## 1. Write the pure repeating (recurring) decimal number as a percentage.

### Approximate to the desired number of decimal places (14):

#### 0.555555556 ≈ 0.55555555655556

### Multiply the number by ^{100}/_{100}.

#### The value of the number does not change when multiplying by ^{100}/_{100}.

#### Note: ^{100}/_{100} = 1

#### 0.55555555655556 =

#### 0.55555555655556 × ^{100}/_{100} =

^{(0.55555555655556 × 100)}/_{100} =

#### ^{55.555555655556}/_{100} =

#### 55.555555655556% ≈

#### 55.56%

#### (rounded off to a maximum of 2 decimal places)

#### In other words:

#### Approximate to the desired number of decimal places...

#### Multiply the number by 100...

#### ... And then add the percent sign, %

#### 0.555555556 ≈ 55.56%

## 2. Write the pure repeating (recurring) decimal number as a proper fraction.

### 0.555555556 can be written as a proper fraction.

#### The numerator is smaller than the denominator.

### Set up the first equation.

#### Let y equal the decimal number:

#### y = 0.555555556

### Set up the second equation.

#### Number of decimal places repeating: 9

#### Multiply both sides of the first equation by 10^{9} = 1,000,000,000

#### y = 0.555555556

#### 1,000,000,000 × y = 1,000,000,000 × 0.555555556

#### 1,000,000,000 × y = 555,555,556.555555556

### Subtract the first equation from the second equation.

#### Having the same number of decimal places ...

#### The repeating pattern drops off by subtracting the two equations.

#### 1,000,000,000 × y - y = 555,555,556.555555556 - 0.555555556 =>

#### (1,000,000,000 - 1) × y = 555,555,556.555555556 - 0.555555556 =>

#### We now have a new equation:

#### 999,999,999 × y = 555,555,556

### Solve for y in the new equation.

#### 999,999,999 × y = 555,555,556 =>

#### y = ^{555,555,556}/_{999,999,999}

#### Let the result written as a fraction.

### Write the number as a fraction.

#### According to our first equation:

#### y = 0.555555556

#### According to our calculations:

#### y = ^{555,555,556}/_{999,999,999}

#### => 0.555555556 = ^{555,555,556}/_{999,999,999}

## 3. Reduce (simplify) the fraction above: ^{555,555,556}/_{999,999,999}

(to the lowest terms, to its simplest equivalent form, irreducible).

### To reduce a fraction divide the numerator and the denominator by their greatest (highest) common factor (divisor), GCF.

### Factor both the numerator and the denominator (prime factorization).

#### In exponential notation (a^{n}):

#### 555,555,556 = 2^{2} × 107 × 1,298,027

#### 999,999,999 = 3^{4} × 37 × 333,667

### Calculate the greatest (highest) common factor (divisor), GCF.

#### Multiply all the common prime factors by the lowest exponents.

#### But, the numerator and the denominator have no common factors.

#### GCF (2^{2} × 107 × 1,298,027; 3^{4} × 37 × 333,667) = 1

### The numerator and the denominator are coprime numbers (no common prime factors, GCF = 1). So, the fraction cannot be reduced (simplified): irreducible fraction.

^{555,555,556}/_{999,999,999}: Equivalent fractions.

#### The above fraction cannot be reduced.

#### That is, it has the smallest possible numerator and denominator.

#### By expanding it we can build up equivalent fractions.

#### Multiply the numerator & the denominator by the same number.

### Example 1. By expanding the fraction by 4:

^{555,555,556}/_{999,999,999} = ^{(555,555,556 × 4)}/_{(999,999,999 × 4)} = ^{2,222,222,224}/_{3,999,999,996}

### Example 2. By expanding the fraction by 6:

^{555,555,556}/_{999,999,999} = ^{(555,555,556 × 6)}/_{(999,999,999 × 6)} = ^{3,333,333,336}/_{5,999,999,994}

#### Of course, the above fractions are reducing...

#### ... to the initial fraction: ^{555,555,556}/_{999,999,999}