## 1. Write the number as a percentage.

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

#### 3.45 = 3.45 × ^{100}/_{100} = ^{(3.45 × 100)}/_{100} = ^{345}/_{100} = 345%

#### In other words: multiply the number by 100... and then add the percent sign, %: 3.45 = 345%

## 2. Write the number as an improper fraction.

#### (The numerator is larger than or equal to the denominator).

### Write down the number divided by 1, as a fraction:

#### 3.45 = ^{3.45}/_{1}

### Turn the top number into a whole number.

#### Multiply both the top and the bottom by the same number: 100 (1 followed by as many 0s as the number of digits after the decimal point).

^{3.45}/_{1} = ^{(3.45 × 100)}/_{(1 × 100)} = ^{345}/_{100}

## 3. Reduce (simplify) the fraction above: ^{345}/_{100}

(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).

#### 345 = 3 × 5 × 23; 100 = 2^{2} × 5^{2};

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

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

#### GCF (3 × 5 × 23; 2^{2} × 5^{2}) = 5

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

^{345}/_{100} = ^{(3 × 5 × 23)}/_{(22 × 52)} = ^{((3 × 5 × 23) ÷ 5)} / _{((22 × 52) ÷ 5)} = ^{(3 × 23)}/_{(22 × 5)} = ^{69}/_{20}

## 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.

#### 69 ÷ 20 = 3, remainder = 9 => 69 = 3 × 20 + 9 =>

#### ^{69}/_{20} = ^{(3 × 20 + 9)} / _{20} = ^{(3 × 20)} / _{20} + ^{9}/_{20} = 3 + ^{9}/_{20} = 3 ^{9}/_{20}

^{69}/_{20}: Equivalent fractions.

#### By expanding the reduced fraction we can build up equivalent fractions (multiply the numerator & the denominator by the same number).

### Example 1. By expanding the fraction by 2:

^{69}/_{20} = ^{(69 × 2)}/_{(20 × 2)} = ^{138}/_{40}

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

^{69}/_{20} = ^{(69 × 6)}/_{(20 × 6)} = ^{414}/_{120}

#### Of course, the above fractions are reducing... to the initial fraction: ^{69}/_{20}

## :: Final answer ::

Written in 4 different ways

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

3.45 = ^{69}/_{20}

## As a mixed number:

3.45 = 3 ^{9}/_{20}

## As a percentage:

3.45 = 345%

## As equivalent fractions:

3.45 = ^{69}/_{20} = ^{138}/_{40} = ^{414}/_{120}