{"id":4500,"date":"2020-04-13T15:18:46","date_gmt":"2020-04-13T15:18:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4500"},"modified":"2021-02-06T00:02:34","modified_gmt":"2021-02-06T00:02:34","slug":"operations-on-fractions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/operations-on-fractions\/","title":{"raw":"Operations on Fractions","rendered":"Operations on Fractions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add or Subtract fractions.<\/li>\r\n \t<li>Simplify fractions.<\/li>\r\n \t<li>Multiply fractions.<\/li>\r\n \t<li>Divide fractions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nMath students and working adults often find their knowledge of how to add, subtract, multiply, and divide fractions has grown rusty with disuse.\u00a0 We tend to rely on calculators to do a lot of the work of fractions for us. College Algebra, though, builds up some important techniques for handling expressions and equations that are based on operations on fractions. So it is important to refamiliarize yourself with these skills. This section will remind you how to do operations on fractions. As you work through the rest of the course, you can return this section as needed for a quick reminder of operations on fractions.\r\n\r\nBefore we get started, let's define some terminology.\r\n<ul>\r\n \t<li><strong>product:\u00a0<\/strong>the result of \u00a0multiplication<\/li>\r\n \t<li><strong>factor:<\/strong> something being multiplied - for \u00a0[latex]3 \\cdot 2 = 6[\/latex] , both [latex]3[\/latex] and [latex]2[\/latex] are factors of [latex]6[\/latex]<\/li>\r\n \t<li><strong>numerator:<\/strong> the top part of a fraction - the numerator in the fraction\u00a0[latex]\\Large\\frac{2}{3}[\/latex] is [latex]2[\/latex]<\/li>\r\n \t<li><strong>denominator:<\/strong> the bottom part of a fraction - the denominator in the fraction\u00a0[latex]\\Large\\frac{2}{3}[\/latex] is [latex]3[\/latex]<\/li>\r\n<\/ul>\r\n<h2>A Note About Instructions<\/h2>\r\nCertain words are used in math textbooks and by teachers to provide students with instructions on what to do with a given problem. For example, you may see instructions such as <em>find<\/em> or <em>simplify.<\/em>\u00a0It is important to understand what these words mean so you can successfully work through the problems in this course. Here is a short list of some problem instructions along with their descriptions as they will be used in this module.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Instruction<\/th>\r\n<th>Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Find<\/td>\r\n<td>Perform the indicated mathematical operations which may include addition, subtraction, multiplication, division (later, use of the word\u00a0<em>find<\/em> will be expanded to solving equations, as in\u00a0<em>find the value of variable).<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Simplify<\/td>\r\n<td>1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division\r\n\r\n2) Write a mathematical statement in smallest terms so there are no other mathematical operations that can be performed\u2014often found in problems related to fractions and the order of operations<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Evaluate<\/td>\r\n<td>1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division\r\n\r\n2) Substitute a given value for a variable in an expression and then perform the indicated mathematical operations<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reduce<\/td>\r\n<td>Write a mathematical statement in smallest or lowest terms so there are no other mathematical operations that can be performed\u2014often found in problems related to fractions or division<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Add Fractions<\/h2>\r\nWhen you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.\r\n\r\nThe \u201cparts of a whole\u201d concept can be modeled with pizzas and pizza slices. For example, imagine a pizza is cut into [latex]4[\/latex] pieces, and someone takes [latex]1[\/latex] piece. Now, [latex]\\Large\\frac{1}{4}[\/latex]\u00a0of the pizza is gone and [latex]\\Large\\frac{3}{4}[\/latex] remains. Note that both of these fractions have a denominator of [latex]4[\/latex], which refers to the number of slices the whole pizza has been cut into. What if you have another pizza that had been cut into [latex]8[\/latex] equal parts and [latex]3[\/latex] of those parts were gone, leaving [latex]\\Large\\frac{5}{8}[\/latex]?\r\n\r\n<img class=\"aligncenter wp-image-2861 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182610\/Screen-Shot-2016-04-19-at-8.59.40-AM.png\" alt=\"A pizza divided into four slices, with one slice missing.\" width=\"240\" height=\"207\" \/>\r\n\r\nHow can you describe the total amount of pizza that is left with one number rather than two different fractions? You need a common denominator,\u00a0technically\u00a0called\u00a0the <strong>least common multiple. <\/strong>Remember that\u00a0if a number is a multiple of another, you can divide them and have no remainder.\r\n\r\nOne way to find the least common multiple of two or more numbers is to first multiply each\u00a0by [latex]1, 2, 3, 4[\/latex], etc. \u00a0For example, find the least common multiple of [latex]2[\/latex] and [latex]5[\/latex].\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td scope=\"col\">First, list all the multiples of [latex]2[\/latex]:<\/td>\r\n<td scope=\"col\">Then list all the multiples of 5:<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 1 = 2[\/latex]<\/td>\r\n<td>[latex]5\\cdot 1 = 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 2 = 4[\/latex]<\/td>\r\n<td>[latex]5\\cdot 2 = 10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 3 = 6[\/latex]<\/td>\r\n<td>[latex]5\\cdot 3 = 15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 4 = 8[\/latex]<\/td>\r\n<td>[latex]5\\cdot 4 = 20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 5 = 10[\/latex]<\/td>\r\n<td>[latex]5\\cdot 5 = 25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe smallest multiple they have in common will be the common denominator to use to rewrite each fraction into equivalent fractions. See the example below for a demonstration of our pizza problem.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nOne pizza, cut into four slices, has one missing. Another pizza of the same size has been cut into eight pieces, of which three have been removed. Describe the total amount of pizza left in the two pizzas using common terms.