To subtract two or more fractions we must follow a procedure similar to that of addition; First we make sure that they all have the same denominator and if not, we have to find equivalent fractions that share a common denominator. Then we subtract their numerators.

In general, there are two types of subtraction of fractions:

- Subtraction of fractions with the same denominator: the numerators are subtracted and the denominator is kept.
- Subtraction of fractions with different denominators: first, the denominators are reduced to a common denominator and the numerators of the equivalent fractions obtained are subtracted.

**General rules for subtracting fractions**

**If the fractions have the same denominator**

- The given fractions are simplified if possible.
- The numerators of the fractions are subtracted and the common denominator is maintained.
- The resulting fraction is simplified, if possible.

**Example**: subtract the following fractions: 3/4; 7/4

Both fractions have the same denominator, therefore, we subtract the numerators and keep the common denominator:

3/4 - 7/4 = - 4/4

We simplify the fraction:

– 4/4 = – 1/1 = – 1

**If the fractions have a different denominator**

- The given fractions are simplified if possible.
- The fractions given are reduced to the lowest common denominator.
- The numerators of the resulting fractions are subtracted and the denominator obtained in step 2 is maintained.
- The resulting fraction is simplified, if possible.

**Example**: subtract the following fractions: 1/6; 2/3

Both fractions have different denominators, we must reduce them to the lowest common denominator. For this, we find the lcm of the denominators: lcm (6; 3) = 6. We divide the lcm between the denominators of the fractions: 6 ÷ 6 = 1 and 6 ÷ 3 = 2

The quotients obtained are multiplied by the respective numerators, that is:

1/6 - 2/3 = 1 (1) / 6 - 2 (2) / 6 = 1/6 - 4/6

We subtract the numerators

1/6 – 4/6 = – 3/6

Simplifying

– 3/6 = – 1/2

**If the fractions are mixed**

- Mixed fractions are converted to improper fractions.
- The given fractions are simplified if possible.
- The fractions given are reduced to the lowest common denominator, if they have a different denominator.
- The numerators of the resulting fractions are subtracted and the denominator obtained in step 3 is maintained.
- The resulting fraction is simplified, if possible.

** Example**: Subtract the following fractions: 1 2/5; 4 3/5

We convert fractions to improper fractions:

1 2/5 = (1 × 5 + 2) / 5 = 7/5

4 3/5 = (4 × 5 + 3) / 5 = 23/5

Both fractions have the same denominator, therefore, we subtract the numerators and keep the common denominator:

7/5 – 23/5 = – 16/5

**Exercises**

Perform the following operations:

- [(a + 5b) / a
^{2}] - [(b - 3) / ab]

Let's see that fractions have a different denominator (and are monomials), therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm (a^{2}; ab) = a^{2}b. We divide^{2}b between each denominator and multiply the quotients by the respective numerator, thus:

[(a + 5b) / a^{2}] - [(b - 3) / ab] = [b (a + 5b) / a^{2}b] - [a (b - 3) / a^{2}b] =

= [(ab + 5b^{2}) / to^{2}b] - [(ab - 3a) / a^{2}b]

We subtract the numerators

[(ab + 5b^{2}) / to^{2}b] - [(ab - 3a) / a^{2}b] = = [(ab + 5b^{2} - ab + 3a) / a^{2}b] =

= (5b^{2} + 3a) / a^{2}b

- [1 / (x-4)] - [1 / (x-3)]

Let's see that fractions have different denominators (and are compounds), therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm = (x - 4) (x - 3). We divide (x - 4) (x - 3) between each denominator and multiplying the quotients by the respective numerator, we have:

[1 / (x-4)] - [1 / (x-3)] = {(x - 3) / [(x - 4) (x - 3)]} - {(x - 4) / [( x - 4) (x - 3)]} =

= (x - 3 - x + 4) / [(x - 4) (x - 3)] = 1 / [(x - 4) (x - 3)]

- (1 4/5) - (2 3/4)

We convert fractions to improper fractions:

1 4/5 = (1 × 5 + 4) / 5 = 9/5

2 3/4 = (2 × 4 + 3) / 4 = 11/4

Let's see that the fractions have a different denominator, therefore, we must reduce them to the lowest common denominator. The lcm of the denominators is: lcm = 20. We divide 20 between each denominator and multiplying the quotients by the respective numerator, we have:

5/9 - 4/11 = [4 (9) / 20] - [5 (11) / 20] = (36/20) - (55/20) = - 19/20