# Convert the mixed repeating (recurring) decimal number 3.1419. 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 3.1419

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

### Approximate to the desired number of decimal places:

#### 3.1419 ≈ 3.142

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

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

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

#### 3.142 =

#### 3.142 × ^{100}/_{100} =

^{(3.142 × 100)}/_{100} =

#### ^{314.2}/_{100} =

#### 314.2%

#### In other words:

#### Approximate to the desired number of decimal places...

#### Multiply the number by 100...

#### ... And then add the percent sign, %

#### 3.1419 ≈ 314.2%

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

### 3.1419 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 = 3.1419

### Set up the second equation.

#### Number of decimal places repeating: 1

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

#### y = 3.1419

#### 10 × y = 10 × 3.1419

#### 10 × y = 31.419

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

#### 10 × y = 31.4199

#### Note: 31.4199 = 31.419

### 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 = 31.4199 - 3.1419 =>

#### (10 - 1) × y = 31.4199 - 3.1419 =>

#### We now have a new equation:

#### 9 × y = 28.278

### Solve for y in the new equation.

#### 9 × y = 28.278 =>

#### y = ^{28.278}/_{9}

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

### Write the number as a fraction.

#### According to our first equation:

#### y = 3.1419

#### According to our calculations:

#### y = ^{28.278}/_{9}

#### => 3.1419 = ^{28.278}/_{9}

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

#### Multiply the top and the bottom number by 1,000.

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

#### 3.1419 = ^{(28.278 × 1,000)}/_{(9 × 1,000)}

#### 3.1419 = ^{28,278}/_{9,000}

## 3. Reduce (simplify) the fraction above: ^{28,278}/_{9,000}

(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}):

#### 28,278 = 2 × 3^{2} × 1,571

#### 9,000 = 2^{3} × 3^{2} × 5^{3}

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

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

#### GCF (2 × 3^{2} × 1,571; 2^{3} × 3^{2} × 5^{3}) = 2 × 3^{2}

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

^{28,278}/_{9,000} =

^{(2 × 32 × 1,571)}/_{(23 × 32 × 53)} =

^{((2 × 32 × 1,571) ÷ (2 × 32))} / _{((23 × 32 × 53) ÷ (2 × 32))} =

^{1,571}/_{(22 × 53)} =

^{1,571}/_{500}

## 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.

#### 1,571 ÷ 500 = 3, remainder = 71 =>

#### 1,571 = 3 × 500 + 71 =>

#### ^{1,571}/_{500} =

#### ^{(3 × 500 + 71)} / _{500} =

^{(3 × 500)} / _{500} + ^{71}/_{500} =

#### 3 + ^{71}/_{500} =

#### 3 ^{71}/_{500}

^{1,571}/_{500}: 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 2:

^{1,571}/_{500} = ^{(1,571 × 2)}/_{(500 × 2)} = ^{3,142}/_{1,000}

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

^{1,571}/_{500} = ^{(1,571 × 4)}/_{(500 × 4)} = ^{6,284}/_{2,000}

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

#### ... to the initial fraction: ^{1,571}/_{500}

## :: Final answer ::

Written in 4 different ways

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

3.1419 = ^{1,571}/_{500}

## As a mixed number:

3.1419 = 3 ^{71}/_{500}

## As a percentage:

3.1419 = 314.2%

## As equivalent fractions:

3.1419 = ^{1,571}/_{500} = ^{3,142}/_{1,000} = ^{6,284}/_{2,000}

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## Learn how to turn a decimal number into a fraction and a percentage. Steps.

### 1. How to write the number as a percentage:

- Multiply the number by 100. Then add the percent sign, %.

### 2. How to write the number as a fraction:

- Write down the number divided by 1, as a fraction.
- Turn the top number into a whole number: multiply both the top and the bottom by the same number.
- Reduce (simplify) the above fraction 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.
- If the fraction is an improper one, rewrite it as a mixed number (mixed fraction).
- Calculate equivalent fractions. By expanding it we can build up equivalent fractions: multiply the numerator & the denominator by the same number.

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