# Multiplying the common ordinary fractions: ^{32}/_{85} × ^{20}/_{77} = ? The multiplication process explained. The result written: As a positive proper fraction (the numerator < the denominator). As a decimal number. As a percentage

^{32}/_{85} × ^{20}/_{77} = ?

## Simplify the operation

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

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

#### * By reducing the values of the numerators and the denominators of the fractions the calculations are easier to make.

#### A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.

#### * In order to easily reduce a fraction, factor its numerator and denominator. This way all the common prime factors are easily identified and crossed out, without calculating the GCF.

^{32}/_{85} is already reduced to the lowest terms.

#### The numerator and denominator have no common prime factors.

#### Their prime factorization:

32 = 2^{5}

85 = 5 × 17

^{20}/_{77} is already reduced to the lowest terms.

#### The numerator and denominator have no common prime factors.

#### Their prime factorization:

20 = 2^{2} × 5

77 = 7 × 11

## Perform the operation of calculating the fractions

### Multiply the fractions:

#### 1) Multiply the numerators, that is, all the numbers above the fractions bars, separately.

#### 2) Multiply the denominators, that is, all the numbers below the fractions bars, separately.

#### * Factor all the numerators and all the denominators in order to easily reduce (simplify) the end fraction.

^{32}/_{85} × ^{20}/_{77} =

^{(32 × 20)} / _{(85 × 77)} =

^{(25 × 22 × 5)} / _{(5 × 17 × 7 × 11)} =

^{(27 × 5)} / _{(5 × 7 × 11 × 17)}

## Fully reduce (simplify) the end fraction to its lowest terms equivalent:

### Calculate the greatest common factor, GCF,

of the numerator and denominator of the fraction:

#### A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.

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

#### * To calculate the GCF, we need to factor the numerator and the denominator of the fraction into prime factors.

#### Then multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).

#### GCF (2^{7} × 5; 5 × 7 × 11 × 17) = 5

### Divide the numerator and the denominator by their GCF:

^{(27 × 5)} / _{(5 × 7 × 11 × 17)} =

^{((27 × 5) ÷ 5)} / _{((5 × 7 × 11 × 17) ÷ 5)} =

^{(27 × 5 ÷ 5)}/_{(5 ÷ 5 × 7 × 11 × 17)} =

^{(27 × 1)}/_{(1 × 7 × 11 × 17)} =

^{27}/_{(7 × 11 × 17)} =

^{128}/_{(7 × 11 × 17)} =

^{128}/_{1,309}

## Rewrite the fraction

### As a decimal number:

#### Simply divide the numerator by the denominator, without a remainder, as shown below:

^{128}/_{1,309} =

#### 128 ÷ 1,309 ≈

#### 0.097784568373 ≈

#### 0.1

### As a percentage:

#### A percentage value p% is equal to the fraction: ^{p}/_{100}, for any decimal number p. So, we need to change the form of the number calculated above, to show a denominator of 100.

#### To do that, multiply the number by the fraction ^{100}/_{100}.

#### The value of the fraction ^{100}/_{100} = 1, so by multiplying the number by this fraction the result is not changing, only the form.

#### 0.097784568373 =

#### 0.097784568373 × ^{100}/_{100} =

^{(0.097784568373 × 100)}/_{100} =

^{9.77845683728}/_{100} ≈

#### 9.77845683728% ≈

#### 9.78%

## The final answer:

written in three ways

## As a positive proper fraction:

(the numerator < the denominator)

^{32}/_{85} × ^{20}/_{77} = ^{128}/_{1,309}

## As a decimal number:

^{32}/_{85} × ^{20}/_{77} ≈ 0.1

## As a percentage:

^{32}/_{85} × ^{20}/_{77} ≈ 9.78%

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

## Other similar operations

## Multiply common ordinary fractions, online calculator: