To add two or more fractions, we must ensure that they all have the same denominator, if not, we must find equivalent fractions that share a common denominator. Then we add their numerators.

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

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

**General rules for adding fractions**

**If the fractions have the same denominator**

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

**Example**: Add the following fractions: 1/9; 2/9

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

1/9 + 2/9 = 3/9

We simplify the fraction:

3/9 = 1/3

**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 added and the denominator obtained in step 2 is maintained.
- The resulting fraction is simplified, if possible.

** Example**: Add the following fractions: 1/6; 1/3

Both fractions have different denominators, we must reduce them to the lowest common denominator. To do 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 + 1/3 = [1(1)] / 6 + [1(2)] / 6 = 1/6 + 2/6

We add the numerators

1/6 + 2/6 = 3/6

Simplifying

3/6 = 1/2

**If the fractions are mixed**

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

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

We convert fractions to improper fractions:

3 2/5 = (3 × 5 + 2) / 5 = 17/5

1 3/5 = (1 × 5 + 3) / 5 = 8/5

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

17/5 + 8/5 = 25/5

We simplify the fraction:

25/5 = 5/1 = 1

**Exercises**

Perform the following operations:

- (3 / 2x) + [(a - 2) / 10x] + (1 / 5x
^{2})

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 (2x; 10x; 5x^{2}) = 10x^{2}. We divide 10x^{2 }between each denominator and multiply the quotients by the respective numerator, like this:

(3 / 2x) + [(a - 2) / 10x] + (1 / 5x^{2}) = [(3 · 5) / 10x^{2}] + {[(a - 2) · x] / 10x^{2}} + [(2 · 1) / 10x^{2}] =

= (15 / 10x^{2}) + [(a - 2) x / 10x^{2}] + (2 / 10x^{2})

We add the numerators

(15 / 10x^{2}) + [(a - 2) x / 10x^{2}] + (2 / 10x^{2}) = [15 + (a - 2) x + 2] / 10x^{2 }=

= [17 + (a - 2) x] / 10x^{2}=

- [1 / (2x - 2)] + [1 / (3x + 3)] + [1 / (x
^{2 }- one)]

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 = 6 (x + 1) (x - 1). We divide 6 (x + 1) (x - 1) between each denominator and multiplying the quotients by the respective numerator, we have:

[1 / (2x - 2)] + [1 / (3x + 3)] + [1 / (x^{2 }- 1)] =

= {[3 (x + 1)] / [6 (x + 1) (x - 1)]} + {[2 (x - 1)] / [6 (x + 1) (x - 1)]} + {6 / [6 (x + 1) (x - 1)]} =

= (3x + 3) / 6 (x + 1) (x-1) + (2x-2) / 6 (x + 1) (x-1) +6/6 (x + 1) (x-1) = (3x + 3 + 2x-2 + 6) / 6 (x + 1) (x-1) =

= (5x + 7) / (6 (x + 1) (x-1))

- (1 4/6) + (2 1/4)

We convert fractions to improper fractions:

1 4/6 = (1 × 6 + 4) / 6 = 10/6

2 1/4 = (2 × 4 + 1) / 4 = 9/4

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 = 12. We divide 12 between each denominator and multiplying the quotients by the respective numerator, we have:

10/6 + 9/4 = [2(10)/12] + [3(9)/12] = 20/12 + 27/12 = 47/12