## How to: Adding ordinary (simple, common) fractions. Steps.

When we add common ordinary fractions we can find ourselves in one of the two possible cases, if we look at the denominators of the fractions:

- A. the fractions have like (equal) denominators;
- B. the fractions have unlike (different) denominators.

### A. How to add ordinary fractions that have like denominators?

- Simply add the numerators of the fractions.
- The denominator of the resulting fraction will be the common denominator of the fractions.
- Reduce the resulting fraction.

### An example of adding ordinary fractions that have like denominators, with explanations

^{3}/_{18}+^{4}/_{18}+^{5}/_{18}=^{(3 + 4 + 5)}/_{18}=^{12}/_{18};- We simply added the numerators of the fractions: 3 + 4 + 5 = 12;
- The denominator of the resulting fraction is: 18;
#### The resulting fraction is being reduced:

^{12}/_{18}=^{(12 ÷ 6)}/_{(18 ÷ 6)}=^{2}/_{3}.#### How to reduce (simplify) the common (ordinary) fraction

^{12}/_{18}?

### B. To add fractions with different denominators (unlike denominators), build up the fractions to the same denominator. How is it done?

#### 1. Reduce the fractions to the lowest terms (simplifying).

- Factor the numerator and the denominator of each fraction down to prime factors (prime factorization).
#### Factor numbers online down to their prime factors.

- Calculate GCF, the greatest common factor (also called GCD, greatest common divisor, HCF, greatest common factor) of each fraction's numerator and denominator.
- GCF is the product of all the unique common prime factors of the numerator and the denominator, taken by the lowest exponents.
#### Calculate online the greatest common factor, GCF.

- Divide the numerator and the denominator of each fraction by their greatest common factor, GCF - after this operation the fraction is reduced to its lowest terms equivalent.
#### Reduce (simplify) fractions online to their lowest terms, with explanations.

#### 2. Calculate the least common multiple, LCM, of all the fractions' new denominators:

- LCM is going to be the common denominator of the added fractions.
- Factor all the new denominators of the reduced fractions (run the prime factorization).
- The least common multiple, LCM, is the product of all the unique prime factors of the denominators, taken by the largest exponents.
#### Calculate LCM, the least common multiple of numbers.

#### 3. Calculate each fraction's expanding number:

- The expanding number is the non-zero number that will be used to multiply both the numerator and the denominator of each fraction, in order to build all the fractions up to the same common denominator.
- Divide the least common multiple, LCM, calculated above, by each fraction's denominator, in order to calculate each fraction's expanding number.

#### 4. Expand each fraction:

- Multiply each fraction's both numerator and denominator by expanding number.
- At this point, fractions are built up to the same denominator.

#### 5. Add the fractions:

- In order to add all the fractions simply add all the fractions' numerators.
- The end fraction will have as a denominator the least common multiple, LCM, calculated above.

#### 6. Reduce the end fraction to the lowest terms, if needed.

### An example of adding fractions that have different denominators (unlike denominators), step by step explanations

^{6}/_{90}+^{16}/_{24}+^{30}/_{75}= ?#### 1. Reduce the fractions to the lowest terms (simplifying):

^{6}/_{90}=^{(2 × 3)}/_{(2 × 32 × 5)}=^{((2 × 3) ÷ (2 × 3))}/_{((2 × 32 × 5) ÷ (2 × 3))}=^{1}/_{(3 × 5)}=^{1}/_{15}^{16}/_{24}=^{24}/_{(23 × 3)}=^{(24 ÷ 23)}/_{((23 × 3) ÷ 23)}=^{2}/_{3}^{30}/_{75}=^{(2 × 3 × 5)}/_{(3 × 52)}=^{((2 × 3 × 5) ÷ (3 × 5))}/_{((3 × 25) ÷ (3 × 5))}=^{2}/_{5}#### The reduced fractions:

^{6}/_{90}+^{16}/_{24}+^{30}/_{75}=^{1}/_{15}+^{2}/_{3}+^{2}/_{5}

#### 2. Calculate the least common multiple, LCM, of all the fractions' new denominators

- Factor all the denominators down to their prime factors then multiply ALL the unique prime factors found, by the largest exponents.
#### 15 = 3 × 5

#### 3 is already a prime number, it cannot be prime factorized anymore

#### 5 is a prime number, it cannot be prime factorized anymore

#### LCM (15, 3, 5) = LCM (3 × 5, 3, 5) = 3 × 5 = 15

#### 3. Calculate each fraction's expanding number:

- Divide the least common multiple LCM by each fraction's denominator.
#### For the first fraction: 15 ÷ 15 = 1

#### For the second fraction: 15 ÷ 3 = 5

#### For the third fraction: 15 ÷ 5 = 3

#### 4. Expand each fraction:

- Multiply both the numerator and the denominator of each fraction by their expanding number.
#### The first fraction stays unchanged:

^{1}/_{15}=^{(1 × 1)}/_{(1 × 15)}=^{1}/_{15}#### The second fraction expands to:

^{2}/_{3}=^{(5 × 2)}/_{(5 × 3)}=^{10}/_{15}#### The third fraction expands to:

^{2}/_{5}=^{(3 × 2)}/_{(3 × 5)}=^{6}/_{15}

#### 5. Add the fractions:

- Simply add fractions' numerators. The denominator = LCM.
^{6}/_{90}+^{16}/_{24}+^{30}/_{75}=^{1}/_{15}+^{2}/_{3}+^{2}/_{5}=^{1}/_{15}+^{10}/_{15}+^{6}/_{15}=^{(1 + 10 + 6)}/_{15}=^{17}/_{15}

#### 6. Reduce the end fraction to the lowest terms, if needed.

- In this particular case it was no longer needed to
**reduce**the fraction, since the numerator and the denominator are coprime numbers (prime to each other, no other common factors than 1).

- In this particular case it was no longer needed to
#### 7. Extra step - rewrite the end fraction:

- Since the final fraction is an
**improper one (also called a top-heavy fraction)**, in other words the absolute value of the numerator is larger than the absolute value of the denominator, it can be written as a**mixed number (also called a mixed fraction)**: ^{17}/_{15}=^{(15 + 2)}/_{15}=^{15}/_{15}+^{2}/_{15}= 1 +^{2}/_{15}= 1^{2}/_{15}

- Since the final fraction is an