## Detailed calculations below:

### Introduction. Fractions

#### A fraction consists of two numbers and a fraction bar: ^{31}/_{80}

#### The number above the bar is the numerator: 31

#### The number below the bar is the denominator: 80

#### The fraction bar means that the two numbers are dividing themselves:

^{31}/_{80} = 31 ÷ 80

#### Divide the numerator by the denominator to get fraction's value:

Value = 31 ÷ 80

### Introduction. Percent

#### 'Percent (%)' means 'out of one hundred':

#### p% = p 'out of one hundred',

#### p% = ^{p}/_{100} = p ÷ 100

### Note:

#### The fraction ^{100}/_{100} = 100 ÷ 100 = 100% = 1

#### Multiply a number by the fraction ^{100}/_{100},

... and its value doesn't change.

## To reduce a fraction, divide both its numerator and denominator by their greatest common factor, GCF.

#### To calculate the greatest common factor, we build the prime factorization of the two numbers.

### Integer numbers prime factorization:

#### Prime Factorization of a number: finding the prime numbers that multiply together to make that number.

#### 31 is a prime number, it cannot be broken down to other prime factors;

#### 80 = 2^{4} × 5;

80 is not a prime, is a composite number;

** Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself. *

* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.

### Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

#### Multiply all the common prime factors, by the lowest exponents (if any).

#### But the two numbers have no common prime factors.

#### gcf, hcf, gcd (31; 80) = 1

coprime numbers (relatively prime)

## Fraction's numerator and denominator are coprime numbers (no common prime factors). Fraction cannot be reduced (simplified) - irreducible.

^{31}/_{80} is a proper fraction.

#### A proper fraction: numerator smaller than denominator.

## Rewrite the fraction:

### As a decimal number:

^{31}/_{80} =

#### 31 ÷ 80 =

#### 0.3875 ≈

#### 0.39

### As a percentage:

#### 0.3875 =

#### 0.3875 × ^{100}/_{100} =

#### ^{38.75}/_{100} =

#### 38.75%

#### In other words:

#### 1) Calculate fraction's value.

#### 2) Multiply that number by 100.

#### 3) Add the percent sign % to it.

## Final answer

continued below...