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Expression Evaluation with Different Denominators

Evaluating combinations of unlike fractions

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Expression Evaluation with Different Denominators
Credit: jayneandd
Source: https://www.flickr.com/photos/jayneandd/4078152188/
License: CC BY-NC 3.0

Leslie is combining the leftover cookies from the bake sale. One pan had \begin{align*}\frac{1}{4}\end{align*} of a dozen left. Another pan had \begin{align*}\frac{1}{3}\end{align*} of a dozen left. Just then, a teacher came by and bought a half dozen cookies. How many cookies were left?

In this concept, you will learn how to evaluate numerical expressions involving fractions with different denominators.

Evaluating Expressions with Different Denominators

To evaluate an expression that involves adding and subtracting fractions with like denominators, follow the order of operations. Resolve the operations with parenthesis first, and then add or subtract the fractions going from left to right. The process is similar when evaluating an expression with different denominators. Before evaluating the expression, rewrite the fractions with the lowest common denominator.

Here is an expression of fractions.

\begin{align*}\frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The denominators are 2, 3, and 6. Find the lowest common denominator by finding the lowest common multiple (LCM) of 2, 3, and 6. 

2: 2, 4, 6, 8, 10, 12, . . . 

3: 3, 6, 9, 12, . . . 

6: 6, 12, . . .

The lowest common multiple for all three denominators is 6. Find the equivalent fractions of \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{1}{3}\end{align*} with the denominator 6. The fraction \begin{align*}\frac{1}{6}\end{align*} is already in terms of sixths.

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

Rewrite the expression with the equivalent fractions.

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

Evaluate the expression in order from left to right. Add or subtract the numerators and write the result over the common denominator.

 \begin{align*} \frac{3}{6} + \frac{2}{6} = \frac {5}{6} - \frac{1}{6} = \frac{4}{6}\end{align*}

Finally, write the fraction in simplest form. The greatest common factor (GCF) of 4 and 6 is 2. Divide the numerator and denominator by 2. 

 \begin{align*}\frac{4 \div 2}{6 \div 2}= \frac {2}{3}\end{align*}

The answer is \begin{align*} \frac {2}{3}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Leslie and the cookies.

Leslie combine one-fourth and one-third of a dozen cookies, and then sold a half dozen cookies to a teacher. Add and subtract the fractions to find out how many cookies were left. 

First, write an expression to solve the problem.

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

Then, rewrite the fractions using the lowest common denominator. The LCD is 12.

\begin{align*} \frac{1}{4} + \frac{1}{3} - \frac{1}{2} = \frac{3}{12} + \frac{4}{12} - \frac{6}{12}\end{align*}

Next, add or subtract in order from left to right.

 \begin{align*} \frac{3}{12} + \frac{4}{12} = \frac{7}{12} - \frac{6}{12} = \frac {1}{12}\end{align*}

There was \begin{align*} \frac{1}{12}\end{align*} of a dozen cookies left. A dozen is equal to 12. Therefore, there was 1 cookie left.

Example 2

Evaluate the expression: \begin{align*} \frac{6}{9} + \frac{1}{3} - \frac{4}{5}\end{align*}. Answer in simplest form.

First, find the lowest common denominator using the lowest common multiple of 9, 3, and 5. The LCD is 45.

Then, rewrite the fractions with the common denominator.

 \begin{align*} \frac{6}{9} &=\frac{30}{45} \\ \frac{1}{3} &=\frac{15}{45} \\ \frac{4}{5}&= \frac{36}{45}\end{align*}

\begin{align*} \frac{6}{9} + \frac{1}{3} - \frac{4}{5} = \frac{30}{45} + \frac{15}{45}- \frac{36}{45}\end{align*}

Next, add or subtract in order from left to right.

\begin{align*} \frac{30}{45} + \frac{15}{45}=\frac{45}{45} - \frac{36}{45} = \frac{9}{45}\end{align*}

Finally, simplify the fraction.

\begin{align*} \frac{9}{45} = \frac{1}{5}\end{align*}

The answer is \begin{align*}\frac{1}{5}\end{align*}.

Example 3

Evaluate the expression: \begin{align*} \frac{4}{5} + \frac{2}{10} - \frac{1}{2}\end{align*}. Answer in simplest form.

First, find the LCD of 5, 10, and 2. The LCD is 10. 

Then, rewrite the fractions with the common denominator.

 \begin{align*} \frac{4}{5} + \frac{2}{10} - \frac{1}{2}= \frac{8}{10} + \frac{2}{10} - \frac{5}{10}\end{align*}

Next, add or subtract in order from left to right.

\begin{align*} \frac{8}{10} + \frac{2}{10} = \frac {10}{10}- \frac{5}{10} = \frac {5}{10}\end{align*}

Finally, simplify the fraction.

 \begin{align*} \frac {5}{10} = \frac {1}{2}\end{align*}

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

Example 4

Evaluate the expression: \begin{align*} \frac{4}{8} - \frac{1}{2} + \frac{1}{4}\end{align*}. Answer in simplest form.

First, find the LCD of 8, 2, and 4. The LCD is 8.

Then, rewrite the fractions with the common denominator.

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

Next, add or subtract in order from left to right.

 \begin{align*}\frac{4}{8} - \frac{4}{8} = \frac{0}{8} + \frac{2}{8} = \frac{2}{8}\end{align*}

Finally, simplify the fraction.

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

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

Example 5

Evaluate the expression: \begin{align*} \frac{3}{4} - \frac{5}{8} + \frac{1}{4}\end{align*}. Answer in simplest form.

First, find the LCD of 4, 8, and 4. The LCD is 8.

Then, rewrite the fractions with the common denominator.

 \begin{align*} \frac{3}{4} - \frac{5}{8} + \frac{1}{4} = \frac{6}{8} - \frac{5}{8} + \frac{2}{8}\end{align*}

Next, add or subtract in order from left to right.

 \begin{align*} \frac{6}{8} - \frac{5}{8} = \frac {1}{8} + \frac{2}{8} = \frac{3}{8}\end{align*}

The fraction is in simplest form.

The answer is \begin{align*} \frac{3}{8}\end{align*}.

Review

Evaluate the following expressions. Answer in simplest form.

  1. \begin{align*}\frac{1}{2} + \frac{1}{3} + \frac{2}{4}\end{align*}
  2. \begin{align*}\frac{6}{9} + \frac{1}{3} - \frac{2}{3}\end{align*}
  3. \begin{align*}\frac{4}{5} + \frac{1}{3} - \frac{1}{5}\end{align*}
  4. \begin{align*}\frac{8}{9} + \frac{1}{2} - \frac{1}{3}\end{align*}
  5. \begin{align*}\frac{3}{4} + \frac{1}{3} - \frac{2}{10}\end{align*}
  6. \begin{align*}\frac{3}{4} + \frac{1}{3} + \frac{1}{2}\end{align*}
  7. \begin{align*}\frac{1}{5} + \frac{2}{5} - \frac{2}{7}\end{align*}
  8. \begin{align*}\frac{5}{6} + \frac{1}{3} - \frac{1}{2}\end{align*}
  9. \begin{align*}\frac{8}{9} + \frac{1}{3} - \frac{2}{9}\end{align*}
  10. \begin{align*}\frac{8}{11} + \frac{1}{3} - \frac{2}{3}\end{align*}
  11. \begin{align*}\frac{6}{7} + \frac{1}{2} - \frac{2}{7}\end{align*}
  12. \begin{align*}\frac{4}{9} + \frac{2}{9} - \frac{2}{3}\end{align*}
  13. \begin{align*}\frac {11}{12} + \frac{1}{12} - \frac{6}{8}\end{align*}
  14. \begin{align*}\frac{13}{14} + \frac{1}{28} - \frac{4}{7}\end{align*}
  15. \begin{align*}\frac{17}{18} + \frac{2}{18} - \frac{5}{9}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.9. 

Resources

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Vocabulary

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.

Image Attributions

  1. [1]^ Credit: jayneandd; Source: https://www.flickr.com/photos/jayneandd/4078152188/; License: CC BY-NC 3.0

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