{"id":9489,"date":"2017-05-02T23:16:12","date_gmt":"2017-05-02T23:16:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9489"},"modified":"2018-05-03T16:16:08","modified_gmt":"2018-05-03T16:16:08","slug":"dividing-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/dividing-fractions\/","title":{"raw":"Dividing Fractions","rendered":"Dividing Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use a model to describe the result of dividing a fraction by a fraction<\/li>\r\n \t<li>Use an algorithm to divide fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>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?<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220928\/CNX_BMath_Figure_04_02_015_img.png\" alt=\"Four red ovals are shown. Inside each oval are three grey circles.\" \/>\r\nThere 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].\r\nWhat about dividing fractions? Suppose we want to find the quotient: [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}[\/latex]. We need to figure out how many [latex]\\Large\\frac{1}{6}\\normalsize\\text{s}[\/latex] there are in [latex]\\Large\\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]\\Large\\frac{1}{6}[\/latex] tiles in [latex]\\Large\\frac{1}{2}[\/latex], so [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}\\normalsize=3[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220930\/CNX_BMath_Figure_04_02_016.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nModel: [latex]\\Large\\frac{1}{4}\\normalsize\\div\\Large\\frac{1}{8}[\/latex]\r\n\r\nSolution:\r\nWe want to determine how many [latex]\\Large\\frac{1}{8}\\normalsize\\text{s}[\/latex] are in [latex]\\Large\\frac{1}{4}[\/latex]. Start with one [latex]\\Large\\frac{1}{4}[\/latex] tile. Line up [latex]\\Large\\frac{1}{8}[\/latex] tiles underneath the [latex]\\Large\\frac{1}{4}[\/latex] tile.\r\n<p style=\"text-align: center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220931\/CNX_BMath_Figure_04_02_017_img.png\" alt=\"A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.\" \/>\r\nThere are two [latex]\\Large\\frac{1}{8}[\/latex]s in [latex]\\Large\\frac{1}{4}[\/latex].\r\nSo, [latex]\\Large\\frac{1}{4}\\normalsize\\div\\Large\\frac{1}{8}\\normalsize=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nModel: [latex]\\Large\\frac{1}{3}\\normalsize\\div\\Large\\frac{1}{6}[\/latex]\r\n[reveal-answer q=\"218091\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"218091\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220932\/CNX_BMath_Figure_04_02_018_img.png\" alt=\"A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.\" \/>\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nModel: [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"763601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"763601\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220934\/CNX_BMath_Figure_04_02_019_img.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.\" \/>\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another way to model division of two fractions.\r\n\r\nhttps:\/\/youtu.be\/pk-K5JF9iMo\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nModel: [latex]2\\div\\Large\\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"391699\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391699\"]\r\n\r\nSolution:\r\nWe are trying to determine how many [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] there are in [latex]2[\/latex]. We can model this as shown.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220935\/CNX_BMath_Figure_04_02_020_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.\" \/>\r\nBecause there are eight [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] in [latex]2,2\\div\\Large\\frac{1}{4}\\normalsize=8[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nModel: [latex]2\\div\\Large\\frac{1}{3}[\/latex]\r\n[reveal-answer q=\"73567\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"73567\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220936\/CNX_BMath_Figure_04_02_021_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.\" \/>\r\n\r\n[latex]6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nModel: [latex]3\\div\\Large\\frac{1}{2}[\/latex]\r\n[reveal-answer q=\"354856\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"354856\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220938\/CNX_BMath_Figure_04_02_022_img.png\" alt=\"Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.\" \/>\r\n\r\n[latex]6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[ohm_question height=\"800\"]117216[\/ohm_question]\r\n\r\n<\/div>\r\nThe next video shows more examples of how to divide a whole number by a fraction.\r\n\r\nhttps:\/\/youtu.be\/JKsfdK1WT1s\r\n\r\nLet\u2019s use money to model [latex]2\\div\\Large\\frac{1}{4}[\/latex] in another way. We often read [latex]\\Large\\frac{1}{4}[\/latex] as a \u2018quarter\u2019, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of [latex]2\\div\\Large\\frac{1}{4}[\/latex] as, \"How many quarters are there in two dollars?\" One dollar is [latex]4[\/latex] quarters, so [latex]2[\/latex] dollars would be [latex]8[\/latex] quarters. So again, [latex]2\\div\\Large\\frac{1}{4}\\normalsize=8[\/latex].\r\n\r\nThe U.S. coin called a quarter is worth one-fourth of a dollar.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220939\/CNX_BMath_Figure_04_02_023.png\" alt=\"A picture of a United States quarter is shown.