{"id":16062,"date":"2019-09-26T23:10:15","date_gmt":"2019-09-26T23:10:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-dividing-fractions-2\/"},"modified":"2020-09-03T10:47:56","modified_gmt":"2020-09-03T10:47:56","slug":"read-dividing-fractions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-dividing-fractions-2\/","title":{"raw":"2.2.d - Dividing Fractions","rendered":"2.2.d &#8211; Dividing Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide a fraction by a whole number<\/li>\r\n \t<li>Divide a fraction by a fraction<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Divide Fractions<\/h2>\r\nThere are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires [latex]3[\/latex] quarts of paint and you have\u00a0a bucket that contains [latex]6[\/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[\/latex] by [latex]3[\/latex] for an answer of [latex]2[\/latex]\u00a0coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\\dfrac{1}{2}[\/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide [latex]6[\/latex]\u00a0by the fraction, [latex]\\dfrac{1}{2}[\/latex].\r\n<h2>Divide a Fraction by a Whole Number<\/h2>\r\nWhen you divide by a whole number, you are also multiplying by the reciprocal.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-the-reciprocal-of-a-number\/\">Review how to find a reciprocal here<\/a>. In the painting example where you need [latex]3[\/latex] quarts of paint for a coat and have [latex]6[\/latex] quarts of paint, you can find the total number of coats that can be painted by dividing [latex]6[\/latex] by [latex]3[\/latex], [latex]6\\div3=2[\/latex]. You can also multiply [latex]6[\/latex] by the reciprocal of [latex]3[\/latex], which is [latex]\\dfrac{1}{3}[\/latex], so the multiplication problem becomes\r\n<p style=\"text-align: center\">[latex]\\dfrac{6}{1}\\cdot\\dfrac{1}{3}=\\dfrac{6}{3}=\\normalsize2[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Dividing is Multiplying by the Reciprocal<\/h3>\r\nFor all division, you can turn the operation\u00a0into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.\r\n\r\n<\/div>\r\nIf you have a recipe that needs to be divided in half, you can divide each ingredient by [latex]2[\/latex], or you can multiply each ingredient by [latex]\\dfrac{1}{2}[\/latex]\u00a0to find the new amount.\r\n\r\nIf you have [latex]\\dfrac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\dfrac{1}{5}[\/latex] of the available candy:\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{5}\\normalsize\\text{ of }\\dfrac{3}{4}=\\dfrac{1}{5}\\cdot\\dfrac{3}{4}=\\dfrac{3}{20}[\/latex]<\/p>\r\n<p style=\"text-align: center\">Each person gets [latex]\\dfrac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\r\nIf you have [latex]\\dfrac{3}{2}[\/latex] of a pizza left over, how can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182616\/image143.gif\" alt=\"Two pizzas divided into fourths. One pizza has all four pieces shaded, and the other pizza has two of the four slices shaded. 3\/2 divided by 6 is equal to 3\/2 times 1\/6. This is 3\/2 times 1\/6 equals 1\/4.\" width=\"360\" height=\"239\" \/>\r\n\r\nEach person gets one piece, so each person gets [latex]\\dfrac{1}{4}[\/latex] of a pizza.\r\n\r\nDividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind [latex]\\dfrac{2}{3}\\div \\normalsize 4[\/latex]\r\n\r\n[reveal-answer q=\"769187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769187\"]\r\n\r\nWrite your answer in lowest terms.\r\n\r\nDividing by [latex]4[\/latex] or [latex]\\dfrac{4}{1}[\/latex] is the same as multiplying by the reciprocal of [latex]4[\/latex], which is [latex]\\dfrac{1}{4}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}\\normalsize\\div 4=\\dfrac{2}{3}\\cdot\\dfrac{1}{4}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\dfrac{2\\cdot 1}{3\\cdot 4}=\\dfrac{2}{12}[\/latex]<\/p>\r\nSimplify to lowest terms by dividing numerator and denominator by the common factor [latex]2[\/latex].\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{6}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{2}{3}\\div4=\\dfrac{1}{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe same idea will work when the divisor (the number being divided) is a fraction.\r\n<h2>Divide a Whole Number by a Fraction<\/h2>\r\nLet\u2019s use money to model [latex]2\\div\\Large\\frac{1}{4}[\/latex]. 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,\u00a0[latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=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\n\r\nLet's look at another way to model [latex]2\\div\\Large\\frac{1}{4}[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide: [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<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\nWe can see that there are eight [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] in [latex]2[\/latex].\u00a0 Because dividing by a fraction is the same as multiplying by its reciprocal, [latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]2\\div\\dfrac{1}{4}=\\normalsize 8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDivide: [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&nbsp;\r\n\r\nDividing by [latex]\\dfrac{1}{3}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{3}[\/latex], which is [latex]\\dfrac{3}{1}[\/latex].\r\n<p style=\"text-align: center\">[latex]2\\div\\dfrac{1}{3}=\\dfrac{2}{1}\\cdot\\dfrac{3}{1}=6[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]2\\div\\dfrac{1}{3}=\\normalsize 6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nDivide: [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\nDividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].\r\n<p style=\"text-align: center\">[latex]3\\div\\dfrac{1}{2}=\\dfrac{3}{1}\\cdot\\dfrac{2}{1}=6[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]3\\div\\dfrac{1}{2}=\\normalsize 6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\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<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] 9\\div\\dfrac{1}{2}[\/latex]\r\n\r\n[reveal-answer q=\"269187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"269187\"]\r\n\r\nWrite your answer in lowest terms.\r\n\r\nDividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].\r\n<p style=\"text-align: center\">[latex]9\\div\\dfrac{1}{2}=\\dfrac{9}{1}\\cdot\\dfrac{2}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\dfrac{9\\cdot 2}{1\\cdot 1}=\\dfrac{18}{1}=\\normalsize 18[\/latex]<\/p>\r\nThis answer is already simplified to lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex]9\\div\\dfrac{1}{2}=\\normalsize 18[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Divide a Fraction by a Fraction<\/h2>\r\nSometimes you need to solve a problem that requires dividing a fraction by a fraction. 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\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.\u00a0 Suppose you have a pizza that is already cut into [latex]4[\/latex] slices. How many [latex]\\dfrac{1}{2}[\/latex] slices are there?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182618\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182619\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.\" width=\"180\" height=\"179\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex]\\dfrac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex].\r\n\r\nWhat would happen if you needed to divide each slice into thirds?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182621\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/>\r\n\r\nYou would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].\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<div class=\"textbox shaded\">\r\n<h3>Dividing with Fractions<\/h3>\r\n<ol>\r\n \t<li>Find the reciprocal of the divisor (the number that follows the division symbol).<\/li>\r\n \t<li>Multiply the dividend (the number before the division symbol) by the reciprocal of the divisor (the number after the division symbol).<\/li>\r\n<\/ol>\r\n<\/div>\r\nAny easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to <strong>KEEP<\/strong> the first number, <strong>CHANGE<\/strong> the division sign to multiplication, and then <strong>FLIP<\/strong> (use the reciprocal) of the second number.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide [latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]\r\n\r\n[reveal-answer q=\"569112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569112\"]\r\n\r\nMultiply by the reciprocal.\r\n\r\n<strong>KEEP<\/strong> [latex]\\dfrac{2}{3}[\/latex]\r\n\r\n<strong>CHANGE<\/strong>\u00a0 [latex] \\div [\/latex] to \u00a0[latex]\\cdot[\/latex]\r\n\r\n<strong>FLIP\u00a0<\/strong> [latex]\\dfrac{1}{6}[\/latex]\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide [latex]\\dfrac{3}{5}\\div\\dfrac{2}{3}[\/latex]\r\n\r\n[reveal-answer q=\"950676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950676\"]\r\n\r\nMultiply by the reciprocal.\u00a0Keep [latex]\\dfrac{3}{5}[\/latex], change [latex] \\div [\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\dfrac{2}{3}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{5}\\cdot\\dfrac{3}{2}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\dfrac{3\\cdot 3}{5\\cdot 2}=\\dfrac{9}{10}[\/latex]<\/p>\r\nThere are no common factors, so the fraction is simplified.\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{3}{5}\\div\\dfrac{2}{3}=\\dfrac{9}{10}[\/latex]\r\n\r\n[\/hidden-answer]\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\r\nWhen solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions before doing calculations\u00a0 [latex](\\text{i.e. } 5=\\dfrac{5}{1}[\/latex]\u00a0 and\u00a0 [latex]1\\dfrac{3}{4}=\\dfrac{7}{4})[\/latex].\u00a0 You can <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/converting-between-improper-fractions-and-mixed-numbers\/\">review how to convert mixed numbers to improper fractions here<\/a>. The final answer should always be simplified and written as a mixed number if larger than [latex]1[\/latex].\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>\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<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<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{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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide a fraction by a whole number<\/li>\n<li>Divide a fraction by a fraction<\/li>\n<\/ul>\n<\/div>\n<h2>Divide Fractions<\/h2>\n<p>There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires [latex]3[\/latex] quarts of paint and you have\u00a0a bucket that contains [latex]6[\/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[\/latex] by [latex]3[\/latex] for an answer of [latex]2[\/latex]\u00a0coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\\dfrac{1}{2}[\/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide [latex]6[\/latex]\u00a0by the fraction, [latex]\\dfrac{1}{2}[\/latex].<\/p>\n<h2>Divide a Fraction by a Whole Number<\/h2>\n<p>When you divide by a whole number, you are also multiplying by the reciprocal.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-the-reciprocal-of-a-number\/\">Review how to find a reciprocal here<\/a>. In the painting example where you need [latex]3[\/latex] quarts of paint for a coat and have [latex]6[\/latex] quarts of paint, you can find the total number of coats that can be painted by dividing [latex]6[\/latex] by [latex]3[\/latex], [latex]6\\div3=2[\/latex]. You can also multiply [latex]6[\/latex] by the reciprocal of [latex]3[\/latex], which is [latex]\\dfrac{1}{3}[\/latex], so the multiplication problem becomes<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{6}{1}\\cdot\\dfrac{1}{3}=\\dfrac{6}{3}=\\normalsize2[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Dividing is Multiplying by the Reciprocal<\/h3>\n<p>For all division, you can turn the operation\u00a0into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.<\/p>\n<\/div>\n<p>If you have a recipe that needs to be divided in half, you can divide each ingredient by [latex]2[\/latex], or you can multiply each ingredient by [latex]\\dfrac{1}{2}[\/latex]\u00a0to find the new amount.<\/p>\n<p>If you have [latex]\\dfrac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\dfrac{1}{5}[\/latex] of the available candy:<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{5}\\normalsize\\text{ of }\\dfrac{3}{4}=\\dfrac{1}{5}\\cdot\\dfrac{3}{4}=\\dfrac{3}{20}[\/latex]<\/p>\n<p style=\"text-align: center\">Each person gets [latex]\\dfrac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\n<p>If you have [latex]\\dfrac{3}{2}[\/latex] of a pizza left over, how can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182616\/image143.gif\" alt=\"Two pizzas divided into fourths. One pizza has all four pieces shaded, and the other pizza has two of the four slices shaded. 3\/2 divided by 6 is equal to 3\/2 times 1\/6. This is 3\/2 times 1\/6 equals 1\/4.\" width=\"360\" height=\"239\" \/><\/p>\n<p>Each person gets one piece, so each person gets [latex]\\dfrac{1}{4}[\/latex] of a pizza.<\/p>\n<p>Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find [latex]\\dfrac{2}{3}\\div \\normalsize 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769187\">Show Solution<\/span><\/p>\n<div id=\"q769187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]4[\/latex] or [latex]\\dfrac{4}{1}[\/latex] is the same as multiplying by the reciprocal of [latex]4[\/latex], which is [latex]\\dfrac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}\\normalsize\\div 4=\\dfrac{2}{3}\\cdot\\dfrac{1}{4}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2\\cdot 1}{3\\cdot 4}=\\dfrac{2}{12}[\/latex]<\/p>\n<p>Simplify to lowest terms by dividing numerator and denominator by the common factor [latex]2[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{2}{3}\\div4=\\dfrac{1}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The same idea will work when the divisor (the number being divided) is a fraction.<\/p>\n<h2>Divide a Whole Number by a Fraction<\/h2>\n<p>Let\u2019s use money to model [latex]2\\div\\Large\\frac{1}{4}[\/latex]. 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,\u00a0[latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=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.\" \/><\/p>\n<p>Let&#8217;s look at another way to model [latex]2\\div\\Large\\frac{1}{4}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide: [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.<br \/>\n<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 \/>\nWe can see that there are eight [latex]\\Large\\frac{1}{4}\\normalsize\\text{s}[\/latex] in [latex]2[\/latex].\u00a0 Because dividing by a fraction is the same as multiplying by its reciprocal, [latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2\\div\\dfrac{1}{4}=\\normalsize 8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Divide: [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>&nbsp;<\/p>\n<p>Dividing by [latex]\\dfrac{1}{3}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{3}[\/latex], which is [latex]\\dfrac{3}{1}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]2\\div\\dfrac{1}{3}=\\dfrac{2}{1}\\cdot\\dfrac{3}{1}=6[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2\\div\\dfrac{1}{3}=\\normalsize 6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Divide: [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>Dividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]3\\div\\dfrac{1}{2}=\\dfrac{3}{1}\\cdot\\dfrac{2}{1}=6[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3\\div\\dfrac{1}{2}=\\normalsize 6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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-1\" 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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]9\\div\\dfrac{1}{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q269187\">Show Solution<\/span><\/p>\n<div id=\"q269187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]9\\div\\dfrac{1}{2}=\\dfrac{9}{1}\\cdot\\dfrac{2}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{9\\cdot 2}{1\\cdot 1}=\\dfrac{18}{1}=\\normalsize 18[\/latex]<\/p>\n<p>This answer is already simplified to lowest terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]9\\div\\dfrac{1}{2}=\\normalsize 18[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Divide a Fraction by a Fraction<\/h2>\n<p>Sometimes you need to solve a problem that requires dividing a fraction by a fraction. 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-2\" 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<p>Using 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.\u00a0 Suppose you have a pizza that is already cut into [latex]4[\/latex] slices. How many [latex]\\dfrac{1}{2}[\/latex] slices are there?<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182618\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182619\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.\" width=\"180\" height=\"179\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex]\\dfrac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex].<\/p>\n<p>What would happen if you needed to divide each slice into thirds?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182621\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/><\/p>\n<p>You would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].<\/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<div class=\"textbox shaded\">\n<h3>Dividing with Fractions<\/h3>\n<ol>\n<li>Find the reciprocal of the divisor (the number that follows the division symbol).<\/li>\n<li>Multiply the dividend (the number before the division symbol) by the reciprocal of the divisor (the number after the division symbol).<\/li>\n<\/ol>\n<\/div>\n<p>Any easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to <strong>KEEP<\/strong> the first number, <strong>CHANGE<\/strong> the division sign to multiplication, and then <strong>FLIP<\/strong> (use the reciprocal) of the second number.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q569112\">Show Solution<\/span><\/p>\n<div id=\"q569112\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply by the reciprocal.<\/p>\n<p><strong>KEEP<\/strong> [latex]\\dfrac{2}{3}[\/latex]<\/p>\n<p><strong>CHANGE<\/strong>\u00a0 [latex]\\div[\/latex] to \u00a0[latex]\\cdot[\/latex]<\/p>\n<p><strong>FLIP\u00a0<\/strong> [latex]\\dfrac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\dfrac{3}{5}\\div\\dfrac{2}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950676\">Show Solution<\/span><\/p>\n<div id=\"q950676\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply by the reciprocal.\u00a0Keep [latex]\\dfrac{3}{5}[\/latex], change [latex]\\div[\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\dfrac{2}{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{3}{5}\\cdot\\dfrac{3}{2}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{3\\cdot 3}{5\\cdot 2}=\\dfrac{9}{10}[\/latex]<\/p>\n<p>There are no common factors, so the fraction is simplified.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{3}{5}\\div\\dfrac{2}{3}=\\dfrac{9}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\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<p>When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions before doing calculations\u00a0 [latex](\\text{i.e. } 5=\\dfrac{5}{1}[\/latex]\u00a0 and\u00a0 [latex]1\\dfrac{3}{4}=\\dfrac{7}{4})[\/latex].\u00a0 You can <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/converting-between-improper-fractions-and-mixed-numbers\/\">review how to convert mixed numbers to improper fractions here<\/a>. The final answer should always be simplified and written as a mixed number if larger than [latex]1[\/latex].<\/p>\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<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<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<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{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\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-16062\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Divide Fractions (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/F5YSNLel3n8\">https:\/\/youtu.be\/F5YSNLel3n8<\/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>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions\">http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions<\/a>. <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@7.3<\/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":169554,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Divide Fractions (Basic)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/F5YSNLel3n8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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