\r\n\r\n[reveal-answer q=\"155500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155500\"]Rewrite the fractions [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{5}{8}[\/latex]\u00a0as fractions with a least common denominator.\r\n\r\nFind the least common multiple of the denominators. This is the least common denominator.\r\n\r\nMultiples of [latex]4: 4, \\textbf{8},12,16, \\textbf{24}[\/latex]\r\n\r\nMultiples of [latex]8: \\textbf{8},16, \\textbf{24}[\/latex]\r\n\r\nThe least common denominator is [latex]8[\/latex]\u2014the smallest multiple they have in common.\r\n\r\nRewrite [latex]\\dfrac{3}{4}[\/latex] with a denominator of [latex]8[\/latex]. You have to multiply both the top and bottom by [latex]2[\/latex] so you don't change the relationship between them.\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}\\cdot\\dfrac{2}{2}=\\dfrac{6}{8}[\/latex]<\/p>\r\nWe don't need to rewrite [latex]\\dfrac{5}{8}[\/latex] since it already has the common denominator.\r\n<h4>Answer<\/h4>\r\nBoth [latex]\\dfrac{6}{8}[\/latex]\u00a0and\u00a0[latex]\\dfrac{5}{8}[\/latex] have the same denominator, and you can describe how much pizza is left\u00a0with common terms. Add the numerators and put them over the common denominator.\r\n\r\nWe have [latex]\\dfrac{6}{8}[\/latex] of the first pizza and\u00a0[latex]\\dfrac{5}{8}[\/latex] of the second pizza left. That's\u00a0[latex]\\dfrac{11}{8}[\/latex] of an identically sized pizza, or [latex]1[\/latex] and\u00a0[latex]\\dfrac{3}{8}[\/latex], pizza still on the table.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo add fractions with unlike denominators, first rewrite them with like denominators. Then add or subtract the numerators over the common denominator.\r\n<div class=\"textbox shaded\">\r\n<h3>Adding Fractions with Unlike Denominators<\/h3>\r\n<ol>\r\n \t<li>Find a common denominator.<\/li>\r\n \t<li>Rewrite each fraction as an equivalent fraction using the common denominator.<\/li>\r\n \t<li>Now that the fractions have a common denominator, you can add the numerators.<\/li>\r\n \t<li>Simplify by canceling out all common factors in the numerator and denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Simplify a Fraction<\/h2>\r\nIt is a common convention in mathematics to present fractions in lowest terms. We call this practice\u00a0<em>simplifying or\u00a0<\/em><em>reducing <\/em>the fraction, and it can be accomplished by canceling (dividing) the common factors in a fraction's numerator and denominator. \u00a0We can do this because a fraction represents division, and for any number [latex]a[\/latex], [latex]\\dfrac{a}{a}=1[\/latex].\r\n\r\nFor example, to simplify [latex]\\dfrac{6}{9}[\/latex] you can rewrite [latex]6[\/latex]and [latex]9[\/latex]\u00a0using the smallest factors possible as follows:\r\n<p style=\"text-align: center\">[latex]\\dfrac{6}{9}=\\dfrac{2\\cdot3}{3\\cdot3}[\/latex]<\/p>\r\nSince there is a [latex]3[\/latex] in both the numerator and denominator, and fractions can be considered division, we can divide the [latex]3[\/latex] in the top by the [latex]3[\/latex] in the bottom to reduce to [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\dfrac{6}{9}=\\dfrac{2\\cdot\\cancel{3}}{3\\cdot\\cancel{3}}=\\dfrac{2\\cdot1}{3}=\\dfrac{2}{3}[\/latex]<\/p>\r\nIn the next example you are shown how to add two\u00a0fractions with different denominators, then simplify the answer.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd [latex]\\Large\\frac{2}{3}+\\Large\\frac{1}{5}[\/latex].\u00a0Simplify the answer.\r\n\r\n[reveal-answer q=\"797488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797488\"]Since the denominators are not alike, find a common denominator by multiplying the denominators.\r\n<p style=\"text-align: center\">[latex]3\\cdot5=15[\/latex]<\/p>\r\nRewrite each fraction with a denominator of [latex]15[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2}{3}\\cdot\\Large\\frac{5}{5}=\\Large\\frac{10}{15}\\\\\\\\\\Large\\frac{1}{5}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{3}{15}\\end{array}[\/latex]<\/p>\r\nAdd the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{10}{15}+\\Large\\frac{3}{15}=\\Large\\frac{13}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{2}{3}+\\Large\\frac{1}{5}=\\Large\\frac{13}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd\u00a0[latex]\\Large\\frac{3}{7}+\\Large\\frac{2}{21}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"520906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"520906\"]Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21.\r\n\r\nMultiples of [latex]7: 7, 14, \\textbf{21}[\/latex]\r\n\r\nMultiples of [latex]21:\\textbf{21}[\/latex]\r\n\r\nRewrite each fraction with a denominator of [latex]21[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{3}{7}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{9}{21}\\\\\\\\\\Large\\frac{2}{21}\\end{array}[\/latex]<\/p>\r\nAdd the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{9}{21}+\\Large\\frac{2}{21}=\\Large\\frac{11}{21}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{3}{7}+\\Large\\frac{2}{21}=\\Large\\frac{11}{21}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see an example of how to add two fractions with different denominators.\r\n\r\nhttps:\/\/youtu.be\/zV4q7j1-89I\r\n\r\nYou can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nAdd [latex]\\Large\\frac{3}{4}+\\Large\\frac{1}{6}+\\Large\\frac{5}{8}[\/latex].\u00a0 Simplify the answer and write as a mixed number.\r\n\r\nWhat makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"680977\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680977\"]Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of [latex]4, 6[\/latex], and [latex]8[\/latex].\r\n<p style=\"text-align: center\">[latex]4=2\\cdot2\\\\6=3\\cdot2\\\\8=2\\cdot2\\cdot2\\\\\\text{LCM}:\\,\\,2\\cdot2\\cdot2\\cdot3=24[\/latex]<\/p>\r\nRewrite each fraction with a denominator of [latex]24[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{3}{4}\\cdot\\Large\\frac{6}{6}=\\Large\\frac{18}{24}\\\\\\\\\\Large\\frac{1}{6}\\cdot\\Large\\frac{4}{4}=\\Large\\frac{4}{24}\\\\\\\\\\Large\\frac{5}{8}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{15}{24}\\end{array}[\/latex]<\/p>\r\nAdd the fractions by adding the numerators and keeping the denominator the same.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{18}{24}+\\Large\\frac{4}{24}+\\Large\\frac{15}{24}=\\Large\\frac{37}{24}[\/latex]<\/p>\r\nWrite the improper fraction as a mixed number and simplify the fraction.