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 [latex]a\text{ and }b[/latex] is the result you get from dividing [latex]a\text{ by }b[/latex], or [latex]\frac{a}{b}[/latex]. Let’s practice translating some phrases into algebraic expressions using these terms.Example
Translate the phrase into an algebraic expression: "the quotient of [latex]3x[/latex] and [latex]8[/latex]." 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. [latex-display]\text{The quotient }\text{of }3x\text{ and }8\text{.}[/latex-display] This tells us that we need to divide [latex]3x[/latex] by [latex]8[/latex]. [latex]\frac{3x}{8}[/latex]try it
#146101 [ohm_question height="270"]146101[/ohm_question]Example
Translate the phrase into an algebraic expression: the quotient of the difference of [latex]m[/latex] and [latex]n[/latex], and [latex]p[/latex].Answer: Solution: We are looking for the quotient of the difference of [latex]m[/latex] and , and [latex]p[/latex]. This means we want to divide the difference of [latex]m[/latex] and [latex]n[/latex] by [latex]p[/latex] [latex-display]\frac{m-n}{p}[/latex-display]
Try it
#146103 [ohm_question height="270"]146103[/ohm_question]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: [latex-display]\frac{\frac{6}{7}}{3}\frac{\frac{3}{4}}{\frac{5}{8}}\frac{\frac{x}{2}}{\frac{5}{6}}[/latex-display] To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction [latex]\frac{\frac{3}{4}}{\frac{5}{8}}[/latex] can be written as [latex]\frac{3}{4}\div \frac{5}{8}[/latex].Example
Simplify: [latex]\frac{\frac{3}{4}}{\frac{5}{8}}[/latex]Answer: Solution:
| [latex]\frac{\frac{3}{4}}{\frac{5}{8}}[/latex] | |
| Rewrite as division. | [latex]\frac{3}{4}\div \frac{5}{8}[/latex] |
| Multiply the first fraction by the reciprocal of the second. | [latex]\frac{3}{4}\cdot \frac{8}{5}[/latex] |
| Multiply. | [latex]\frac{3\cdot 8}{4\cdot 5}[/latex] |
| Look for common factors. | [latex]\frac{3\cdot\color{red}{4}\cdot 2}{\color{red}{4} \cdot 5}[/latex] |
| Remove common factors and simplify. | [latex]\frac{6}{5}[/latex] |
Try it
#146109 [ohm_question height="270"]146109[/ohm_question]Simplify a complex fraction.
- Rewrite the complex fraction as a division problem.
- Follow the rules for dividing fractions.
- Simplify if possible.
Example
Simplify: [latex]\frac{-\frac{6}{7}}{3}[/latex]Answer: Solution:
| [latex]\frac{-\frac{6}{7}}{3}[/latex] | |
| Rewrite as division. | [latex]-\frac{6}{7}\div 3[/latex] |
| Multiply the first fraction by the reciprocal of the second. | [latex]-\frac{6}{7}\cdot \frac{1}{3}[/latex] |
| Multiply; the product will be negative. | [latex]-\frac{6\cdot 1}{7\cdot 3}[/latex] |
| Look for common factors. | [latex]-\frac{\color{red}{3} \cdot 2\cdot 1}{7\cdot \color{red}{3} }[/latex] |
| Remove common factors and simplify. | [latex]-\frac{2}{7}[/latex] |
Try it
#146110 [ohm_question height="270"]146110[/ohm_question] #146111 [ohm_question height="270"]146111[/ohm_question]Example
Simplify: [latex]\frac{\frac{x}{2}}{\frac{xy}{6}}[/latex]Answer: Solution:
| [latex]\frac{\frac{x}{2}}{\frac{xy}{6}}[/latex] | |
| Rewrite as division. | [latex]\frac{x}{2}\div \frac{xy}{6}[/latex] |
| Multiply the first fraction by the reciprocal of the second. | [latex]\frac{x}{2}\cdot \frac{6}{xy}[/latex] |
| Multiply. | [latex]\frac{x\cdot 6}{2\cdot xy}[/latex] |
| Look for common factors. | [latex]\frac{\color{red}{x}\cdot 3\cdot \color{red}{2}}{\color{red}{2}\cdot \color{red}{x}\cdot y}[/latex] |
| Remove common factors and simplify. | [latex]\frac{3}{y}[/latex] |
Try it
#146112 [ohm_question height="270"]146112[/ohm_question]Example
Simplify: [latex]\frac{2\frac{3}{4}}{\frac{1}{8}}[/latex]Answer: Solution:
| [latex]\frac{2\frac{3}{4}}{\frac{1}{8}}[/latex] | |
| Rewrite as division. | [latex]2\frac{3}{4}\div \frac{1}{8}[/latex] |
| Change the mixed number to an improper fraction. | [latex]\frac{11}{4}\div \frac{1}{8}[/latex] |
| Multiply the first fraction by the reciprocal of the second. | [latex]\frac{11}{4}\cdot \frac{8}{1}[/latex] |
| Multiply. | [latex]\frac{11\cdot 8}{4\cdot 1}[/latex] |
| Look for common factors. | [latex]\frac{11\cdot \color{red}{4} \cdot 2}{\color{red}{4} \cdot 1}[/latex] |
| Remove common factors and simplify. | [latex]22[/latex] |
Try it
#146161 [ohm_question height="270"]146161[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Simplify Complex Fractions (2/3)/(5/6) and (-8/7)/4. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Complex Fractions (8/5)/(3 1/2) and (a/8)/((ab)/8). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex: Evaluate a Complex Fraction. Authored by: Sousa, James(mathispower4u.com) . License: CC BY: Attribution.
- The Language of Division. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
