{"id":264,"date":"2016-06-01T20:49:53","date_gmt":"2016-06-01T20:49:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=264"},"modified":"2019-06-17T22:51:24","modified_gmt":"2019-06-17T22:51:24","slug":"read-dividing-fractions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-dividing-fractions-2\/","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>Find the reciprocal of a number<\/li>\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\r\nBefore we begin dividing fractions, let's cover some important terminology.\r\n<ul>\r\n \t<li><strong>reciprocal:<\/strong> two fractions are reciprocals if their product is [latex]1[\/latex]\u00a0(Don't worry; we will show you examples of what this means.)<\/li>\r\n \t<li><strong>quotient:<\/strong> the result\u00a0of division<\/li>\r\n<\/ul>\r\nDividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get [latex]1[\/latex]\u00a0as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Original number<\/th>\r\n<th>Reciprocal<\/th>\r\n<th>Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] 3=\\dfrac{3}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\dfrac{1}{3}=\\dfrac{7}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}=\\normalsize 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nYou can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex]. Make sure that if it's a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.\r\n<h2>Division by Zero<\/h2>\r\nYou know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction\r\n<p style=\"text-align: center;\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\r\nWe can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\r\nNow let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have \"what times [latex]0[\/latex] equals [latex]8[\/latex]?\"\r\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\r\nThis doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better defined, all division by zero is undefined.\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\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. 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\nThe same idea will work when the divisor (the thing being divided) is a fraction.\r\n\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\n&nbsp;\r\n\r\nFor example, dividing by [latex]6[\/latex] is the same as multiplying by the reciprocal of [latex]6[\/latex], which is [latex]\\dfrac{1}{6}[\/latex]. Look at the diagram of two pizzas below. \u00a0How 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]4[\/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\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 by a fraction. 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>Dividing with Fractions<\/h3>\r\n<ol>\r\n \t<li>Find the reciprocal of the number that follows the division symbol.<\/li>\r\n \t<li>Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one 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\n\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\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]. The final answer should always be simplified and written as a mixed number if larger than [latex]1[\/latex].\r\n\r\nIn the following video you will see an example of how to divide an integer by a fraction, as well as an example of how to divide a fraction by another fraction.\r\n\r\nhttps:\/\/youtu.be\/F5YSNLel3n8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the reciprocal of a number<\/li>\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<p>Before we begin dividing fractions, let&#8217;s cover some important terminology.<\/p>\n<ul>\n<li><strong>reciprocal:<\/strong> two fractions are reciprocals if their product is [latex]1[\/latex]\u00a0(Don&#8217;t worry; we will show you examples of what this means.)<\/li>\n<li><strong>quotient:<\/strong> the result\u00a0of division<\/li>\n<\/ul>\n<p>Dividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get [latex]1[\/latex]\u00a0as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:<\/p>\n<table>\n<thead>\n<tr>\n<th>Original number<\/th>\n<th>Reciprocal<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3=\\dfrac{3}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\dfrac{1}{3}=\\dfrac{7}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}=\\normalsize 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>You can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex]. Make sure that if it&#8217;s a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.<\/p>\n<h2>Division by Zero<\/h2>\n<p>You know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\n<p>We can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\n<p style=\"text-align: left;\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\n<p>Now let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have &#8220;what times [latex]0[\/latex] equals [latex]8[\/latex]?&#8221;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\n<p>This doesn&#8217;t make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better defined, all division by zero is undefined.<\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\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. 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>The same idea will work when the divisor (the thing being divided) is a fraction.<\/p>\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>&nbsp;<\/p>\n<p>For example, dividing by [latex]6[\/latex] is the same as multiplying by the reciprocal of [latex]6[\/latex], which is [latex]\\dfrac{1}{6}[\/latex]. Look at the diagram of two pizzas below. \u00a0How 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]4[\/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<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 by a fraction. 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>Dividing with Fractions<\/h3>\n<ol>\n<li>Find the reciprocal of the number that follows the division symbol.<\/li>\n<li>Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one 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<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>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]. The final answer should always be simplified and written as a mixed number if larger than [latex]1[\/latex].<\/p>\n<p>In the following video you will see an example of how to divide an integer by a fraction, as well as an example of how to divide a fraction by another fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Divide Fractions (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F5YSNLel3n8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-264\">\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":21,"menu_order":8,"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 http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@7.3\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"c5da9274-7eca-4240-9d28-6c1992d476e7","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-264","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions"}],"predecessor-version":[{"id":5674,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions\/5674"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/media?parent=264"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=264"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/contributor?post=264"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/license?post=264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}