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{37}{24}=\\normalsize 1\\,\\,\\Large\\frac{13}{24}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{3}{4}+\\Large\\frac{1}{6}+\\Large\\frac{5}{8}=\\normalsize 1\\Large\\frac{13}{24}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Subtract Fractions<\/h2>\r\nSubtracting fractions follows the same technique as adding them. First, determine whether or not the denominators are alike. If not, rewrite each fraction as an equivalent fraction, all having the same denominator.\u00a0 Below are some examples of subtracting fractions whose denominators are not alike.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract\u00a0[latex]\\Large\\frac{1}{5}-\\Large\\frac{1}{6}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"155692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155692\"]The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.\r\n<p style=\"text-align: center\">[latex]5\\cdot6=30[\/latex]<\/p>\r\nRewrite each fraction as an equivalent fraction with a denominator of [latex]30[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{1}{5}\\cdot\\Large\\frac{6}{6}=\\Large\\frac{6}{30}\\\\\\\\\\Large\\frac{1}{6}\\cdot\\Large\\frac{5}{5}=\\Large\\frac{5}{30}\\end{array}[\/latex]<\/p>\r\nSubtract the numerators. Simplify the answer if needed.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{6}{30}-\\Large\\frac{5}{30}=\\Large\\frac{1}{30}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{1}{5}-\\Large\\frac{1}{6}=\\Large\\frac{1}{30}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe example below shows how to use\u00a0multiples to find the least common multiple, which will be the least common denominator.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract [latex]\\Large\\frac{5}{6}-\\Large\\frac{1}{4}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"984596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"984596\"]Find the least common multiple of the denominators\u2014this is the least common denominator.\r\n\r\nMultiples of \u00a0[latex]6: 6, \\textbf{12}, 18, 24[\/latex]\r\n\r\nMultiples of \u00a0[latex]4: 4, 8, \\textbf{12},16, 20[\/latex]\r\n\r\n[latex]12[\/latex] is the least common multiple of [latex]6[\/latex] and [latex]4[\/latex].\r\n\r\nRewrite each fraction with a denominator of [latex]12[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{5}{6}\\cdot\\Large\\frac{2}{2}=\\Large\\frac{10}{12}\\\\\\\\\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{3}{12}\\end{array}[\/latex]<\/p>\r\nSubtract the fractions. Simplify the answer if needed.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{10}{12}-\\Large\\frac{3}{12}=\\Large\\frac{7}{12}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{5}{6}-\\Large\\frac{1}{4}=\\Large\\frac{7}{12}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see an example of how to subtract fractions with unlike denominators.\r\n\r\nhttps:\/\/youtu.be\/RpHtOMjeI7g\r\n\r\n<!---MULTIPLY FRACTIONS-->\r\n<h2>Multiply Fractions<\/h2>\r\nJust as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. \u00a0\u00a0There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.\r\n\r\nWhen you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\Large\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\Large\\frac{1}{2}[\/latex]\u00a0of the [latex]\\Large\\frac{3}{4}[\/latex]:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182611\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/>\r\n\r\nBy dividing each fourth in half, you can divide the candy bar into eighths.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182612\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/>\r\n\r\nThen, choose half of those to get [latex]\\Large\\frac{3}{8}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182613\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/>\r\n\r\nIn both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two Fractions<\/h3>\r\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{a\\cdot c}{b\\cdot d}=\\Large\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Fractions<\/h3>\r\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}\\cdot\\Large\\frac{e}{f}=\\Large\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{4}{5}[\/latex]\r\n\r\n[reveal-answer q=\"368042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368042\"]Multiply the numerators and multiply the denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\r\nSimplify, if possible. This fraction is already in lowest terms.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{8}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{8}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo review: if a fraction has\u00a0common factors in the numerator and denominator, we can <strong>reduce<\/strong> the fraction to its simplified form by removing the common factors.\r\n\r\nFor example,\r\n<ul>\r\n \t<li>Given [latex]\\Large\\frac{8}{15}[\/latex], the factors of [latex]8[\/latex] are: [latex]1, 2, 4, 8[\/latex] and the factors of [latex]15[\/latex] are: [latex]1, 3, 5, 15[\/latex]. \u00a0[latex]\\Large\\frac{8}{15}[\/latex] is simplified because there are no common factors of [latex]8[\/latex] and [latex]15[\/latex].<\/li>\r\n \t<li>Given [latex]\\Large\\frac{10}{15}[\/latex], the factors of [latex]10[\/latex] are: [latex]1, 2, 5, 10[\/latex] and the factors of [latex]15[\/latex] are: [latex]1, 3, 5, 15[\/latex]. [latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex]\u00a0is a common factor of [latex]10[\/latex]\u00a0and [latex]15[\/latex].<\/li>\r\n<\/ul>\r\nYou can\u00a0simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.\r\n\r\nIn the following video you will see an example of how to multiply two fractions, then simplify the answer.\r\n\r\nhttps:\/\/youtu.be\/f_L-EFC8Z7c\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nMultiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex]. Simplify the answer.\r\n\r\nWhat makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"385641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385641\"]Multiply the numerators and multiply the denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\r\nSimplify first by canceling (dividing) the\u00a0common factors of [latex]3[\/latex] and [latex]2[\/latex]. [latex]3[\/latex] divided by [latex]3[\/latex] is [latex]1[\/latex], and [latex]2[\/latex] divided by \u00a0[latex]2[\/latex] is [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\Large\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\Large\\frac{1}{10}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex] = [latex]\\Large\\frac{1}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<!---DIVIDE FRACTIONS-->\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]\\Large\\frac{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]2[\/latex]\u00a0by the fraction, [latex]\\Large\\frac{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]\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{4}{3}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}\\cdot\\Large\\frac{4}{3}=\\Large\\frac{3\\cdot 4}{4\\cdot 3}=\\Large\\frac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\Large\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{2}{1}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{1}{2}\\cdot\\Large\\frac{2}{1}=\\Large\\frac{1\\cdot2}{2\\cdot1}=\\Large\\frac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] 3=\\Large\\frac{3}{1}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{1}{3}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{3}{1}\\cdot\\Large\\frac{1}{3}=\\Large\\frac{3\\cdot 1}{1\\cdot 3}=\\Large\\frac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\Large\\frac{1}{3}=\\Large\\frac{7}{3}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{3}{7}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{7}{3}\\cdot\\Large\\frac{3}{7}=\\Large\\frac{7\\cdot3}{3\\cdot7}=\\Large\\frac{21}{21}=\\normalsize 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex]\\Large\\frac{2}{5}[\/latex] to get the reciprocal [latex]\\Large\\frac{5}{2}[\/latex].\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? Can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{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.\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]\\Large\\frac{0}{8}[\/latex] which would be [latex]\\Large\\frac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have\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. The reciprocal of\u00a0[latex]\\Large\\frac{8}{0}[\/latex] is undefined, and in fact, 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]\\Large\\frac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\Large\\frac{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 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]\\Large\\frac{1}{3}[\/latex], so the multiplication problem becomes\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{6}{1}\\cdot\\Large\\frac{1}{3}=\\Large\\frac{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. If you have [latex]\\Large\\frac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\Large\\frac{1}{5}[\/latex] of the available candy:\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{1}{5}\\normalsize\\text{ of }\\Large\\frac{3}{4}=\\Large\\frac{1}{5}\\cdot\\Large\\frac{3}{4}=\\Large\\frac{3}{20}[\/latex]<\/p>\r\n<p style=\"text-align: center\">Each person gets [latex]\\Large\\frac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\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]\\Large\\frac{1}{2}[\/latex]\u00a0to find the new amount.\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]\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{2}{3}\\div \\normalsize 4[\/latex]\r\n\r\n[reveal-answer q=\"769187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769187\"]Write your answer in lowest terms.\r\n\r\nDividing by [latex]4[\/latex] or [latex]\\Large\\frac{4}{1}[\/latex] is the same as multiplying by the reciprocal of [latex]4[\/latex], which is [latex]\\Large\\frac{1}{4}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\normalsize\\div 4=\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 1}{3\\cdot 4}=\\Large\\frac{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]\\Large\\frac{1}{6}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{2}{3}\\normalsize\\div4=\\Large\\frac{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\\Large\\frac{1}{2}[\/latex]\r\n\r\n[reveal-answer q=\"269187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"269187\"]Write your answer in lowest terms.\r\n\r\nDividing by [latex]\\Large\\frac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\Large\\frac{1}{2}[\/latex], which is [latex]\\Large\\frac{2}{1}[\/latex].\r\n<p style=\"text-align: center\">[latex]9\\div\\Large\\frac{1}{2}=\\Large\\frac{9}{1}\\cdot\\Large\\frac{2}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{9\\cdot 2}{1\\cdot 1}=\\Large\\frac{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\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{2}{3}\\div\\Large\\frac{1}{6}[\/latex]\r\n\r\n[reveal-answer q=\"569112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569112\"]Multiply by the reciprocal.\r\n\r\n<strong>KEEP<\/strong> [latex]\\Large\\frac{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]\\Large\\frac{1}{6}[\/latex]\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{6}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot6}{3\\cdot1}=\\Large\\frac{12}{3}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{12}{3}=\\normalsize 4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{2}{3}\\div\\Large \\frac{1}{6}=\\normalsize 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]\\Large\\frac{3}{5}\\div\\Large\\frac{2}{3}[\/latex]\r\n\r\n[reveal-answer q=\"950676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950676\"]Multiply by the reciprocal.\u00a0Keep [latex]\\Large\\frac{3}{5}[\/latex], change [latex] \\div [\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\Large\\frac{2}{3}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{3}{5}\\cdot\\Large\\frac{3}{2}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{3\\cdot 3}{5\\cdot 2}=\\Large\\frac{9}{10}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{3}{5}\\div\\Large\\frac{2}{3}=\\Large\\frac{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. The final answer should be simplified and written as a mixed number.\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=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add or Subtract fractions.<\/li>\n<li>Simplify fractions.<\/li>\n<li>Multiply fractions.<\/li>\n<li>Divide fractions.<\/li>\n<\/ul>\n<\/div>\n<p>Math students and working adults often find their knowledge of how to add, subtract, multiply, and divide fractions has grown rusty with disuse.\u00a0 We tend to rely on calculators to do a lot of the work of fractions for us. College Algebra, though, builds up some important techniques for handling expressions and equations that are based on operations on fractions. So it is important to refamiliarize yourself with these skills. This section will remind you how to do operations on fractions. As you work through the rest of the course, you can return this section as needed for a quick reminder of operations on fractions.<\/p>\n<p>Before we get started, let&#8217;s define some terminology.<\/p>\n<ul>\n<li><strong>product:\u00a0<\/strong>the result of \u00a0multiplication<\/li>\n<li><strong>factor:<\/strong> something being multiplied &#8211; for \u00a0[latex]3 \\cdot 2 = 6[\/latex] , both [latex]3[\/latex] and [latex]2[\/latex] are factors of [latex]6[\/latex]<\/li>\n<li><strong>numerator:<\/strong> the top part of a fraction &#8211; the numerator in the fraction\u00a0[latex]\\Large\\frac{2}{3}[\/latex] is [latex]2[\/latex]<\/li>\n<li><strong>denominator:<\/strong> the bottom part of a fraction &#8211; the denominator in the fraction\u00a0[latex]\\Large\\frac{2}{3}[\/latex] is [latex]3[\/latex]<\/li>\n<\/ul>\n<h2>A Note About Instructions<\/h2>\n<p>Certain words are used in math textbooks and by teachers to provide students with instructions on what to do with a given problem. For example, you may see instructions such as <em>find<\/em> or <em>simplify.<\/em>\u00a0It is important to understand what these words mean so you can successfully work through the problems in this course. Here is a short list of some problem instructions along with their descriptions as they will be used in this module.<\/p>\n<table>\n<thead>\n<tr>\n<th>Instruction<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Find<\/td>\n<td>Perform the indicated mathematical operations which may include addition, subtraction, multiplication, division (later, use of the word\u00a0<em>find<\/em> will be expanded to solving equations, as in\u00a0<em>find the value of variable).<\/em><\/td>\n<\/tr>\n<tr>\n<td>\u00a0Simplify<\/td>\n<td>1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division<\/p>\n<p>2) Write a mathematical statement in smallest terms so there are no other mathematical operations that can be performed\u2014often found in problems related to fractions and the order of operations<\/td>\n<\/tr>\n<tr>\n<td>Evaluate<\/td>\n<td>1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division<\/p>\n<p>2) Substitute a given value for a variable in an expression and then perform the indicated mathematical operations<\/td>\n<\/tr>\n<tr>\n<td>Reduce<\/td>\n<td>Write a mathematical statement in smallest or lowest terms so there are no other mathematical operations that can be performed\u2014often found in problems related to fractions or division<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Add Fractions<\/h2>\n<p>When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.<\/p>\n<p>The \u201cparts of a whole\u201d concept can be modeled with pizzas and pizza slices. For example, imagine a pizza is cut into [latex]4[\/latex] pieces, and someone takes [latex]1[\/latex] piece. Now, [latex]\\Large\\frac{1}{4}[\/latex]\u00a0of the pizza is gone and [latex]\\Large\\frac{3}{4}[\/latex] remains. Note that both of these fractions have a denominator of [latex]4[\/latex], which refers to the number of slices the whole pizza has been cut into. What if you have another pizza that had been cut into [latex]8[\/latex] equal parts and [latex]3[\/latex] of those parts were gone, leaving [latex]\\Large\\frac{5}{8}[\/latex]?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2861 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182610\/Screen-Shot-2016-04-19-at-8.59.40-AM.png\" alt=\"A pizza divided into four slices, with one slice missing.\" width=\"240\" height=\"207\" \/><\/p>\n<p>How can you describe the total amount of pizza that is left with one number rather than two different fractions? You need a common denominator,\u00a0technically\u00a0called\u00a0the <strong>least common multiple. <\/strong>Remember that\u00a0if a number is a multiple of another, you can divide them and have no remainder.<\/p>\n<p>One way to find the least common multiple of two or more numbers is to first multiply each\u00a0by [latex]1, 2, 3, 4[\/latex], etc. \u00a0For example, find the least common multiple of [latex]2[\/latex] and [latex]5[\/latex].<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td scope=\"col\">First, list all the multiples of [latex]2[\/latex]:<\/td>\n<td scope=\"col\">Then list all the multiples of 5:<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 1 = 2[\/latex]<\/td>\n<td>[latex]5\\cdot 1 = 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 2 = 4[\/latex]<\/td>\n<td>[latex]5\\cdot 2 = 10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 3 = 6[\/latex]<\/td>\n<td>[latex]5\\cdot 3 = 15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 4 = 8[\/latex]<\/td>\n<td>[latex]5\\cdot 4 = 20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 5 = 10[\/latex]<\/td>\n<td>[latex]5\\cdot 5 = 25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The smallest multiple they have in common will be the common denominator to use to rewrite each fraction into equivalent fractions. See the example below for a demonstration of our pizza problem.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>One pizza, cut into four slices, has one missing. Another pizza of the same size has been cut into eight pieces, of which three have been removed. Describe the total amount of pizza left in the two pizzas using common terms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155500\">Show Solution<\/span><\/p>\n<div id=\"q155500\" class=\"hidden-answer\" style=\"display: none\">Rewrite the fractions [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{5}{8}[\/latex]\u00a0as fractions with a least common denominator.<\/p>\n<p>Find the least common multiple of the denominators. This is the least common denominator.<\/p>\n<p>Multiples of [latex]4: 4, \\textbf{8},12,16, \\textbf{24}[\/latex]<\/p>\n<p>Multiples of [latex]8: \\textbf{8},16, \\textbf{24}[\/latex]<\/p>\n<p>The least common denominator is [latex]8[\/latex]\u2014the smallest multiple they have in common.<\/p>\n<p>Rewrite [latex]\\dfrac{3}{4}[\/latex] with a denominator of [latex]8[\/latex]. You have to multiply both the top and bottom by [latex]2[\/latex] so you don&#8217;t change the relationship between them.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}\\cdot\\dfrac{2}{2}=\\dfrac{6}{8}[\/latex]<\/p>\n<p>We don&#8217;t need to rewrite [latex]\\dfrac{5}{8}[\/latex] since it already has the common denominator.<\/p>\n<h4>Answer<\/h4>\n<p>Both [latex]\\dfrac{6}{8}[\/latex]\u00a0and\u00a0[latex]\\dfrac{5}{8}[\/latex] have the same denominator, and you can describe how much pizza is left\u00a0with common terms. Add the numerators and put them over the common denominator.<\/p>\n<p>We have [latex]\\dfrac{6}{8}[\/latex] of the first pizza and\u00a0[latex]\\dfrac{5}{8}[\/latex] of the second pizza left. That&#8217;s\u00a0[latex]\\dfrac{11}{8}[\/latex] of an identically sized pizza, or [latex]1[\/latex] and\u00a0[latex]\\dfrac{3}{8}[\/latex], pizza still on the table.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To add fractions with unlike denominators, first rewrite them with like denominators. Then add or subtract the numerators over the common denominator.<\/p>\n<div class=\"textbox shaded\">\n<h3>Adding Fractions with Unlike Denominators<\/h3>\n<ol>\n<li>Find a common denominator.<\/li>\n<li>Rewrite each fraction as an equivalent fraction using the common denominator.<\/li>\n<li>Now that the fractions have a common denominator, you can add the numerators.<\/li>\n<li>Simplify by canceling out all common factors in the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<h2>Simplify a Fraction<\/h2>\n<p>It is a common convention in mathematics to present fractions in lowest terms. We call this practice\u00a0<em>simplifying or\u00a0<\/em><em>reducing <\/em>the fraction, and it can be accomplished by canceling (dividing) the common factors in a fraction&#8217;s numerator and denominator. \u00a0We can do this because a fraction represents division, and for any number [latex]a[\/latex], [latex]\\dfrac{a}{a}=1[\/latex].<\/p>\n<p>For example, to simplify [latex]\\dfrac{6}{9}[\/latex] you can rewrite [latex]6[\/latex]and [latex]9[\/latex]\u00a0using the smallest factors possible as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{6}{9}=\\dfrac{2\\cdot3}{3\\cdot3}[\/latex]<\/p>\n<p>Since there is a [latex]3[\/latex] in both the numerator and denominator, and fractions can be considered division, we can divide the [latex]3[\/latex] in the top by the [latex]3[\/latex] in the bottom to reduce to [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{6}{9}=\\dfrac{2\\cdot\\cancel{3}}{3\\cdot\\cancel{3}}=\\dfrac{2\\cdot1}{3}=\\dfrac{2}{3}[\/latex]<\/p>\n<p>In the next example you are shown how to add two\u00a0fractions with different denominators, then simplify the answer.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add [latex]\\Large\\frac{2}{3}+\\Large\\frac{1}{5}[\/latex].\u00a0Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797488\">Show Solution<\/span><\/p>\n<div id=\"q797488\" class=\"hidden-answer\" style=\"display: none\">Since the denominators are not alike, find a common denominator by multiplying the denominators.<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot5=15[\/latex]<\/p>\n<p>Rewrite each fraction with a denominator of [latex]15[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2}{3}\\cdot\\Large\\frac{5}{5}=\\Large\\frac{10}{15}\\\\\\\\\\Large\\frac{1}{5}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{3}{15}\\end{array}[\/latex]<\/p>\n<p>Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{10}{15}+\\Large\\frac{3}{15}=\\Large\\frac{13}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{2}{3}+\\Large\\frac{1}{5}=\\Large\\frac{13}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add\u00a0[latex]\\Large\\frac{3}{7}+\\Large\\frac{2}{21}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520906\">Show Solution<\/span><\/p>\n<div id=\"q520906\" class=\"hidden-answer\" style=\"display: none\">Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21.<\/p>\n<p>Multiples of [latex]7: 7, 14, \\textbf{21}[\/latex]<\/p>\n<p>Multiples of [latex]21:\\textbf{21}[\/latex]<\/p>\n<p>Rewrite each fraction with a denominator of [latex]21[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{3}{7}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{9}{21}\\\\\\\\\\Large\\frac{2}{21}\\end{array}[\/latex]<\/p>\n<p>Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{9}{21}+\\Large\\frac{2}{21}=\\Large\\frac{11}{21}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{3}{7}+\\Large\\frac{2}{21}=\\Large\\frac{11}{21}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see an example of how to add two fractions with different denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Add Fractions with Unlike Denominators (Basic with Model)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zV4q7j1-89I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Add [latex]\\Large\\frac{3}{4}+\\Large\\frac{1}{6}+\\Large\\frac{5}{8}[\/latex].\u00a0 Simplify the answer and write as a mixed number.<\/p>\n<p>What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680977\">Show Solution<\/span><\/p>\n<div id=\"q680977\" class=\"hidden-answer\" style=\"display: none\">Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of [latex]4, 6[\/latex], and [latex]8[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]4=2\\cdot2\\\\6=3\\cdot2\\\\8=2\\cdot2\\cdot2\\\\\\text{LCM}:\\,\\,2\\cdot2\\cdot2\\cdot3=24[\/latex]<\/p>\n<p>Rewrite each fraction with a denominator of [latex]24[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{3}{4}\\cdot\\Large\\frac{6}{6}=\\Large\\frac{18}{24}\\\\\\\\\\Large\\frac{1}{6}\\cdot\\Large\\frac{4}{4}=\\Large\\frac{4}{24}\\\\\\\\\\Large\\frac{5}{8}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{15}{24}\\end{array}[\/latex]<\/p>\n<p>Add the fractions by adding the numerators and keeping the denominator the same.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{18}{24}+\\Large\\frac{4}{24}+\\Large\\frac{15}{24}=\\Large\\frac{37}{24}[\/latex]<\/p>\n<p>Write the improper fraction as a mixed number and simplify the fraction.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{37}{24}=\\normalsize 1\\,\\,\\Large\\frac{13}{24}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{3}{4}+\\Large\\frac{1}{6}+\\Large\\frac{5}{8}=\\normalsize 1\\Large\\frac{13}{24}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Subtract Fractions<\/h2>\n<p>Subtracting fractions follows the same technique as adding them. First, determine whether or not the denominators are alike. If not, rewrite each fraction as an equivalent fraction, all having the same denominator.\u00a0 Below are some examples of subtracting fractions whose denominators are not alike.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract\u00a0[latex]\\Large\\frac{1}{5}-\\Large\\frac{1}{6}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155692\">Show Solution<\/span><\/p>\n<div id=\"q155692\" class=\"hidden-answer\" style=\"display: none\">The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.<\/p>\n<p style=\"text-align: center\">[latex]5\\cdot6=30[\/latex]<\/p>\n<p>Rewrite each fraction as an equivalent fraction with a denominator of [latex]30[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{1}{5}\\cdot\\Large\\frac{6}{6}=\\Large\\frac{6}{30}\\\\\\\\\\Large\\frac{1}{6}\\cdot\\Large\\frac{5}{5}=\\Large\\frac{5}{30}\\end{array}[\/latex]<\/p>\n<p>Subtract the numerators. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{6}{30}-\\Large\\frac{5}{30}=\\Large\\frac{1}{30}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{1}{5}-\\Large\\frac{1}{6}=\\Large\\frac{1}{30}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The example below shows how to use\u00a0multiples to find the least common multiple, which will be the least common denominator.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract [latex]\\Large\\frac{5}{6}-\\Large\\frac{1}{4}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q984596\">Show Solution<\/span><\/p>\n<div id=\"q984596\" class=\"hidden-answer\" style=\"display: none\">Find the least common multiple of the denominators\u2014this is the least common denominator.<\/p>\n<p>Multiples of \u00a0[latex]6: 6, \\textbf{12}, 18, 24[\/latex]<\/p>\n<p>Multiples of \u00a0[latex]4: 4, 8, \\textbf{12},16, 20[\/latex]<\/p>\n<p>[latex]12[\/latex] is the least common multiple of [latex]6[\/latex] and [latex]4[\/latex].<\/p>\n<p>Rewrite each fraction with a denominator of [latex]12[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{5}{6}\\cdot\\Large\\frac{2}{2}=\\Large\\frac{10}{12}\\\\\\\\\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{3}=\\Large\\frac{3}{12}\\end{array}[\/latex]<\/p>\n<p>Subtract the fractions. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{10}{12}-\\Large\\frac{3}{12}=\\Large\\frac{7}{12}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{5}{6}-\\Large\\frac{1}{4}=\\Large\\frac{7}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see an example of how to subtract fractions with unlike denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Subtract Fractions with Unlike Denominators (Basic with Model)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RpHtOMjeI7g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><!---MULTIPLY FRACTIONS --><\/p>\n<h2>Multiply Fractions<\/h2>\n<p>Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. \u00a0\u00a0There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.<\/p>\n<p>When you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\Large\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\Large\\frac{1}{2}[\/latex]\u00a0of the [latex]\\Large\\frac{3}{4}[\/latex]:<\/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\/01182611\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/><\/p>\n<p>By dividing each fourth in half, you can divide the candy bar into eighths.<\/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\/01182612\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/><\/p>\n<p>Then, choose half of those to get [latex]\\Large\\frac{3}{8}[\/latex].<\/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\/01182613\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/><\/p>\n<p>In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two Fractions<\/h3>\n<p>[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{a\\cdot c}{b\\cdot d}=\\Large\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Fractions<\/h3>\n<p>[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}\\cdot\\Large\\frac{e}{f}=\\Large\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{4}{5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368042\">Show Solution<\/span><\/p>\n<div id=\"q368042\" class=\"hidden-answer\" style=\"display: none\">Multiply the numerators and multiply the denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\n<p>Simplify, if possible. This fraction is already in lowest terms.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{8}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{8}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To review: if a fraction has\u00a0common factors in the numerator and denominator, we can <strong>reduce<\/strong> the fraction to its simplified form by removing the common factors.<\/p>\n<p>For example,<\/p>\n<ul>\n<li>Given [latex]\\Large\\frac{8}{15}[\/latex], the factors of [latex]8[\/latex] are: [latex]1, 2, 4, 8[\/latex] and the factors of [latex]15[\/latex] are: [latex]1, 3, 5, 15[\/latex]. \u00a0[latex]\\Large\\frac{8}{15}[\/latex] is simplified because there are no common factors of [latex]8[\/latex] and [latex]15[\/latex].<\/li>\n<li>Given [latex]\\Large\\frac{10}{15}[\/latex], the factors of [latex]10[\/latex] are: [latex]1, 2, 5, 10[\/latex] and the factors of [latex]15[\/latex] are: [latex]1, 3, 5, 15[\/latex]. [latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex]\u00a0is a common factor of [latex]10[\/latex]\u00a0and [latex]15[\/latex].<\/li>\n<\/ul>\n<p>You can\u00a0simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.<\/p>\n<p>In the following video you will see an example of how to multiply two fractions, then simplify the answer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1: Multiply Fractions (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f_L-EFC8Z7c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Multiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex]. Simplify the answer.<\/p>\n<p>What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385641\">Show Solution<\/span><\/p>\n<div id=\"q385641\" class=\"hidden-answer\" style=\"display: none\">Multiply the numerators and multiply the denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\n<p>Simplify first by canceling (dividing) the\u00a0common factors of [latex]3[\/latex] and [latex]2[\/latex]. [latex]3[\/latex] divided by [latex]3[\/latex] is [latex]1[\/latex], and [latex]2[\/latex] divided by \u00a0[latex]2[\/latex] is [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\Large\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\Large\\frac{1}{10}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex] = [latex]\\Large\\frac{1}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><!---DIVIDE FRACTIONS --><\/p>\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]\\Large\\frac{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]2[\/latex]\u00a0by the fraction, [latex]\\Large\\frac{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]\\Large\\frac{3}{4}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{4}{3}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{3}{4}\\cdot\\Large\\frac{4}{3}=\\Large\\frac{3\\cdot 4}{4\\cdot 3}=\\Large\\frac{12}{12}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\Large\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{2}{1}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{1}{2}\\cdot\\Large\\frac{2}{1}=\\Large\\frac{1\\cdot2}{2\\cdot1}=\\Large\\frac{2}{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3=\\Large\\frac{3}{1}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{1}{3}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{3}{1}\\cdot\\Large\\frac{1}{3}=\\Large\\frac{3\\cdot 1}{1\\cdot 3}=\\Large\\frac{3}{3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\Large\\frac{1}{3}=\\Large\\frac{7}{3}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{3}{7}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{7}{3}\\cdot\\Large\\frac{3}{7}=\\Large\\frac{7\\cdot3}{3\\cdot7}=\\Large\\frac{21}{21}=\\normalsize 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Sometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex]\\Large\\frac{2}{5}[\/latex] to get the reciprocal [latex]\\Large\\frac{5}{2}[\/latex].<\/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? Can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{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.<\/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]\\Large\\frac{0}{8}[\/latex] which would be [latex]\\Large\\frac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have<\/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. The reciprocal of\u00a0[latex]\\Large\\frac{8}{0}[\/latex] is undefined, and in fact, 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]\\Large\\frac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\Large\\frac{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 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]\\Large\\frac{1}{3}[\/latex], so the multiplication problem becomes<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{6}{1}\\cdot\\Large\\frac{1}{3}=\\Large\\frac{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. If you have [latex]\\Large\\frac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\Large\\frac{1}{5}[\/latex] of the available candy:<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{1}{5}\\normalsize\\text{ of }\\Large\\frac{3}{4}=\\Large\\frac{1}{5}\\cdot\\Large\\frac{3}{4}=\\Large\\frac{3}{20}[\/latex]<\/p>\n<p style=\"text-align: center\">Each person gets [latex]\\Large\\frac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/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]\\Large\\frac{1}{2}[\/latex]\u00a0to find the new amount.<\/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]\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{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\">Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]4[\/latex] or [latex]\\Large\\frac{4}{1}[\/latex] is the same as multiplying by the reciprocal of [latex]4[\/latex], which is [latex]\\Large\\frac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\normalsize\\div 4=\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot 1}{3\\cdot 4}=\\Large\\frac{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]\\Large\\frac{1}{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{2}{3}\\normalsize\\div4=\\Large\\frac{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\\Large\\frac{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\">Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]\\Large\\frac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\Large\\frac{1}{2}[\/latex], which is [latex]\\Large\\frac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]9\\div\\Large\\frac{1}{2}=\\Large\\frac{9}{1}\\cdot\\Large\\frac{2}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{9\\cdot 2}{1\\cdot 1}=\\Large\\frac{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\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{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]\\Large\\frac{2}{3}\\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=\"q569112\">Show Solution<\/span><\/p>\n<div id=\"q569112\" class=\"hidden-answer\" style=\"display: none\">Multiply by the reciprocal.<\/p>\n<p><strong>KEEP<\/strong> [latex]\\Large\\frac{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]\\Large\\frac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2\\cdot6}{3\\cdot1}=\\Large\\frac{12}{3}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{12}{3}=\\normalsize 4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{2}{3}\\div\\Large \\frac{1}{6}=\\normalsize 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\Large\\frac{3}{5}\\div\\Large\\frac{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\">Multiply by the reciprocal.\u00a0Keep [latex]\\Large\\frac{3}{5}[\/latex], change [latex]\\div[\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\Large\\frac{2}{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{3}{5}\\cdot\\Large\\frac{3}{2}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{3\\cdot 3}{5\\cdot 2}=\\Large\\frac{9}{10}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{3}{5}\\div\\Large\\frac{2}{3}=\\Large\\frac{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. The final answer should be simplified and written as a mixed number.<\/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-4\" 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-4500\">\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><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><li>Revision and Adaption. <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>Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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: Add Fractions with Unlike Denominators (Basic with Model). <strong>Authored by<\/strong>: James Sousa (Mathispower4u). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zV4q7j1-89I\">https:\/\/youtu.be\/zV4q7j1-89I<\/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><strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RpHtOMjeI7g\">https:\/\/youtu.be\/RpHtOMjeI7g<\/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 1: Multiply Fractions (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/f_L-EFC8Z7c\">https:\/\/youtu.be\/f_L-EFC8Z7c<\/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 1: Diviide Fractions (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <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><\/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":25777,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open 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