\" \/>\r\nUsing fraction tiles, we showed that [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}\\normalsize=3[\/latex]. Notice that [latex]\\Large\\frac{1}{2}\\cdot \\frac{6}{1}\\normalsize=3[\/latex] also. How are [latex]\\Large\\frac{1}{6}[\/latex] and [latex]\\Large\\frac{6}{1}[\/latex] related? They are reciprocals. This leads us to the procedure for fraction division.\r\n<div class=\"textbox shaded\">\r\n<h3>Fraction Division<\/h3>\r\nIf [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\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\normalsize\\div\\Large\\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\r\nTo divide fractions, multiply the first fraction by the reciprocal of the second.\r\n\r\nWe need to say [latex]b\\ne 0,c\\ne 0\\text{ and }d\\ne 0[\/latex] to be sure we don\u2019t divide by zero.\r\n\r\n<span style=\"color: #3366ff\">Tip:\u00a0 Here's a rhyme to help you with dividing fractions.\u00a0 When dividing fractions don't ask why, just flip the second and multiply. <\/span>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\Large\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]\r\n[reveal-answer q=\"261121\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261121\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468274991\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\Large\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large\\frac{2}{5}\\left(-\\frac{7}{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply. The product is negative.<\/td>\r\n<td>[latex]\\Large-\\frac{14}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146066[\/ohm_question]\r\n\r\n[ohm_question height=\"270\"]146067[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video for more examples of dividing fractions using a reciprocal.\r\n\r\nhttps:\/\/youtu.be\/fnaRnEXlUvs\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\Large\\frac{2}{3}\\normalsize\\div\\Large\\frac{n}{5}[\/latex]\r\n[reveal-answer q=\"853209\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"853209\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466192195\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{2}{3}\\normalsize\\div\\Large\\frac{n}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large\\frac{2}{3}\\cdot \\frac{5}{n}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{10}{3n}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146089[\/ohm_question]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\Large-\\frac{3}{4}\\normalsize\\div\\Large\\left(-\\frac{7}{8}\\right)[\/latex]\r\n[reveal-answer q=\"873547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"873547\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467263034\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\Large-\\frac{3}{4}\\normalsize\\div\\Large\\left(-\\frac{7}{8}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large-\\frac{3}{4}\\cdot \\left(-\\frac{8}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply. Remember to determine the sign first.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot 8}{4\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite to show common factors.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot 4\\cdot 2}{4\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors and simplify.<\/td>\r\n<td>[latex]\\Large\\frac{6}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146066[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows more examples of dividing fractions that are negative.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\Large\\frac{7}{18}\\normalsize\\div\\Large\\frac{14}{27}[\/latex]\r\n[reveal-answer q=\"987562\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"987562\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466022407\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"No Alt Text\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{7}{18}\\normalsize\\div\\Large\\frac{14}{27}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large\\frac{7}{18}\\cdot \\frac{27}{14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{7\\cdot 27}{18\\cdot 14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing common factors.<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{7}\\cdot\\color{blue}{9}\\cdot3}{\\color{blue}{9}\\cdot2\\cdot\\color{red}{7}\\cdot2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large\\frac{3}{2\\cdot 2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146091[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use a model to describe the result of dividing a fraction by a fraction<\/li>\n<li>Use an algorithm to divide fractions<\/li>\n<\/ul>\n<\/div>\n<p>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?<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220928\/CNX_BMath_Figure_04_02_015_img.png\" alt=\"Four red ovals are shown. Inside each oval are three grey circles.\" \/><br \/>\nThere 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].<br \/>\nWhat about dividing fractions? Suppose we want to find the quotient: [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}[\/latex]. We need to figure out how many [latex]\\Large\\frac{1}{6}\\normalsize\\text{s}[\/latex] there are in [latex]\\Large\\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]\\Large\\frac{1}{6}[\/latex] tiles in [latex]\\Large\\frac{1}{2}[\/latex], so [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}\\normalsize=3[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220930\/CNX_BMath_Figure_04_02_016.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Model: [latex]\\Large\\frac{1}{4}\\normalsize\\div\\Large\\frac{1}{8}[\/latex]<\/p>\n<p>Solution:<br \/>\nWe want to determine how many [latex]\\Large\\frac{1}{8}\\normalsize\\text{s}[\/latex] are in [latex]\\Large\\frac{1}{4}[\/latex]. Start with one [latex]\\Large\\frac{1}{4}[\/latex] tile. Line up [latex]\\Large\\frac{1}{8}[\/latex] tiles underneath the [latex]\\Large\\frac{1}{4}[\/latex] tile.<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220931\/CNX_BMath_Figure_04_02_017_img.png\" alt=\"A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.\" \/><br \/>\nThere are two [latex]\\Large\\frac{1}{8}[\/latex]s in [latex]\\Large\\frac{1}{4}[\/latex].<br \/>\nSo, [latex]\\Large\\frac{1}{4}\\normalsize\\div\\Large\\frac{1}{8}\\normalsize=2[\/latex].<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Model: [latex]\\Large\\frac{1}{3}\\normalsize\\div\\Large\\frac{1}{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q218091\">Show Solution<\/span><\/p>\n<div id=\"q218091\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220932\/CNX_BMath_Figure_04_02_018_img.png\" alt=\"A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.\" \/><\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Model: [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q763601\">Show Solution<\/span><\/p>\n<div id=\"q763601\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220934\/CNX_BMath_Figure_04_02_019_img.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.\" \/><\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another way to model division of two fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Using a Fraction Wall to Find the Quotient of Two Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/pk-K5JF9iMo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Model: [latex]2\\div\\Large\\frac{1}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391699\">Show Solution<\/span><\/p>\n<div id=\"q391699\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe are trying to determine how many [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] there are in [latex]2[\/latex]. We can model this as shown.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220935\/CNX_BMath_Figure_04_02_020_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.\" \/><br \/>\nBecause there are eight [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] in [latex]2,2\\div\\Large\\frac{1}{4}\\normalsize=8[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Model: [latex]2\\div\\Large\\frac{1}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q73567\">Show Solution<\/span><\/p>\n<div id=\"q73567\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220936\/CNX_BMath_Figure_04_02_021_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.\" \/><\/p>\n<p>[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Model: [latex]3\\div\\Large\\frac{1}{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q354856\">Show Solution<\/span><\/p>\n<div id=\"q354856\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220938\/CNX_BMath_Figure_04_02_022_img.png\" alt=\"Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.\" \/><\/p>\n<p>[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm117216\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=117216&theme=oea&iframe_resize_id=ohm117216&show_question_numbers\" width=\"100%\" height=\"800\"><\/iframe><\/p>\n<\/div>\n<p>The next video shows more examples of how to divide a whole number by a fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Find the Quotient of a Whole Number and Fraction using Fraction Strips\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JKsfdK1WT1s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let\u2019s use money to model [latex]2\\div\\Large\\frac{1}{4}[\/latex] in another way. We often read [latex]\\Large\\frac{1}{4}[\/latex] as a \u2018quarter\u2019, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of [latex]2\\div\\Large\\frac{1}{4}[\/latex] as, &#8220;How many quarters are there in two dollars?&#8221; One dollar is [latex]4[\/latex] quarters, so [latex]2[\/latex] dollars would be [latex]8[\/latex] quarters. So again, [latex]2\\div\\Large\\frac{1}{4}\\normalsize=8[\/latex].<\/p>\n<p>The U.S. coin called a quarter is worth one-fourth of a dollar.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220939\/CNX_BMath_Figure_04_02_023.png\" alt=\"A picture of a United States quarter is shown.\" \/><br \/>\nUsing fraction tiles, we showed that [latex]\\Large\\frac{1}{2}\\normalsize\\div\\Large\\frac{1}{6}\\normalsize=3[\/latex]. Notice that [latex]\\Large\\frac{1}{2}\\cdot \\frac{6}{1}\\normalsize=3[\/latex] also. How are [latex]\\Large\\frac{1}{6}[\/latex] and [latex]\\Large\\frac{6}{1}[\/latex] related? They are reciprocals. This leads us to the procedure for fraction division.<\/p>\n<div class=\"textbox shaded\">\n<h3>Fraction Division<\/h3>\n<p>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<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\normalsize\\div\\Large\\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\n<p>To divide fractions, multiply the first fraction by the reciprocal of the second.<\/p>\n<p>We need to say [latex]b\\ne 0,c\\ne 0\\text{ and }d\\ne 0[\/latex] to be sure we don\u2019t divide by zero.<\/p>\n<p><span style=\"color: #3366ff\">Tip:\u00a0 Here&#8217;s a rhyme to help you with dividing fractions.\u00a0 When dividing fractions don&#8217;t ask why, just flip the second and multiply. <\/span><\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\Large\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261121\">Show Solution<\/span><\/p>\n<div id=\"q261121\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468274991\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\Large\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{2}{5}\\left(-\\frac{7}{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply. The product is negative.