Several examples with detailed solutions and exercises with answers on the addition of fractions, are presented.

    

A fraction represents a part of a whole. Take a whole divide into b equal parts and take a parts; this is represented by the fraction $$\dfrac{a}{b}$$. a is called the numerator and b is called the denominator and must be non zero.
Examples of fractions
$$\dfrac{2}{3}$$ , $$\dfrac{3}{4}$$ , and $$\dfrac{7}{2}$$ are examples of fractions.

Examples of Mixed Numbers
A combination of a whole number and a fraction is called a mixed number.
$$2\dfrac{1}{3}$$ , $$5\dfrac{3}{5}$$ , and $$1\dfrac{7}{2}$$ are examples of mixed numbers.

NOTES:
1) Any fraction with denominator equal to 1 is equal to its numerator.
Examples of Fractions with Denominator Equal to 1
$$\dfrac{3}{1} = 3$$ , $$\dfrac{10}{1} = 10$$

2) Fraction with the same denominator are called like fractions.
Examples of like Fractions
$$\dfrac{2}{5}$$ , $$\dfrac{ - 3}{5}$$ , $$\dfrac{21}{5}$$ are like fractions

3) Fraction with different denominators are called unlike fractions.
Examples of unlike Fractions
$$\dfrac{2}{6}$$ , $$\dfrac{ - 3}{7}$$ , $$\dfrac{21}{2}$$ are unlike fractions

4) Any fraction with denominator equal to zero is undefined.
Examples of Fractions with Denominator Equal to Zero
$$\dfrac{2}{0}$$ is undefined because it deniminator is equal to zero..

A fraction calculator that helps you develop further the skills of how to reduce, add and multiply fractions with steps is included.

## Adding Fractions: Examples with Solutions

### 1. Add fractions with same denominators (like fractions)

Example 1
$$\dfrac{2}{3} + \dfrac{4}{3}$$

Solution to Example 1
To add fractions with the same denominator, you add the numerators, keep the same denominator

$$\dfrac{2}{3} + \dfrac{4}{3} = \dfrac{2+4}{3} = \dfrac{6}{3}$$

3 is a common factor to the numerator 6 and the denominator 3, hence we reduce (or simplify) the fraction by dividing the numerator and denominator by the common factor 3 as follows:

$$= \dfrac{6\div3}{3\div3} = \dfrac{2}{1} = 2$$

Example 2
$$\dfrac{12}{14} + \dfrac{34}{14}$$

Solution to Example 2
The two fractions have the same denominator, hence we add the numerators and keep the common denominator as follows:

$$\dfrac{12}{14} + \dfrac{34}{14} = \dfrac{12+34}{14} = \dfrac{46}{14}$$

divide numerator and denominator by $$2$$ and simplify

$$= \dfrac{46\div 2}{14 \div 2} = \dfrac{23}{7}$$

### 2. Add fractions with different denominators (unlike fractions)

Example 3

$$\dfrac{2}{9} + \dfrac{4}{6}$$

Solution to Example 3
We first need to find the lowest common multiple (LCM) of the two denominators 9 and 6 by factoring into prime factors.

$$9 = 3 \times 3$$
$$6 = 2 \times 3$$
The LCM of 9 and 6 $$= 3 \times 3 \times 2 = 18$$

We next convert the two given fractions so that they have common denominator equal to the LCM = 18. The denominator of the fraction $$\dfrac{2}{9}$$ is 9 and we need to multiply numerator and denominator by 2 in order to change the denominator to 18

$$\dfrac{2}{9} = \dfrac{2 \times 2}{9 \times 2} = \dfrac{4}{18}$$

The denominator of the fraction $$\dfrac{4}{6}$$ is 6 and we need to multiply numerator and denominator by 3 in order to change the denominator to 18

$$\dfrac{4}{6} = \dfrac{4 \times 3}{6 \times 3} = \dfrac{12}{18}$$

Now that we have converted the two fractions so that they have common denominator, we can easily add them as follows

$$\dfrac{2}{9} + \dfrac{4}{6} = \dfrac{4}{18} + \dfrac{12}{18} = \dfrac{16}{18}$$

The numerator 16 and denominator 18 have a common factor 2, hence we divide numrator and denominator by 2 to reduce (simplify) the fraction as follows:

$$= \dfrac{16\div2}{18\div2} = \dfrac{8}{9}$$

Example 4
Add, simplify and express the final answer as a fraction and as a mixed number.
$$\dfrac{23}{15} + \dfrac{27}{55}$$

Solution to Example 4
Find the LCM of the denominators 15 and 55

$$15 = 3 \times 5$$
$$55 = 5 \times 11$$
The LCM of 5 and 55 $$= 3 \times 5 \times 11 = 165$$

We convert the given fractions so the they have common denominator equal to the LCM = 165. The denominator of the fraction $$\dfrac{23}{15}$$ is 15 and both numerator and denominator need to be multiplied by 11 in order to change the denominator to 165

$$\dfrac{23}{15} = \dfrac{23\times11}{15\times11} = \dfrac{253}{165}$$

The second fraction $$\dfrac{27}{55}$$ is converted to one with the denominator equal to 165 by multiplying numerator and denominator by 3 in order to change the denominator to 165

$$\dfrac{27}{55} = \dfrac{27\times3}{55\times3} = \dfrac{81}{165}$$

$$\dfrac{23}{15} + \dfrac{27}{55} = \dfrac{253}{165} + \dfrac{81}{165} = \dfrac{334}{165}$$

Because the numerator is larger than the denominator, it is possible to write the above fraction as a mixed number using the long division as follows

$$= \dfrac{334}{165} = 2 + \dfrac{4}{165} = 2 \dfrac{4}{165}$$

2 is the quotient and 4 is the remainder of the divison 334 by 165.

Example 5

$$2 + \dfrac{1}{5}$$

Solution to Example 5

We first convert 2 into a fraction

$$2 = \dfrac{2}{1}$$

The common denominator will be 5

$$2 = \dfrac{2}{1} = \dfrac{10}{5}$$

$$2 + \dfrac{1}{5} = \dfrac{10}{5} + \dfrac{1}{5} = \dfrac{11}{5}$$

Example 6
Add the mixed numbers and simplify

$$4\dfrac{5}{6} + 5\dfrac{7}{8}$$

Solution to Example 6

We first rewrite the two fractions included in the mixed numbers to the same denominator. The LCM of 6 and 8 is 24. Hence
$$\dfrac{5}{6} = \dfrac{20}{24}$$ and $$\dfrac{7}{8} = \dfrac{21}{24}$$

We now substitute the fractions by those with equal denominator

$$4\dfrac{5}{6} + 5\dfrac{7}{8} = 4\dfrac{20}{24} + 5\dfrac{21}{24}$$

We now add the whole numbers and the fractions

$$= (4 + 5) + (\dfrac{20}{24} + \dfrac{21}{24}) = 9 + \dfrac{41}{24}$$

We now rewrite the improper fraction $$\dfrac{41}{24}$$ as a mixed number.

$$\dfrac{41}{24} = 1\dfrac{17}{24}$$

Substitute $$\dfrac{41}{24}$$ by $$1\dfrac{17}{24}$$ in the expression $$9 + \dfrac{41}{24}$$ to obtain the final answer.

$$= 9 + 1\dfrac{17}{24} = 9 + 1 + \dfrac{17}{24} = 10\dfrac{17}{24}$$

Example 7

$$\dfrac{5}{-6} + \dfrac{7}{6}$$

Solution to Example 7
We first change the fraction $$\dfrac{5}{-6}$$ so that its denominator is positive.
$$\dfrac{5}{-6} = \dfrac{-5}{6}$$

$$\dfrac{5}{-6} + \dfrac{7}{6} = \dfrac{-5}{6} + \dfrac{7}{6} = \dfrac{-5+7}{6} = \dfrac{2}{6}$$

$$\dfrac{1}{3}$$

## Execises

Add, simplify and express the answer as a proper fraction or a mixed number.
1. $$\dfrac{3}{7} + \dfrac{6}{7}$$

2. $$\dfrac{2}{3} + \dfrac{5}{7}$$

3. $$4\dfrac{23}{30} + 1\dfrac{8}{25}$$

4. $$5 - \dfrac{2}{6}$$

5. $$\dfrac{11}{8} + \dfrac{3}{-8}$$

1. $$1\dfrac{2}{7}$$

2. $$1\dfrac{8}{21}$$

3. $$6\dfrac{13}{150}$$

4. $$4\dfrac{2}{3}$$

5. $$1$$