## 1. Write the number as a percentage.

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

#### 0.77 =

#### 0.77 × ^{100}/_{100} =

^{(0.77 × 100)}/_{100} =

#### ^{77}/_{100} =

#### 77%

#### In other words:

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

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

#### 0.77 = 77%

## 2. Write the number as a proper fraction.

#### 0.77 can be written as a proper fraction.

#### (The numerator is smaller than the denominator).

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

#### 0.77 = ^{0.77}/_{1}

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

#### Multiply both the top and the bottom by the same number.

#### This number is: 100.

#### (1 followed by as many 0s as the number of digits after the decimal point)

^{0.77}/_{1} =

^{(0.77 × 100)}/_{(1 × 100)} =

^{77}/_{100}

## 3. Reduce (simplify) the fraction above: ^{77}/_{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).

#### In exponential notation (a^{n}):

#### 77 = 7 × 11

#### 100 = 2^{2} × 5^{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 (7 × 11; 2^{2} × 5^{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.

^{77}/_{100}: Equivalent fractions.

#### The above fraction cannot be reduced.

#### That is, it has the smallest numerator and denominator possible.

#### 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:

^{77}/_{100} = ^{(77 × 4)}/_{(100 × 4)} = ^{308}/_{400}

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

^{77}/_{100} = ^{(77 × 5)}/_{(100 × 5)} = ^{385}/_{500}

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

#### ... to the initial fraction: ^{77}/_{100}