# Convert the pure repeating (recurring) decimal number - 0.145. Turn it into a reduced (simplified) proper fraction and write it as a percentage. Calculate other equivalent fractions to the decimal number, by expanding

## Convert - 0.145

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

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

#### - 0.145 ≈ - 0.14514514514515

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

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

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

#### - 0.14514514514515 =

#### - 0.14514514514515 × ^{100}/_{100} =

#### - ^{(0.14514514514515 × 100)}/_{100} =

#### - ^{14.514514514515}/_{100} =

#### - 14.514514514515% ≈

#### - 14.51%

#### (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, %

#### - 0.145 ≈ - 14.51%

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

### - 0.145 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 = - 0.145

### Set up the second equation.

#### Number of decimal places repeating: 3

#### Multiply both sides of the first equation by 10^{3} = 1,000

#### y = - 0.145

#### 1,000 × y = 1,000 × - 0.145

#### 1,000 × y = - 145.145

### 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 × y - y = - 145.145 - (- 0.145) ⇒

#### 1,000 × y - y = - 145.145 + 0.145 ⇒

#### (1,000 - 1) × y = - 145.145 + 0.145 ⇒

#### We now have a new equation:

#### 999 × y = - 145

### Solve for y in the new equation.

#### 999 × y = - 145 ⇒

#### y = - ^{145}/_{999}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = - 0.145

#### According to our calculations:

#### y = - ^{145}/_{999}

#### ⇒ - 0.145 = - ^{145}/_{999}

## 3. Reduce (simplify) the fraction above: - ^{145}/_{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}):

#### 145 = 5 × 29

#### 999 = 3^{3} × 37

### 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 (5 × 29; 3^{3} × 37) = 1

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

## - ^{145}/_{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:

#### - ^{145}/_{999} = - ^{(145 × 4)}/_{(999 × 4)} = - ^{580}/_{3,996}

### Example 2. By expanding the fraction by 7:

#### - ^{145}/_{999} = - ^{(145 × 7)}/_{(999 × 7)} = - ^{1,015}/_{6,993}

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

#### ... to the initial fraction: - ^{145}/_{999}

## :: Final answer ::

Written in 3 different ways

## As a reduced (simplified) negative proper fraction:

- 0.145 = - ^{145}/_{999}

## As a percentage:

- 0.145 ≈ - 14.51%

## As equivalent fractions:

- 0.145 = - ^{145}/_{999} = - ^{580}/_{3,996} = - ^{1,015}/_{6,993}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator