Dividing Fractions
Learning Outcomes
- Use a model to describe the result of dividing a fraction by a fraction
- Use an algorithm to divide fractions
Why is [latex]12\div 3=4?[/latex] We previously modeled this with counters. How many groups of [latex]3[/latex] counters can be made from a group of [latex]12[/latex] counters?
There are [latex]4[/latex] groups of [latex]3[/latex] counters. In other words, there are four [latex]3\text{s}[/latex] in [latex]12[/latex]. So, [latex]12\div 3=4[/latex]. What about dividing fractions? Suppose we want to find the quotient: [latex]\frac{1}{2}\div \frac{1}{6}[/latex]. We need to figure out how many [latex]\frac{1}{6}\text{s}[/latex] there are in [latex]\frac{1}{2}[/latex]. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three [latex]\frac{1}{6}[/latex] tiles in [latex]\frac{1}{2}[/latex], so [latex]\frac{1}{2}\div \frac{1}{6}=3[/latex]. Doing the Manipulative Mathematics activity "Model Fraction Division" will help you develop a better understanding of dividing fractions.Example
Model: [latex]\frac{1}{4}\div \frac{1}{8}[/latex] Solution: We want to determine how many [latex]\frac{1}{8}\text{s}[/latex] are in [latex]\frac{1}{4}[/latex]. Start with one [latex]\frac{1}{4}[/latex] tile. Line up [latex]\frac{1}{8}[/latex] tiles underneath the [latex]\frac{1}{4}[/latex] tile. There are two [latex]\frac{1}{8}\text{s}[/latex] in [latex]\frac{1}{4}[/latex]. So, [latex]\frac{1}{4}\div \frac{1}{8}=2[/latex].Try It
Model: [latex]\frac{1}{3}\div \frac{1}{6}[/latex]Answer:
Model: [latex]\frac{1}{2}\div \frac{1}{4}[/latex]Answer:
**Don't delete these** #117916 [ohm_question height="270"]117916[/ohm_question]Example
Model: [latex]2\div \frac{1}{4}[/latex]Answer: Solution: We are trying to determine how many [latex]\frac{1}{4}\text{s}[/latex] there are in [latex]2[/latex]. We can model this as shown. Because there are eight [latex]\frac{1}{4}\text{s}[/latex] in [latex]2,2\div \frac{1}{4}=8[/latex].
Try It
Model: [latex]2\div \frac{1}{3}[/latex]Answer:
Model: [latex]3\div \frac{1}{2}[/latex]Answer:
**Don't delete these. #117216 [ohm_question height="270"]117216[/ohm_question]Fraction Division
If [latex]a,b,c,\text{ and }d[/latex] are numbers where [latex]b\ne 0,c\ne 0,\text{ and }d\ne 0[/latex], then [latex-display]\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}[/latex-display] To divide fractions, multiply the first fraction by the reciprocal of the second. We need to say [latex]b\ne 0,c\ne 0\text{ and }d\ne 0[/latex] to be sure we don’t divide by zero.Example
Divide, and write the answer in simplified form: [latex]\frac{2}{5}\div \left(-\frac{3}{7}\right)[/latex]Answer: Solution:
[latex]\frac{2}{5}\div \left(-\frac{3}{7}\right)[/latex] | |
Multiply the first fraction by the reciprocal of the second. | [latex]\frac{2}{5}\left(-\frac{7}{3}\right)[/latex] |
Multiply. The product is negative. | [latex]-\frac{14}{15}[/latex] |
Try It
#146066 [ohm_question height="270"]146066[/ohm_question] #146067 [ohm_question height="270"]146067[/ohm_question]Example
Divide, and write the answer in simplified form: [latex]\frac{2}{3}\div \frac{n}{5}[/latex]Answer: Solution:
[latex]\frac{2}{3}\div \frac{n}{5}[/latex] | |
Multiply the first fraction by the reciprocal of the second. | [latex]\frac{2}{3}\cdot \frac{5}{n}[/latex] |
Multiply. | [latex]\frac{10}{3n}[/latex] |
Try It
#146089 [ohm_question height="270"]146089[/ohm_question]Example
Divide, and write the answer in simplified form: [latex]-\frac{3}{4}\div \left(-\frac{7}{8}\right)[/latex]Answer: Solution:
[latex]-\frac{3}{4}\div \left(-\frac{7}{8}\right)[/latex] | |
Multiply the first fraction by the reciprocal of the second. | [latex]-\frac{3}{4}\cdot \left(-\frac{8}{7}\right)[/latex] |
Multiply. Remember to determine the sign first. | [latex]\frac{3\cdot 8}{4\cdot 7}[/latex] |
Rewrite to show common factors. | [latex]\frac{3\cdot \overline{)4}\cdot 2}{\overline{)4}\cdot 7}[/latex] |
Remove common factors and simplify. | [latex]\frac{6}{7}[/latex] |
Try It
#146066 [ohm_question height="270"]146066[/ohm_question]Example
Divide, and write the answer in simplified form: [latex]\frac{7}{18}\div \frac{14}{27}[/latex]Answer: Solution:
[latex]\frac{7}{18}\div \frac{14}{27}[/latex] | |
Multiply the first fraction by the reciprocal of the second. | [latex]\frac{7}{18}\cdot \frac{27}{14}[/latex] |
Multiply. | [latex]\frac{7\cdot 27}{18\cdot 14}[/latex] |
Rewrite showing common factors. | [latex]\frac{\color{red}{7}\cdot\color{blue}{9}\cdot3}{\color{blue}{9}\cdot2\cdot\color{red}{7}\cdot2}[/latex] |
Remove common factors. | [latex]\frac{3}{2\cdot 2}[/latex] |
Simplify. | [latex]\frac{3}{4}[/latex] |
Try It
#146091 [ohm_question height="270"]146091[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Ex: Using a Fraction Wall to Find the Quotient of Two Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex: Find the Quotient of a Whole Number and Fraction using Fraction Strips. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex 2: Divide Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 146066, 146067, 146089, 146091. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Shared previously
- Ex 1: Dividing Signed Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 117216, 117916, . Authored by: Amy Volpe. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
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].