# Sum of fractions

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

1. The given fractions are simplified if possible.
2. The numerators of the fractions are added and the common denominator is maintained.
3. 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

1. The given fractions are simplified if possible.
2. The fractions given are reduced to the lowest common denominator.
3. The numerators of the resulting fractions are added and the denominator obtained in step 2 is maintained.
4. 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

1/6 + 2/6 = 3/6

Simplifying

3/6 = 1/2

### If the fractions are mixed

1. Convert mixed fractions to improper fractions.
2. The given fractions are simplified if possible.
3. The fractions given to the lowest common denominator are reduced if they are of a different denominator.
4. The numerators of the resulting fractions are added and the denominator obtained in step 3 is maintained.
5. 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:

1. (3 / 2x) + [(a - 2) / 10x] + (1 / 5x2)

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; 5x2) = 10x2. We divide 10x2 between each denominator and multiply the quotients by the respective numerator, like this:

(3 / 2x) + [(a - 2) / 10x] + (1 / 5x2) = [(3 · 5) / 10x2] + {[(a - 2) · x] / 10x2} + [(2 · 1) / 10x2] =

= (15 / 10x2) + [(a - 2) x / 10x2] + (2 / 10x2)

(15 / 10x2) + [(a - 2) x / 10x2] + (2 / 10x2) = [15 + (a - 2) x + 2] / 10x2 =

= [17 + (a - 2) x] / 10x2=

1. [1 / (2x - 2)] + [1 / (3x + 3)] + [1 / (x2 - 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 / (x2 - 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. (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