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

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

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

- Simply subtract 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 subtracting ordinary fractions that have like denominators, with explanations

^{3}/_{18}+^{4}/_{18}-^{5}/_{18}=^{(3 + 4 - 5)}/_{18}=^{2}/_{18};- We simply subtracted the numerators of the fractions: 3 + 4 - 5 = 2;
- The denominator of the resulting fraction is: 18;
#### The resulting fraction is being reduced as:

^{2}/_{18}=^{(2 ÷ 2)}/_{(18 ÷ 2)}=^{1}/_{9}.#### How to reduce (simplify) the common fraction

^{2}/_{18}?

### B. To subtract 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 (simplify them).

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

- Calculate GCF, the greatest common factor of the numerator and of the denominator of each fraction.
- GCF is the product of all the unique common prime factors of the numerator and of the denominator, multiplied by the lowest exponents.
#### Calculate the greatest common factor, GCF, online.

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

#### 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, also called
**the lowest common denominator (the least common denominator)**. - 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, multiplied by the largest exponents.
#### Calculate LCM, the least common multiple of numbers, online.

- LCM is going to be the common denominator of the added fractions, also called
#### 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 the denominator of each fraction, in order to calculate each fraction's expanding number.

#### 4. Expand each fraction:

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

#### 5. Subtract the fractions:

- In order to subtract all the fractions simply subtract 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.