# Convert the pure repeating (recurring) decimal number 5.7. Turn it into a reduced (simplified) improper fraction, into a mixed number and write it as a percentage. Calculate other equivalent fractions to the decimal number, by expanding

## Convert 5.7

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

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

#### 5.7 ≈ 5.77777777777778

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

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

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

#### 5.77777777777778 =

#### 5.77777777777778 × ^{100}/_{100} =

^{(5.77777777777778 × 100)}/_{100} =

#### ^{577.777777777778}/_{100} =

#### 577.777777777778% ≈

#### 577.78%

#### (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, %

#### 5.7 ≈ 577.78%

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

### 5.7 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 = 5.7

### Set up the second equation.

#### Number of decimal places repeating: 1

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

#### y = 5.7

#### 10 × y = 10 × 5.7

#### 10 × y = 57.7

### 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 = 57.7 - 5.7 ⇒

#### (10 - 1) × y = 57.7 - 5.7 ⇒

#### We now have a new equation:

#### 9 × y = 52

### Solve for y in the new equation.

#### 9 × y = 52 ⇒

#### y = ^{52}/_{9}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 5.7

#### According to our calculations:

#### y = ^{52}/_{9}

#### ⇒ 5.7 = ^{52}/_{9}

## 3. Reduce (simplify) the fraction above: ^{52}/_{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}):

#### 52 = 2^{2} × 13

#### 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 (2^{2} × 13; 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.

#### 52 ÷ 9 = 5, remainder = 7 ⇒

#### 52 = 5 × 9 + 7 ⇒

#### ^{52}/_{9} =

#### ^{(5 × 9 + 7)} / _{9} =

^{(5 × 9)} / _{9} + ^{7}/_{9} =

#### 5 + ^{7}/_{9} =

#### 5 ^{7}/_{9}

^{52}/_{9}: 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 5:

^{52}/_{9} = ^{(52 × 5)}/_{(9 × 5)} = ^{260}/_{45}

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

^{52}/_{9} = ^{(52 × 6)}/_{(9 × 6)} = ^{312}/_{54}

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

#### ... to the initial fraction: ^{52}/_{9}

## :: Final answer ::

Written in 4 different ways

## As a reduced (simplified) positive improper fraction:

5.7 = ^{52}/_{9}

## As a mixed number:

5.7 = 5 ^{7}/_{9}

## As a percentage:

5.7 ≈ 577.78%

## As equivalent fractions:

5.7 = ^{52}/_{9} = ^{260}/_{45} = ^{312}/_{54}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator