# Compare and sort in ascending order the two common ordinary fractions, which one is larger: ^{64}/_{78} and ^{70}/_{87}. Common ordinary fractions compared and sorted in ascending order, result explained below

## Compare: ^{64}/_{78} and ^{70}/_{87}

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

## The operation of comparing fractions:

^{64}/_{78} and ^{70}/_{87}

### Simplify the operation

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

#### By reducing the values of the numerators and denominators of the fractions, further calculations with these fractions become easier to do.

#### To reduce a fraction to the lowest terms equivalent: divide both the numerator and denominator by their greatest common factor, GCF.

^{64}/_{78} = ^{26}/_{(2 × 3 × 13)} = ^{(26 ÷ 2)}/_{((2 × 3 × 13) ÷ 2)} = ^{32}/_{39}

^{70}/_{87} is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:

70 = 2 × 5 × 7

87 = 3 × 29

## To compare and sort the fractions, make their numerators the same.

### To make the fractions' numerators the same - we have to:

#### 1) calculate their common numerator

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

#### 3) then make 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:

#### 32 = 2^{5}

#### 70 = 2 × 5 × 7

#### 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 (32, 70) = 2^{5} × 5 × 7 = 1,120

### Calculate the expanding number of each fraction:

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

^{32}/_{39} : 1,120 ÷ 32 = (2^{5} × 5 × 7) ÷ 2^{5} = 35

^{70}/_{87} : 1,120 ÷ 70 = (2^{5} × 5 × 7) ÷ (2 × 5 × 7) = 16

### Make the fractions' numerators the same:

#### 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:

^{32}/_{39} = ^{(35 × 32)}/_{(35 × 39)} = ^{1,120}/_{1,365}

^{70}/_{87} = ^{(16 × 70)}/_{(16 × 87)} = ^{1,120}/_{1,392}

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

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

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

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

The final answer:

## The fractions sorted in ascending order:

^{1,120}/_{1,392} < ^{1,120}/_{1,365}

The initial fractions sorted in ascending order:

^{70}/_{87} < ^{64}/_{78}

#### 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: