# Convert the mixed repeating (recurring) decimal number 0.7129. 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 0.7129

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

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

#### 0.7129 ≈ 0.71292929292929

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

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

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

#### 0.71292929292929 =

#### 0.71292929292929 × ^{100}/_{100} =

^{(0.71292929292929 × 100)}/_{100} =

#### ^{71.292929292929}/_{100} =

#### 71.292929292929% ≈

#### 71.29%

#### (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, %

#### 0.7129 ≈ 71.29%

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

### 0.7129 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 = 0.7129

### Set up the second equation.

#### Number of decimal places repeating: 2

#### Multiply both sides of the first equation by 10^{2} = 100

#### y = 0.7129

#### 100 × y = 100 × 0.7129

#### 100 × y = 71.29

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

#### 100 × y = 71.2929

#### Note: 71.2929 = 71.29

### 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 × y - y = 71.2929 - 0.7129 ⇒

#### (100 - 1) × y = 71.2929 - 0.7129 ⇒

#### We now have a new equation:

#### 99 × y = 70.58

### Solve for y in the new equation.

#### 99 × y = 70.58 ⇒

#### y = ^{70.58}/_{99}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 0.7129

#### According to our calculations:

#### y = ^{70.58}/_{99}

#### ⇒ 0.7129 = ^{70.58}/_{99}

### Get rid of the decimal places in the fraction above.

#### Multiply the top and the bottom number by 100.

#### 1 followed by as many 0-s as the number of digits after the decimal point.

#### 0.7129 = ^{(70.58 × 100)}/_{(99 × 100)}

#### 0.7129 = ^{7,058}/_{9,900}

## 3. Reduce (simplify) the fraction above: ^{7,058}/_{9,900}

(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}):

#### 7,058 = 2 × 3,529

#### 9,900 = 2^{2} × 3^{2} × 5^{2} × 11

### Calculate the greatest (highest) common factor (divisor), GCF.

#### Multiply all the common prime factors by the lowest exponents.

#### GCF (2 × 3,529; 2^{2} × 3^{2} × 5^{2} × 11) = 2

### Divide both the numerator and the denominator by their greatest common factor, GCF.

^{7,058}/_{9,900} =

^{(2 × 3,529)}/_{(22 × 32 × 52 × 11)} =

^{((2 × 3,529) ÷ 2)} / _{((22 × 32 × 52 × 11) ÷ 2)} =

^{3,529}/_{(2 × 32 × 52 × 11)} =

^{3,529}/_{4,950}

^{3,529}/_{4,950}: 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:

^{3,529}/_{4,950} = ^{(3,529 × 4)}/_{(4,950 × 4)} = ^{14,116}/_{19,800}

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

^{3,529}/_{4,950} = ^{(3,529 × 6)}/_{(4,950 × 6)} = ^{21,174}/_{29,700}

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

#### ... to the initial fraction: ^{3,529}/_{4,950}

## :: Final answer ::

Written in 3 different ways

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

0.7129 = ^{3,529}/_{4,950}

## As a percentage:

0.7129 ≈ 71.29%

## As equivalent fractions:

0.7129 = ^{3,529}/_{4,950} = ^{14,116}/_{19,800} = ^{21,174}/_{29,700}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator