Adding Fractions

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
Add the fraction and simplfy.
\( \dfrac{2}{3} + \dfrac{4}{3} \)

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
Add, simplify and express the final answer as a fraction.
\( \dfrac{12}{14} + \dfrac{34}{14} \)

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

Add, simplify and express the final answer as a fraction.
\( \dfrac{2}{9} + \dfrac{4}{6} \)

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

\( \dfrac{27}{55} = \dfrac{27\times3}{55\times3} = \dfrac{81}{165}\)

We now add the fractions

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

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
Add, simplify and express the final answer as a fraction.

\( 2 + \dfrac{1}{5} \)

Solution to Example 5

We first convert 2 into a fraction

\( 2 = \dfrac{2}{1} \)

The common denominator will be 5

\( 2 = \dfrac{2}{1} = \dfrac{10}{5} \)

We now add

\( 2 + \dfrac{1}{5} = \dfrac{10}{5} + \dfrac{1}{5} = \dfrac{11}{5}\)

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
Add and simplify.

\( \dfrac{5}{-6} + \dfrac{7}{6} \)

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

We now add the fractions
\( \dfrac{5}{-6} + \dfrac{7}{6} = \dfrac{-5}{6} + \dfrac{7}{6} = \dfrac{-5+7}{6} = \dfrac{2}{6} \)

and reduce the final answer
\( \dfrac{1}{3} \)

Execises

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. \( \dfrac{11}{8} + \dfrac{3}{-8} \)

Answers to Above Exercises:.

1. \( 1\dfrac{2}{7} \)
   
   
2. \( 1\dfrac{8}{21} \)
   
   
3. \( 6\dfrac{13}{150} \)
   
   
4. \( 4\dfrac{2}{3} \)
   
   
5. \( 1 \)

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