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Sums of Fractions with Different Denominators

Adding equivalent fractions with LCD

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Sums of Fractions with Different Denominators
License: CC BY-NC 3.0

Larry is building a skateboard ramp in his backyard. At first, he used a \begin{align*}\frac{3}{8}\end{align*} inch thick piece of plywood. It didn't feel sturdy enough so he is replacing it with something that is \begin{align*}\frac{1}{2}\end{align*} inch thicker that what he originally used. How thick will the ramp be? 

In this concept, you will learn how to add fractions with different denominators. 

Adding Fractions with Different Denominators

Fractions that have the same denominator have a common denominator. To add fractions with a common denominator, you find the sum over the numerators over the common denominator. Not all addition problems will involve fractions with common denominators. 

Here is an addition problem.

\begin{align*}\frac{1}{2} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

One-half and one-fourth have different denominators and represent different quantities of a whole. 

To add fractions with different denominators, you will need to rewrite the fractions so that they have a common denominator before finding the sum.

The first step is to the find the least common multiple (LCM) of the denominators, 2 and 4. Remember that the LCM is the smallest multiple that is shared by the numbers being compared. This LCM will become the lowest common denominator (LCD) for the fractions.

List the multiples of 2 and 4.

2: 2, 4, 6, 8, 10 . . . 

4: 4, 8, 12, 16 . . . 

The least common multiple of 2 and 4 is 4.

Then, rewrite each fraction with the common denominator of 4. Multiply the numerator and the denominator of \begin{align*}\frac{1}{2}\end{align*} by 2 to find the equivalent fraction..

\begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}

The second fraction, \begin{align*}\frac{1}{4}\end{align*}, is already written in terms of fourths.

\begin{align*}\frac{2}{4} + \frac{1}{4}\end{align*}

Now you can add the fractions with common denominators. 

 \begin{align*}\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\end{align*}

Finally, simplify the fraction, if possible. The \begin{align*}\frac{3}{4}\end{align*} is a fraction in simplest form. 

The sum is \begin{align*}\frac{3}{4}\end{align*}

You can add any number of fractions with unlike denominators as long as you rewrite the fractions with a common denominator.

Examples

Example 1

Earlier, you were given a problem about Larry and his ramp.

Larry started with \begin{align*}\frac{3}{8}\end{align*} inch plywood, but wanted something \begin{align*}\frac{1}{2}\end{align*} inch thicker. Add the measurements to find the total thickness of the ramp.

 \begin{align*}\frac{3}{8} +\frac{1}{2}=\underline{\;\;\;\;\;\;\;}\end{align*}

First, check the denominators. The denominators are 8 and 2. The LCD is 8.

Then, rewrite the fractions with the common denominator.

 \begin{align*}\frac{1}{2} = \frac{4}{8}\end{align*}

 \begin{align*}\frac{3}{8} + \frac{4}{8} \end{align*}

Next, add the fractions.

 \begin{align*} \frac {3}{8} + \frac{4}{8} = \frac {7}{8} \end{align*}

The fraction is in simplest form.

The skate ramp will be \begin{align*}\frac{7}{8}\end{align*} inch thick.

Example 2

Find the sum.

\begin{align*}\frac{2}{7} + \frac{3}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, check for a common denominator. The denominators are 7 and 9 and are not common. Find the LCD using the LCM of 7 and 9. 

7 – 7, 14, 21, 28, 35, 42, 49, 56, 63

9 – 9, 18, 27, 36, 45, 54, 63

The LCD is 63. 

Then, rewrite the fractions. Find the equivalent fractions with the denominator 63.

\begin{align*}\frac{2}{7} = \frac{18}{63}\\ \frac{3}{9} = \frac{21}{63} \end{align*}  

\begin{align*}\frac{18}{63} + \frac{21}{63}\end{align*}

Next, add the fractions. Add the numerators over the common denominator.

\begin{align*}\frac{18}{63} + \frac{21}{63} = \frac{39}{63}\end{align*}

Finally, simplify the fraction. The GCF of 39 and 63 is 3.  Divide the numerator and the denominator by 3.

 \begin{align*}\frac {39 \div 3} {63 \div 3} = \frac {13}{21} \end{align*}

The sum is \begin{align*}\frac{13}{21}\end{align*}.

Example 3

Find the sum: \begin{align*}\frac{1}{2} + \frac{2}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check the denominators. The denominators are 2 and 6. The LCD is 6.

Then, rewrite the fractions with the common denominator.

\begin{align*}\frac{1}{2} &= \frac {3}{6} \\ \end{align*}

 \begin{align*}\frac {3}{6} + \frac{2}{6}\end{align*}

Next, add the fractions.

 \begin{align*}\frac {3}{6} + \frac{2}{6} = \frac{5}{6}\end{align*}

The fraction is in simplest form.

The sum is \begin{align*} \frac{5}{6}\end{align*}.

Example 4

Find the sum: \begin{align*}\frac{2}{3} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check the denominators. The denominators are 3 and 9. The LCD is 9.

Then, rewrite the fractions with the common denominator.

 \begin{align*}\frac{2}{3} &= \frac{6}{9}\\ \end{align*}

 \begin{align*}\frac{6}{9} + \frac{1}{9}\end{align*}

Next, add the fractions.

 \begin{align*}\frac{6}{9} + \frac{1}{9} = \frac {7}{9}\end{align*}

The fraction is in simplest form.

The sum is \begin{align*} \frac{7}{9}\end{align*}.

Example 5

Find the sum: \begin{align*}\frac{4}{5} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check the denominators. The denominators are 5 and 3. The LCD is 15.

Then, rewrite the fractions with the common denominator.

 \begin{align*}\frac{4}{5}& =\frac {12}{15} \\ \frac{1}{3} & =\frac {5}{15}\end{align*}

 \begin{align*}\frac {12}{15} + \frac {5}{15}\end{align*}

Next, add the fractions.

 \begin{align*}\frac {12}{15} + \frac {5}{15} = \frac {17}{15}\end{align*}

Finally, simplify the fraction. Convert the improper fraction to a proper fraction.

\begin{align*} \frac{17}{15} = 1 \frac {2}{15}\end{align*}

The sum is \begin{align*}1 \frac{2}{15}\end{align*}.

Review

Find the sum. Answer in simplest form.

  1. \begin{align*}\frac{3}{4} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  2. \begin{align*}\frac{6}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  3. \begin{align*}\frac{2}{3} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  4. \begin{align*}\frac{2}{3} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  5. \begin{align*}\frac{1}{2} + \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  6. \begin{align*}\frac{3}{6} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  7. \begin{align*}\frac{6}{8} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}\frac{4}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  9. \begin{align*}\frac{4}{5} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}\frac{3}{7} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}\frac{5}{8} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}\frac{6}{7} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}\frac{9}{12} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}\frac{8}{10} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}\frac{6}{7} + \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  16. \begin{align*}\frac{3}{4} + \frac{2}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.7. 

Resources

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Vocabulary

TermDefinition
Equivalent Fractions Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
Least Common Multiple The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.
Lowest Common Denominator The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.
Renaming fractions Renaming fractions means rewriting fractions with different denominators, but not changing the value of the fraction.

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