{"id":254,"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=254"},"modified":"2019-06-17T22:12:24","modified_gmt":"2019-06-17T22:12:24","slug":"read-adding-and-subtracting-fractions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-adding-and-subtracting-fractions-2\/","title":{"raw":"Adding and Subtracting Fractions","rendered":"Adding and Subtracting Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the common denominator of two or more fractions<\/li>\r\n \t<li>Use the common denominator to add or subtract fractions<\/li>\r\n \t<li>Simplify a fraction to its lowest terms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Introduction<\/h2>\r\nBefore we get started, here is some important terminology that will help you understand the concepts about working with fractions in this section.\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 \u2014 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 \u2014 the numerator in the fraction\u00a0[latex]\\dfrac{2}{3}[\/latex] is [latex]2[\/latex]<\/li>\r\n \t<li><strong>denominator:<\/strong> the bottom part of a fraction \u2014 the denominator in the fraction\u00a0[latex]\\dfrac{2}{3}[\/latex] is [latex]3[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Math Question Instructions<\/h2>\r\nMany different words are used by math textbooks and teachers to provide students with instructions on what they are to do with a given problem. For example, you may see instructions such as \"Find\" or \"Simplify\" in the example in this module. It 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 the words you may see that can help you know how to work through the problems 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.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify<\/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>Adding 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]\\dfrac{1}{4}[\/latex]\u00a0of the pizza is gone and [latex]\\dfrac{3}{4}[\/latex] remains.\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\nNote 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]\\dfrac{5}{8}[\/latex]?\r\n\r\nHow can you describe the total amount of pizza that is left between the two pizzas? There are now two very different fractions and it's hard to compare the two. You will 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 for the two! In the example above, that is [latex]10[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the amount of pizza left in the examples above using common terms.\r\n\r\n[reveal-answer q=\"155500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155500\"]\r\n\r\nRewrite 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] \u2014 the 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 (because [latex]\\dfrac{2}{2}=1[\/latex]).\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, and compare the values more easily.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo add fractions with unlike denominators, you will need to first rewrite them with common denominators. Then, you know what to do! The steps are shown below.\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 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>Simplifying a Fraction<\/h2>\r\nOften, if the answer to a problem is a fraction, you will be asked to write it in lowest terms. This is a common convention used in mathematics, similar to starting a sentence with a capital letter and ending it with a period. In this course, we will not go into great detail about methods for reducing fractions because there are many. The process of simplifying a fraction is often called <em data-effect=\"italics\">reducing the fraction<\/em>. We can simplify by canceling (dividing) the common factors in a fraction's numerator and denominator. \u00a0We can do this because a fraction represents division.\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}{3}\\cdot1=\\dfrac{2}{3}[\/latex]<\/p>\r\nRewriting fractions with the smallest factors possible is often called prime factorization.\r\n\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]\\dfrac{2}{3}+\\dfrac{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] by multiplying it by [latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{2}{3}\\cdot\\dfrac{5}{5}=\\dfrac{10}{15}\\\\\\\\\\dfrac{1}{5}\\cdot\\dfrac{3}{3}=\\dfrac{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]\\dfrac{10}{15}+\\dfrac{3}{15}=\\dfrac{13}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{2}{3}+\\dfrac{1}{5}=\\dfrac{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]\\dfrac{3}{7}+\\dfrac{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}\\dfrac{3}{7}\\cdot\\dfrac{3}{3}=\\dfrac{9}{21}\\\\\\\\\\dfrac{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]\\dfrac{9}{21}+\\dfrac{2}{21}=\\dfrac{11}{21}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{3}{7}+\\dfrac{2}{21}=\\dfrac{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]\\dfrac{3}{4}+\\dfrac{1}{6}+\\dfrac{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}\\dfrac{3}{4}\\cdot\\dfrac{6}{6}=\\dfrac{18}{24}\\\\\\\\\\dfrac{1}{6}\\cdot\\dfrac{4}{4}=\\dfrac{4}{24}\\\\\\\\\\dfrac{5}{8}\\cdot\\dfrac{3}{3}=\\dfrac{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]\\dfrac{18}{24}+\\dfrac{4}{24}+\\dfrac{15}{24}=\\dfrac{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]\\dfrac{37}{24}=\\normalsize 1\\,\\,\\dfrac{13}{24}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{3}{4}+\\dfrac{1}{6}+\\dfrac{5}{8}=\\normalsize 1\\dfrac{13}{24}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Subtracting Fractions<\/h2>\r\nWhen you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. 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]\\dfrac{1}{5}-\\dfrac{1}{6}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"155692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155692\"]\r\n\r\nThe 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}\\dfrac{1}{5}\\cdot\\dfrac{6}{6}=\\dfrac{6}{30}\\\\\\\\\\dfrac{1}{6}\\cdot\\dfrac{5}{5}=\\dfrac{5}{30}\\end{array}[\/latex]<\/p>\r\nSubtract the numerators. Simplify the answer if needed.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{6}{30}-\\dfrac{5}{30}=\\dfrac{1}{30}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{1}{5}-\\dfrac{1}{6}=\\dfrac{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]\\dfrac{5}{6}-\\dfrac{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}\\dfrac{5}{6}\\cdot\\dfrac{2}{2}=\\dfrac{10}{12}\\\\\\\\\\dfrac{1}{4}\\cdot\\dfrac{3}{3}=\\dfrac{3}{12}\\end{array}[\/latex]<\/p>\r\nSubtract the fractions. Simplify the answer if needed.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{10}{12}-\\dfrac{3}{12}=\\dfrac{7}{12}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{5}{6}-\\dfrac{1}{4}=\\dfrac{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","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the common denominator of two or more fractions<\/li>\n<li>Use the common denominator to add or subtract fractions<\/li>\n<li>Simplify a fraction to its lowest terms<\/li>\n<\/ul>\n<\/div>\n<h2>Introduction<\/h2>\n<p>Before we get started, here is some important terminology that will help you understand the concepts about working with fractions in this section.<\/p>\n<ul>\n<li><strong>product:\u00a0<\/strong>the result of \u00a0multiplication<\/li>\n<li><strong>factor:<\/strong> something being multiplied \u2014 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 \u2014 the numerator in the fraction\u00a0[latex]\\dfrac{2}{3}[\/latex] is [latex]2[\/latex]<\/li>\n<li><strong>denominator:<\/strong> the bottom part of a fraction \u2014 the denominator in the fraction\u00a0[latex]\\dfrac{2}{3}[\/latex] is [latex]3[\/latex]<\/li>\n<\/ul>\n<h2>Math Question Instructions<\/h2>\n<p>Many different words are used by math textbooks and teachers to provide students with instructions on what they are to do with a given problem. For example, you may see instructions such as &#8220;Find&#8221; or &#8220;Simplify&#8221; in the example in this module. It 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 the words you may see that can help you know how to work through the problems 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.<\/td>\n<\/tr>\n<tr>\n<td>Simplify<\/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>Adding 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]\\dfrac{1}{4}[\/latex]\u00a0of the pizza is gone and [latex]\\dfrac{3}{4}[\/latex] remains.<\/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>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]\\dfrac{5}{8}[\/latex]?<\/p>\n<p>How can you describe the total amount of pizza that is left between the two pizzas? There are now two very different fractions and it&#8217;s hard to compare the two. You will 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 for the two! In the example above, that is [latex]10[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the amount of pizza left in the examples above 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\">\n<p>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] \u2014 the 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 (because [latex]\\dfrac{2}{2}=1[\/latex]).<\/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, and compare the values more easily.