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

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

#### 2.5 ≈ 2.55555555555556

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

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

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

#### 2.55555555555556 =

#### 2.55555555555556 × ^{100}/_{100} =

^{(2.55555555555556 × 100)}/_{100} =

#### ^{255.555555555556}/_{100} =

#### 255.555555555556% ≈

#### 255.56%

#### (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, %

#### 2.5 ≈ 255.56%

## 2. Write the pure repeating (recurring) decimal number as an improper fraction.

#### 2.5 can be written as an improper fraction.

#### (The numerator is larger than or equal to the denominator).

### Set up the first equation.

#### Let y equal the decimal number:

#### y = 2.5

### Set up the second equation.

#### Number of decimal places repeating: 1

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

#### y = 2.5

#### 10 × y = 10 × 2.5

#### 10 × y = 25.5

### 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.

#### 10 × y - y = 25.5 - 2.5 =>

#### (10 - 1) × y = 25.5 - 2.5 =>

#### We now have a new equation:

#### 9 × y = 23

### Solve for y in the new equation.

#### 9 × y = 23 =>

#### y = ^{23}/_{9}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 2.5

#### According to our calculations:

#### y = ^{23}/_{9}

#### => 2.5 = ^{23}/_{9}

## 3. Reduce (simplify) the fraction above: ^{23}/_{9}

(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}):

#### 23 is a prime number, it cannot be factored into other prime factors

#### 9 = 3^{2}

### 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 (23; 3^{2}) = 1

### The numerator and the denominator are coprime numbers (no common prime factors, GCF = 1).

So, the fraction cannot be reduced (simplified): irreducible fraction.

## 4. The fraction is an improper one, rewrite it as a mixed number (mixed fraction):

#### A mixed number = an integer number and a proper fraction, of the same sign.

#### Example 1: 2 ^{1}/_{5}; Example 2: - 1 ^{3}/_{7}.

#### A proper fraction = the numerator is smaller than the denominator.

#### 23 ÷ 9 = 2, remainder = 5 =>

#### 23 = 2 × 9 + 5 =>

#### ^{23}/_{9} =

#### ^{(2 × 9 + 5)} / _{9} =

^{(2 × 9)} / _{9} + ^{5}/_{9} =

#### 2 + ^{5}/_{9} =

#### 2 ^{5}/_{9}

^{23}/_{9}: Equivalent fractions.

#### The above fraction cannot be reduced.

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

#### 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 2:

^{23}/_{9} = ^{(23 × 2)}/_{(9 × 2)} = ^{46}/_{18}

### Example 2. By expanding the fraction by 4:

^{23}/_{9} = ^{(23 × 4)}/_{(9 × 4)} = ^{92}/_{36}

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

#### ... to the initial fraction: ^{23}/_{9}