# Convert the pure repeating (recurring) decimal number 0.00017. 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 0.00017

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

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

#### 0.00017 ≈ 0.00017000170002

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

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

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

#### 0.00017000170002 =

#### 0.00017000170002 × ^{100}/_{100} =

^{(0.00017000170002 × 100)}/_{100} =

#### ^{0.017000170002}/_{100} =

#### 0.017000170002% ≈

#### 0.02%

#### (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, %

#### 0.00017 ≈ 0.02%

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

### 0.00017 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 = 0.00017

### Set up the second equation.

#### Number of decimal places repeating: 5

#### Multiply both sides of the first equation by 10^{5} = 100,000

#### y = 0.00017

#### 100,000 × y = 100,000 × 0.00017

#### 100,000 × y = 17.00017

### 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.

#### 100,000 × y - y = 17.00017 - 0.00017 ⇒

#### (100,000 - 1) × y = 17.00017 - 0.00017 ⇒

#### We now have a new equation:

#### 99,999 × y = 17

### Solve for y in the new equation.

#### 99,999 × y = 17 ⇒

#### y = ^{17}/_{99,999}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 0.00017

#### According to our calculations:

#### y = ^{17}/_{99,999}

#### ⇒ 0.00017 = ^{17}/_{99,999}

## 3. Reduce (simplify) the fraction above: ^{17}/_{99,999}

(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}):

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

#### 99,999 = 3^{2} × 41 × 271

### 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 (17; 3^{2} × 41 × 271) = 1

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

^{17}/_{99,999}: 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 3:

^{17}/_{99,999} = ^{(17 × 3)}/_{(99,999 × 3)} = ^{51}/_{299,997}

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

^{17}/_{99,999} = ^{(17 × 4)}/_{(99,999 × 4)} = ^{68}/_{399,996}

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

#### ... to the initial fraction: ^{17}/_{99,999}

## :: Final answer ::

Written in 3 different ways

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

0.00017 = ^{17}/_{99,999}

## As a percentage:

0.00017 ≈ 0.02%

## As equivalent fractions:

0.00017 = ^{17}/_{99,999} = ^{51}/_{299,997} = ^{68}/_{399,996}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator