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Guias de estudo > Prealgebra

Simplifying Complex Fractions

Learning Outcomes

  • Translate phrases into algebraic expressions that involve division
  • Identify a complex fraction
  • Simplify complex fractions

Translate Phrases to Expressions with Fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of a and ba\text{ and }b is the result you get from dividing a by ba\text{ by }b, or ab\frac{a}{b}. Let’s practice translating some phrases into algebraic expressions using these terms.

Example

Translate the phrase into an algebraic expression: "the quotient of 3x3x and 88." Solution: The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide. The quotient of 3x and 8.\text{The quotient }\text{of }3x\text{ and }8\text{.} This tells us that we need to divide 3x3x by 88. 3x8\frac{3x}{8}
 

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#146101 [ohm_question height="270"]146101[/ohm_question]
 

Example

Translate the phrase into an algebraic expression: the quotient of the difference of mm and nn, and pp.

Answer: Solution: We are looking for the quotient of the difference of mm and , and pp. This means we want to divide the difference of mm and nn by pp m−np\frac{m-n}{p}

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#146103 [ohm_question height="270"]146103[/ohm_question]
In the following video we show more examples of translating English expressions into algebraic expressions. https://youtu.be/WxJxY4aJ9Vk

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction. Some examples of complex fractions are: 6733458x256\frac{\frac{6}{7}}{3}\frac{\frac{3}{4}}{\frac{5}{8}}\frac{\frac{x}{2}}{\frac{5}{6}} To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction 3458\frac{\frac{3}{4}}{\frac{5}{8}} can be written as 34÷58\frac{3}{4}\div \frac{5}{8}.

Example

Simplify: 3458\frac{\frac{3}{4}}{\frac{5}{8}}

Answer: Solution:

3458\frac{\frac{3}{4}}{\frac{5}{8}}
Rewrite as division. 34÷58\frac{3}{4}\div \frac{5}{8}
Multiply the first fraction by the reciprocal of the second. 34⋅85\frac{3}{4}\cdot \frac{8}{5}
Multiply. 3⋅84⋅5\frac{3\cdot 8}{4\cdot 5}
Look for common factors. 3⋅4⋅24⋅5\frac{3\cdot\color{red}{4}\cdot 2}{\color{red}{4} \cdot 5}
Remove common factors and simplify. 65\frac{6}{5}

 

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#146109 [ohm_question height="270"]146109[/ohm_question]
The following video shows another example of how to simplify a complex fraction. https://youtu.be/CLmWfkeKBg0

Simplify a complex fraction.

  1. Rewrite the complex fraction as a division problem.
  2. Follow the rules for dividing fractions.
  3. Simplify if possible.

Example

Simplify: −673\frac{-\frac{6}{7}}{3}

Answer: Solution:

−673\frac{-\frac{6}{7}}{3}
Rewrite as division. −67÷3-\frac{6}{7}\div 3
Multiply the first fraction by the reciprocal of the second. −67⋅13-\frac{6}{7}\cdot \frac{1}{3}
Multiply; the product will be negative. −6⋅17⋅3-\frac{6\cdot 1}{7\cdot 3}
Look for common factors. −3⋅2⋅17⋅3-\frac{\color{red}{3} \cdot 2\cdot 1}{7\cdot \color{red}{3} }
Remove common factors and simplify. −27-\frac{2}{7}

 

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#146110 [ohm_question height="270"]146110[/ohm_question] #146111 [ohm_question height="270"]146111[/ohm_question]
https://youtu.be/T1PIfrU3NTw

Example

Simplify: x2xy6\frac{\frac{x}{2}}{\frac{xy}{6}}

Answer: Solution:

x2xy6\frac{\frac{x}{2}}{\frac{xy}{6}}
Rewrite as division. x2÷xy6\frac{x}{2}\div \frac{xy}{6}
Multiply the first fraction by the reciprocal of the second. x2⋅6xy\frac{x}{2}\cdot \frac{6}{xy}
Multiply. x⋅62⋅xy\frac{x\cdot 6}{2\cdot xy}
Look for common factors. x⋅3⋅22⋅x⋅y\frac{\color{red}{x}\cdot 3\cdot \color{red}{2}}{\color{red}{2}\cdot \color{red}{x}\cdot y}
Remove common factors and simplify. 3y\frac{3}{y}

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#146112 [ohm_question height="270"]146112[/ohm_question]
 

Example

Simplify: 23418\frac{2\frac{3}{4}}{\frac{1}{8}}

Answer: Solution:

23418\frac{2\frac{3}{4}}{\frac{1}{8}}
Rewrite as division. 234÷182\frac{3}{4}\div \frac{1}{8}
Change the mixed number to an improper fraction. 114÷18\frac{11}{4}\div \frac{1}{8}
Multiply the first fraction by the reciprocal of the second. 114⋅81\frac{11}{4}\cdot \frac{8}{1}
Multiply. 11⋅84⋅1\frac{11\cdot 8}{4\cdot 1}
Look for common factors. 11⋅4⋅24⋅1\frac{11\cdot \color{red}{4} \cdot 2}{\color{red}{4} \cdot 1}
Remove common factors and simplify. 2222

 

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#146161 [ohm_question height="270"]146161[/ohm_question]
https://youtu.be/H5K4ESHLBks

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