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Differences of Fractions with Different Denominators

Subtracting equivalent fractions with LCD

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

Cindy is painting her room. She has \begin{align*} \frac{6}{8}\end{align*} of a gallon of paint at the start. She used \begin{align*} \frac{1}{2}\end{align*} of a gallon to paint one wall. How much paint does Cindy have left? Will she have enough to paint another wall? 

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

Subtracting Fractions with Different Denominators

You can add fractions with different denominators by rewriting the fractions with a common denominator before adding. The same step is taken to subtract fractions with different denominators. 

To find the difference of two fractions with different denominators, rewrite the fractions with a common denominator before subtracting. 

Here is a subtraction problem.

\begin{align*}\frac{6}{8} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The fractions in this problem have different denominators, 8 and 4. 

Rewrite the fractions so they share a common denominator. The least common denominator (LCD) is the lowest common multiple (LCM) of 8 and 4.

8: 8, 16, 24 . . .

4: 4, 8, 12, 16 . . .

The LCM is 8. Find the equivalent of each fraction with the denominator 8. \begin{align*}\frac{6}{8}\end{align*} is already in terms of eighths. Multiply the numerator and denominator of \begin{align*}\frac{1}{4}\end{align*} by 2.

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

\begin{align*}\frac{6}{8} - \frac{1}{4} =\frac{6}{8} - \frac{2}{8} \end{align*}

Now you can subtract the fractions with the common denominator. Subtract the numerators over the common denominator.

\begin{align*}\frac{6}{8} - \frac{2}{8} = \frac{4}{8}\end{align*}

Finally, simplify the fraction by dividing the numerator and the denominator by the greatest common factor (GCF). The GCF of 4 and 8 is 4. Divide the numerator and the denominator by 4.

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

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

Examples

Example 1

Earlier, you were given a problem about Cindy and her paint project.

Cindy had \begin{align*} \frac{6}{8}\end{align*} of a gallon of paint and used \begin{align*} \frac{1}{2}\end{align*} gallon to paint one wall. Subtract the amount used from the original amount to find the amount of paint Cindy has left.

\begin{align*}\frac{6}{8} - \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, find the LCM of 8 and 2. The LCM is 8.

Then, rewrite the fractions with the common denominator 8.

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

 \begin{align*}\frac{6}{8} - \frac{1}{2} = \frac{6}{8} - \frac{4}{8}\end{align*}

Next, subtract the fractions.

\begin{align*}\frac{6}{8} - \frac{4}{8} =\frac{2}{8}\end{align*}

Finally, simplify the fraction.

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

Cindy has \begin{align*}\frac{1}{4}\end{align*} of a gallon left over. She will not have enough to paint another wall.

Example 2

Find the difference: \begin{align*}\frac{3}{4} - \frac{6}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

The fractions do not have a common denominator. The denominators are 4 and 12. 

First, find the LCM of 4 and 12. The LCM is 12. 

4: 4, 8, 12, . . .

12: 12, 24 . . .

Then, rewrite the fractions with the common denominator of 12. 

 \begin{align*}\frac{3}{4}=\frac{9}{12}\end{align*}

 \begin{align*}\frac{3}{4} - \frac{6}{12} = \frac{9}{12} - \frac{6}{12}\end{align*} 

Next, subtract the fractions. Subtract the numerators over the common denominator. 

\begin{align*}\frac{9}{12} - \frac{6}{12} = \frac{3}{12} \end{align*}

Finally, simplify the fraction.

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

The difference is \begin{align*}\frac{1}{4}\end{align*}.

Example 3

Find the difference: \begin{align*}\frac{5}{6} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, find the LCM of 6 and 3. The LCM is 6.

Then, rewrite the fractions with a common denominator. 

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

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

Next, subtract the fractions. 

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

Finally, simplify the fraction.

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

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

Example 4

Find the difference\begin{align*}\frac{1}{2} - \frac{4}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, find the LCM of 2 and 9. The LCM is 18.

Then, rewrite the fractions with a common denominator. 

 \begin{align*}\frac{1}{2} &= \frac {9}{18} \\ \frac{4}{9} &= \frac {8}{18}\end{align*}

 \begin{align*}\frac{1}{2} - \frac{4}{9} = \frac {9}{18} - \frac {8}{18}\end{align*}

Next, subtract the fractions. 

 \begin{align*}\frac {9}{18} - \frac {8}{18} = \frac{1}{18}\end{align*}

The fraction is in simplest form.

The difference is \begin{align*}\frac{1}{18}\end{align*}.

Example 5

Find the difference\begin{align*}\frac{4}{5} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, find the LCM of 5 and 4. The LCM is 20.

Then, rewrite the fractions with a common denominator. 

 \begin{align*}\frac{4}{5} & = \frac {16}{20} \\ \frac{1}{4} & = \frac{5}{20}\end{align*}

 \begin{align*} \frac{4}{5} - \frac{1}{4} = \frac {16}{20} - \frac{5}{20}\end{align*}

Next, subtract the fractions. 

 \begin{align*} \frac {16}{20} - \frac{5}{20} = \frac{11}{20}\end{align*}

The fraction is in simplest form.

The difference is \begin{align*}\frac{11}{20}\end{align*}.

Review

Subtract the following fractions. Answer in simplest form.

  1. \begin{align*}\frac{4}{8} - \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}\frac{9}{10} - \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  3. \begin{align*}\frac{10}{10} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} 
  4. \begin{align*}\frac{15}{16} - \frac{2}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  5. \begin{align*}\frac{9}{10} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  6. \begin{align*}\frac{3}{5} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  7. \begin{align*}\frac{9}{10} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}\frac{20}{30} - \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  9. \begin{align*}\frac{18}{19} - \frac{2}{19} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}\frac{4}{6} - \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}\frac{7}{8} - \frac{4}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}\frac{1}{2} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}\frac{4}{5} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}\frac{7}{9} - \frac{2}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}\frac{11}{12} - \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  16. \begin{align*}\frac{6}{7} - \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.8. 

Resources

 

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Vocabulary

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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