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