<\/td>\n<td>[latex]\\Large-\\frac{14}{15}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146066\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146066&theme=oea&iframe_resize_id=ohm146066&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146067\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146067&theme=oea&iframe_resize_id=ohm146067&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video for more examples of dividing fractions using a reciprocal.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2: Divide Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fnaRnEXlUvs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\Large\\frac{2}{3}\\normalsize\\div\\Large\\frac{n}{5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q853209\">Show Solution<\/span><\/p>\n<div id=\"q853209\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466192195\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{2}{3}\\normalsize\\div\\Large\\frac{n}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{2}{3}\\cdot \\frac{5}{n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{10}{3n}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146089\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146089&theme=oea&iframe_resize_id=ohm146089&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\Large-\\frac{3}{4}\\normalsize\\div\\Large\\left(-\\frac{7}{8}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q873547\">Show Solution<\/span><\/p>\n<div id=\"q873547\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467263034\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\Large-\\frac{3}{4}\\normalsize\\div\\Large\\left(-\\frac{7}{8}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large-\\frac{3}{4}\\cdot \\left(-\\frac{8}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply. Remember to determine the sign first.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot 8}{4\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite to show common factors.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot 4\\cdot 2}{4\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors and simplify.<\/td>\n<td>[latex]\\Large\\frac{6}{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146066\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146066&theme=oea&iframe_resize_id=ohm146066&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of dividing fractions that are negative.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\Large\\frac{7}{18}\\normalsize\\div\\Large\\frac{14}{27}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987562\">Show Solution<\/span><\/p>\n<div id=\"q987562\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466022407\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"No Alt Text\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{7}{18}\\normalsize\\div\\Large\\frac{14}{27}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{7}{18}\\cdot \\frac{27}{14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{7\\cdot 27}{18\\cdot 14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing common factors.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{7}\\cdot\\color{blue}{9}\\cdot3}{\\color{blue}{9}\\cdot2\\cdot\\color{red}{7}\\cdot2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large\\frac{3}{2\\cdot 2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146091\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146091&theme=oea&iframe_resize_id=ohm146091&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9489\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Ex: Using a Fraction Wall to Find the Quotient of Two Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pk-K5JF9iMo\">https:\/\/youtu.be\/pk-K5JF9iMo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Find the Quotient of a Whole Number and Fraction using Fraction Strips. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JKsfdK1WT1s\">https:\/\/youtu.be\/JKsfdK1WT1s<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: Divide Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/fnaRnEXlUvs\">https:\/\/youtu.be\/fnaRnEXlUvs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 146066, 146067, 146089, 146091. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Dividing Signed Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OPHdadhDJoI\">https:\/\/youtu.be\/OPHdadhDJoI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 117216, 117916, . <strong>Authored by<\/strong>: Amy Volpe. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Ex: Using a Fraction Wall to Find the Quotient of Two Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pk-K5JF9iMo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Find the Quotient of a Whole Number and Fraction using Fraction Strips\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/JKsfdK1WT1s\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex 2: Divide Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/fnaRnEXlUvs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Dividing Signed Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/OPHdadhDJoI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 117216, 117916, \",\"author\":\"Amy Volpe\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Question ID: 146066, 146067, 146089, 146091\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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