# Convert the mixed repeating (recurring) decimal number 0.194. 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 0.194

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

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

#### 0.194 ≈ 0.19444444444444

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

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

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

#### 0.19444444444444 =

#### 0.19444444444444 × ^{100}/_{100} =

^{(0.19444444444444 × 100)}/_{100} =

#### ^{19.444444444444}/_{100} =

#### 19.444444444444% ≈

#### 19.44%

#### (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, %

#### 0.194 ≈ 19.44%

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

### 0.194 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 = 0.194

### Set up the second equation.

#### Number of decimal places repeating: 1

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

#### y = 0.194

#### 10 × y = 10 × 0.194

#### 10 × y = 1.94

#### Get the same number of decimal places as for y:

#### 10 × y = 1.944

#### Note: 1.944 = 1.94

### 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 = 1.944 - 0.194 ⇒

#### (10 - 1) × y = 1.944 - 0.194 ⇒

#### We now have a new equation:

#### 9 × y = 1

### Solve for y in the new equation.

#### 9 × y = 1 ⇒

#### y = ^{1}/_{9}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 0.194

#### According to our calculations:

#### y = ^{1}/_{9}

#### ⇒ 0.194 = ^{1}/_{9}

## 3. Reduce (simplify) the fraction above: ^{1}/_{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}):

#### 1 = one

#### 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 (one; 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.

^{1}/_{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 3:

^{1}/_{9} = ^{(1 × 3)}/_{(9 × 3)} = ^{3}/_{27}

### Example 2. By expanding the fraction by 5:

^{1}/_{9} = ^{(1 × 5)}/_{(9 × 5)} = ^{5}/_{45}

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

#### ... to the initial fraction: ^{1}/_{9}

## :: Final answer ::

Written in 3 different ways

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

0.194 = ^{1}/_{9}

## As a percentage:

0.194 ≈ 19.44%

## As equivalent fractions:

0.194 = ^{1}/_{9} = ^{3}/_{27} = ^{5}/_{45}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator