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

There are two cases regarding the denominators when we add ordinary fractions:

- A. the fractions have like denominators;
- B. the fractions have unlike 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 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