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To add fractions with unlike denominators, you will need to first rewrite them with common denominators. Then, you know what to do! The steps are shown below.<\/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 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>Simplifying a Fraction<\/h2>\n<p>Often, if the answer to a problem is a fraction, you will be asked to write it in lowest terms. This is a common convention used in mathematics, similar to starting a sentence with a capital letter and ending it with a period. In this course, we will not go into great detail about methods for reducing fractions because there are many. The process of simplifying a fraction is often called <em data-effect=\"italics\">reducing the fraction<\/em>. We can simplify by canceling (dividing) the common factors in a fraction&#8217;s numerator and denominator. \u00a0We can do this because a fraction represents division.<\/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}{3}\\cdot1=\\dfrac{2}{3}[\/latex]<\/p>\n<p>Rewriting fractions with the smallest factors possible is often called prime factorization.<\/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]\\dfrac{2}{3}+\\dfrac{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] by multiplying it by [latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{2}{3}\\cdot\\dfrac{5}{5}=\\dfrac{10}{15}\\\\\\\\\\dfrac{1}{5}\\cdot\\dfrac{3}{3}=\\dfrac{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]\\dfrac{10}{15}+\\dfrac{3}{15}=\\dfrac{13}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{2}{3}+\\dfrac{1}{5}=\\dfrac{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]\\dfrac{3}{7}+\\dfrac{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}\\dfrac{3}{7}\\cdot\\dfrac{3}{3}=\\dfrac{9}{21}\\\\\\\\\\dfrac{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]\\dfrac{9}{21}+\\dfrac{2}{21}=\\dfrac{11}{21}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{3}{7}+\\dfrac{2}{21}=\\dfrac{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]\\dfrac{3}{4}+\\dfrac{1}{6}+\\dfrac{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}\\dfrac{3}{4}\\cdot\\dfrac{6}{6}=\\dfrac{18}{24}\\\\\\\\\\dfrac{1}{6}\\cdot\\dfrac{4}{4}=\\dfrac{4}{24}\\\\\\\\\\dfrac{5}{8}\\cdot\\dfrac{3}{3}=\\dfrac{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]\\dfrac{18}{24}+\\dfrac{4}{24}+\\dfrac{15}{24}=\\dfrac{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]\\dfrac{37}{24}=\\normalsize 1\\,\\,\\dfrac{13}{24}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{3}{4}+\\dfrac{1}{6}+\\dfrac{5}{8}=\\normalsize 1\\dfrac{13}{24}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Subtracting Fractions<\/h2>\n<p>When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. 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]\\dfrac{1}{5}-\\dfrac{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\">\n<p>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}\\dfrac{1}{5}\\cdot\\dfrac{6}{6}=\\dfrac{6}{30}\\\\\\\\\\dfrac{1}{6}\\cdot\\dfrac{5}{5}=\\dfrac{5}{30}\\end{array}[\/latex]<\/p>\n<p>Subtract the numerators. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{6}{30}-\\dfrac{5}{30}=\\dfrac{1}{30}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{1}{5}-\\dfrac{1}{6}=\\dfrac{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]\\dfrac{5}{6}-\\dfrac{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}\\dfrac{5}{6}\\cdot\\dfrac{2}{2}=\\dfrac{10}{12}\\\\\\\\\\dfrac{1}{4}\\cdot\\dfrac{3}{3}=\\dfrac{3}{12}\\end{array}[\/latex]<\/p>\n<p>Subtract the fractions. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{10}{12}-\\dfrac{3}{12}=\\dfrac{7}{12}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{5}{6}-\\dfrac{1}{4}=\\dfrac{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\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-254\">\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 Adaptiation. <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: Add Fractions with Unlike Denominators (Basic with Model). <strong>Authored by<\/strong>: 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>Ex: Subtract Fractions with Unlike Denominators (Basic with Model) Mathispower4u . <strong>Authored by<\/strong>: Mathispower4u. <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>Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <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><\/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":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Add Fractions with Unlike Denominators (Basic with 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