# 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

### Solve for y in the new equation.

#### 9 × y = 28 ⇒

#### y = ^{28}/_{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}/_{9}

#### ⇒ 3.1419 = ^{28}/_{9}

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

#### 28 = 2^{2} × 7

#### 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 (2^{2} × 7; 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.

## 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.

#### 28 ÷ 9 = 3, remainder = 1 ⇒

#### 28 = 3 × 9 + 1 ⇒

#### ^{28}/_{9} =

#### ^{(3 × 9 + 1)} / _{9} =

^{(3 × 9)} / _{9} + ^{1}/_{9} =

#### 3 + ^{1}/_{9} =

#### 3 ^{1}/_{9}

^{28}/_{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:

^{28}/_{9} = ^{(28 × 3)}/_{(9 × 3)} = ^{84}/_{27}

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

^{28}/_{9} = ^{(28 × 7)}/_{(9 × 7)} = ^{196}/_{63}

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

#### ... to the initial fraction: ^{28}/_{9}

## :: Final answer ::

Written in 4 different ways

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

3.1419 = ^{28}/_{9}

## As a mixed number:

3.1419 = 3 ^{1}/_{9}

## As a percentage:

3.1419 = 314.2%

## As equivalent fractions:

3.1419 = ^{28}/_{9} = ^{84}/_{27} = ^{196}/_{63}

### More operations of this kind

## Decimal numbers to fractions and percentages, calculator