Comparing ordinary fractions

### How to compare two fractions?

#### 1. Fractions that have different signs:

- Any positive fraction is larger than any negative fraction:
- ie:
^{4}/_{25} > - ^{19}/_{2}

#### 2. A proper and an improper fraction:

- Any positive improper fraction is larger than any positive proper fraction:
- ie:
^{44}/_{25} > 1 > ^{19}/_{200} - Any negative improper fraction is smaller than any negative proper fraction:
- ie: -
^{44}/_{25} < -1 < - ^{19}/_{200}

#### 3. Fractions that have both like numerators and denominators:

- The fractions are equal:
- ie:
^{89}/_{50} = ^{89}/_{50}

#### 4. Fractions that have unlike (different) numerators but like (equal) denominators.

**Positive fractions**: compare the numerators, the larger fraction is the one with the larger numerator: - ie:
^{24}/_{25} > ^{19}/_{25} **Negative fractions**: compare the numerators, the larger fraction is the one with the smaller numerator: - ie: -
^{19}/_{25} < - ^{17}/_{25}

#### 5. Fractions that have unlike (different) denominators but like (equal) numerators.

**Positive fractions**: compare the denominators, the larger fraction is the one with the smaller denominator: - ie:
^{24}/_{25} > ^{24}/_{26} **Negative fractions**: compare the denominators, the larger fraction is the one with the larger denominator: - ie: -
^{17}/_{25} < - ^{17}/_{29}

#### 6. Fractions that have different denominators and numerators (unlike denominators and numerators).

- To compare them, fractions should be built up to the same denominator (or if it's easier, to the same numerator).

## More on ordinary (common) fractions / theory: