{"id":1410,"date":"2021-11-01T18:40:18","date_gmt":"2021-11-01T18:40:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1410"},"modified":"2022-09-01T18:40:24","modified_gmt":"2022-09-01T18:40:24","slug":"1-3-6-adding-and-subtracting-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-3-6-adding-and-subtracting-fractions\/","title":{"raw":"1.3.5: Adding and Subtracting Fractions","rendered":"1.3.5: Adding and Subtracting Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Model fraction addition<\/li>\r\n \t<li>Add and subtract fractions with a common denominator<\/li>\r\n \t<li>Find equivalent fractions with the common denominator<\/li>\r\n \t<li>Add and subtract fractions with different denominators<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Common denominator<\/strong>: a common multiple of all denominators<\/li>\r\n \t<li><strong>Least common denominator<\/strong>: the smallest common multiple of all denominators<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Model Fraction Addition<\/h2>\r\nHow many quarters are pictured? One quarter plus [latex]2[\/latex] quarters equals [latex]3[\/latex] quarters.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220949\/CNX_BMath_Figure_04_04_001.png\" alt=\"Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.\" \/>\r\nRemember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that\r\n<p style=\"text-align: center;\">[latex]{\\frac{1}{4}}+{\\frac{2}{4}}={\\frac{3}{4}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{one quarter }+\\text{ two quarters }=\\text{ three quarters} [\/latex]<\/p>\r\nLet\u2019s use fraction circles to model the same example, [latex]\\frac{1}{4}+\\frac{2}{4}[\/latex].\r\n<table id=\"eip-id1168467352669\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Start with one [latex]\\frac{1}{4}[\/latex] piece.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220951\/CNX_BMath_Figure_04_04_002_img-01.png\" alt=\".\" \/><\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add two more [latex]\\frac{1}{4}[\/latex] pieces.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220953\/CNX_BMath_Figure_04_04_002_img-02.png\" alt=\".\" \/><\/td>\r\n<td>[latex]+\\frac{2}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The result is [latex]\\frac{3}{4}[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220954\/CNX_BMath_Figure_04_04_002_img-03.png\" alt=\".\" \/><\/td>\r\n<td>[latex]\\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo again, we see that\u00a0[latex]\\frac{1}{4}+\\frac{2}{4}=\\frac{3}{4}[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse a model to find each sum. Show a diagram to illustrate your model.\r\n\r\n[latex]\\frac{1}{8}+\\frac{4}{8}[\/latex]\r\n[reveal-answer q=\"304582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304582\"]\r\n\r\n[latex]\\frac{5}{8}[\/latex]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221004\/CNX_BMath_Figure_04_04_004_img.png\" alt=\"A circle divided into 8 sections, 5 of which are shaded.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nUse a model to find each sum. Show a diagram to illustrate your model.\r\n[latex]\\frac{1}{6}+\\frac{4}{6}[\/latex]\r\n[reveal-answer q=\"297291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"297291\"]\r\n\r\n[latex]\\frac{5}{6}[\/latex]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221004\/CNX_BMath_Figure_04_04_005_img.png\" alt=\"A circle divided into 6 sections, 5 of which are shaded.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[ohm_question height=\"270\"]146178[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows more examples of how to use models to add fractions with like denominators.\r\n\r\nhttps:\/\/youtu.be\/GTkY34kl6Kw\r\n<h2>Add Fractions with a Common Denominator<\/h2>\r\nThe example above shows that to add the same-size pieces\u2014meaning that the fractions have the same denominator\u2014we just add the number of pieces.\r\n<div class=\"textbox shaded\">\r\n<h3>Fraction Addition<\/h3>\r\nIf [latex]a,b,\\text{ and }c[\/latex] are integers where [latex]c\\ne 0[\/latex], then\u00a0[latex]\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}[\/latex]\r\n\r\nTo add fractions with a common denominator, add the numerators and place the sum over the common denominator.\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum: [latex]\\frac{3}{5}+\\frac{1}{5}[\/latex]\r\n[reveal-answer q=\"250553\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"250553\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468585410\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{3}{5}+\\frac{1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the numerators and place the sum over the common denominator.<\/td>\r\n<td>[latex]\\frac{3+1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{4}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146182[\/ohm_question]\r\n\r\n<\/div>\r\nTechnically, a negative sign on a fraction can be written in the following locations: by the numerator, by the denominator or out in front of the fraction bar.\r\n<p style=\"text-align: center;\">[latex]-\\frac{2}{3}=\\frac{-2}{3}=\\frac{2}{-3}[\/latex]<\/p>\r\nThis is because [latex]\\frac{(-)}{(+)}=(-)[\/latex] and [latex]\\frac{(+)}{(-)}=(-)[\/latex] due to division of integers. Usually, we avoid putting the negative sign on the denominator and keep it either out front or on the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum: [latex]-\\frac{3}{12}+\\left(-\\frac{5}{12}\\right)[\/latex]\r\n[reveal-answer q=\"855499\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"855499\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467276148\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\frac{3}{12}+\\left(-\\frac{5}{12}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the numerators and place the sum over the common denominator.<\/td>\r\n<td>[latex]\\frac{-3+\\left(-5\\right)}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\frac{-8}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the fraction.<\/td>\r\n<td>[latex]-\\frac{2}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146187[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Adding and Subtracting Fractions with Different Denominators<\/h2>\r\nCan you add one quarter and one dime? You could say there are two coins, but that\u2019s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit\u2014cents. One quarter equals [latex]25[\/latex] cents and one dime equals [latex]10[\/latex] cents, so the sum is [latex]35[\/latex] cents. See the image below.\r\n\r\nTogether, a quarter and a dime are worth [latex]35[\/latex] cents, or [latex]{\\Large\\frac{35}{100}}[\/latex] of a dollar.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221013\/CNX_BMath_Figure_04_05_002_img.png\" alt=\"A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.\" \/>\r\nSimilarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is [latex]100[\/latex]. Since there are [latex]100[\/latex] cents in one dollar, [latex]25[\/latex] cents is [latex]\\frac{25}{100}[\/latex] and [latex]10[\/latex] cents is [latex]\\frac{10}{100}[\/latex]. So we add [latex]\\frac{25}{100}+\\frac{10}{100}[\/latex] to get [latex]\\frac{35}{100}[\/latex], which is [latex]35[\/latex] cents.\r\n\r\nWe have practiced adding and subtracting fractions with common denominators. Now let\u2019s see what we need to do with fractions that have different denominators.\r\n\r\nTo add fractions, they must have the same denominator. If they have different denominators, we must build equivalent fractions so that they have the same denominator. The new denominator must be a multiple of each of the fractions denominators.\r\n\r\nFor example, [latex]\\frac{5}{6}+\\frac{3}{4}[\/latex] has denominators of [latex]6[\/latex] and [latex]4[\/latex]. We need a new denominator that has factors of both\u00a0[latex]6[\/latex] and [latex]4[\/latex]. One way to achieve this is to simply multiply the two denominators to get [latex]24[\/latex]. \u00a0[latex]6[\/latex] and [latex]4[\/latex] both divide exactly into\u00a0[latex]24[\/latex]. Now we can build equivalent fractions with denominators of\u00a0[latex]24[\/latex].\r\n\r\n[latex]\\frac{5}{6}=\\frac{5 \\color{red}{\\cdot 4}}{6 \\color{red}{\\cdot 4}}=\\frac{20}{24}[\/latex]\r\n\r\n[latex]\\frac{3}{4}=\\frac{3 \\color{red}{\\cdot 6}}{4 \\color{red}{\\cdot 6}}=\\frac{18}{24}[\/latex]\r\n\r\nThen we can add: \u00a0\u00a0[latex]\\frac{5}{6}+\\frac{3}{4}=\\frac{20}{24}+\\frac{18}{24}=\\frac{20+18}{24}=\\frac{38}{24}[\/latex]\r\n\r\nFinally, we simplify the sum: \u00a0 [latex]\\large\\frac{38}{24}=\\frac{\\color{red}{\\cancel 2}\\cdot 19}{\\color{red}{\\cancel 2}\\cdot 12}=\\frac{19}{12}[\/latex]\r\n\r\nNotice that although multiplying the denominators yields a common denominator, it is not the smallest number that could be used. Any common multiple of the denominators will work, but the least common multiple (LCM) is typically used for efficiency.\r\n\r\nlet's take another look at\u00a0[latex]\\frac{5}{6}+\\frac{3}{4}[\/latex]. The LCM of the denominators [latex]6[\/latex] and [latex]4[\/latex] is [latex]12[\/latex].\r\n\r\nWe build equivalent fractions with denominator of\u00a0[latex]12[\/latex]:\r\n\r\n[latex]\\frac{5}{6}=\\frac{5 \\color{red}{\\cdot 2}}{6 \\color{red}{\\cdot 2}}=\\frac{10}{12}[\/latex]\r\n\r\n[latex]\\frac{3}{4}=\\frac{3 \\color{red}{\\cdot 3}}{4 \\color{red}{\\cdot 3}}=\\frac{9}{12}[\/latex]\r\n\r\nThen we add: [latex]\\frac{5}{6}+\\frac{3}{4}=\\frac{10}{12}+\\frac{9}{12}=\\frac{10+9}{12}=\\frac{19}{12}[\/latex]\r\n\r\nEither way we get the same answer, but we are using smaller integer values using the LCM.\r\n<div class=\"textbox shaded\">\r\n<h3>Least Common Denominator<\/h3>\r\nThe lowest common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146252[\/ohm_question]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox shaded\">\r\n<h3>Convert two fractions to equivalent fractions with their LCD as the common denominator<\/h3>\r\n<ol id=\"eip-id1168468227692\" class=\"stepwise\">\r\n \t<li>Find the LCD.<\/li>\r\n \t<li>For each fraction, determine the natural number needed to multiply the denominator to get the LCD.<\/li>\r\n \t<li>Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.<\/li>\r\n \t<li>Simplify the numerator and denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146254[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146255[\/ohm_question]\r\n\r\n<\/div>\r\nOnce we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.\r\n<div class=\"textbox shaded\">\r\n<h3>Add or subtract fractions with different denominators<\/h3>\r\n<ol id=\"eip-id1168468303196\" class=\"stepwise\">\r\n \t<li>Find the LCD.<\/li>\r\n \t<li>Convert each fraction to an equivalent form with the LCD as the denominator.<\/li>\r\n \t<li>Add or subtract the fractions.<\/li>\r\n \t<li>Write the result in simplified form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd: [latex]\\frac{1}{2}+\\frac{1}{3}[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466144855\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 1 half plus 1 third. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{1}{2}+\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the LCD of [latex]2[\/latex], [latex]3[\/latex].<\/td>\r\n<td><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221040\/CNX_BMath_Figure_04_05_029_img-01.png\" alt=\".\" width=\"81\" height=\"110\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change into equivalent fractions with the LCD [latex]6[\/latex].<\/td>\r\n<td>[latex]\\frac{1\\cdot\\color{red}{3}}{2\\cdot\\color{red}{3}} +\\frac{1\\cdot\\color{red}{2}}{3\\cdot\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerators and denominators.<\/td>\r\n<td>[latex]\\frac{3}{6}+\\frac{2}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\frac{5}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRemember, always check to see if the answer can be simplified. Since [latex]5[\/latex] and [latex]6[\/latex] have no common factors (other than 1), the fraction [latex]\\frac{5}{6}[\/latex] cannot be simplified.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"230\"]146262[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples and explanation about how to add two fractions with unlike denominators.\r\n\r\nhttps:\/\/youtu.be\/zV4q7j1-89I\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"230\"]146264[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd: [latex]\\frac{7}{12}+\\frac{5}{18}[\/latex]\r\n[reveal-answer q=\"826911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"826911\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468285390\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 7 over 12 plus 5 over 18. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{7}{12}+\\frac{5}{18}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the LCD of [latex]12[\/latex] and [latex]18[\/latex].<\/td>\r\n<td><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221043\/CNX_BMath_Figure_04_05_031_img-01.png\" alt=\".\" width=\"146\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\r\n<td>[latex]\\frac{7\\cdot\\color{red}{3}}{12\\cdot\\color{red}{3}} +\\frac{5\\cdot\\color{red}{2}}{18\\cdot\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerators and denominators.<\/td>\r\n<td>[latex]\\frac{21}{36}+\\frac{10}{36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\frac{31}{36}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBecause [latex]31[\/latex] is a prime number, it has no factors in common with [latex]36[\/latex]. The answer is simplified.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"230\"]146265[\/ohm_question]\r\n\r\n<\/div>\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]\\frac{3}{4}+\\frac{1}{6}+\\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 4, 6, and 8.\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 24.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3}{4}\\cdot\\frac{6}{6}=\\frac{18}{24}\\\\\\\\\\frac{1}{6}\\cdot\\frac{4}{4}=\\frac{4}{24}\\\\\\\\\\frac{5}{8}\\cdot\\frac{3}{3}=\\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]\\frac{18}{24}+\\frac{4}{24}+\\frac{15}{24}=\\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]\\frac{37}{24}=1\\,\\,\\frac{13}{24}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{3}{4}+\\frac{1}{6}+\\frac{5}{8}=1\\frac{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=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract: [latex]\\frac{7}{15}-\\frac{19}{24}[\/latex]\r\n[reveal-answer q=\"169999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169999\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466319808\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 7 over 15 minus 19 over 24. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{7}{15}-\\frac{19}{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the LCD.\r\n\r\n&nbsp;<\/td>\r\n<td><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221046\/CNX_BMath_Figure_04_05_032_img-01.png\" alt=\".\" width=\"170\" height=\"80\" \/>\r\n\r\n[latex]15[\/latex] is 'missing' three factors of [latex]2[\/latex]\r\n\r\n[latex]24[\/latex] is 'missing' a factor of [latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\r\n<td>[latex]\\frac{7\\cdot\\color{red}{8}}{15\\cdot\\color{red}{8}} -\\frac{19\\cdot\\color{red}{5}}{24\\cdot\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each numerator and denominator.<\/td>\r\n<td>[latex]\\frac{56}{120}-\\frac{95}{120}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]-\\frac{39}{120}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing the common factor of [latex]3[\/latex].<\/td>\r\n<td>[latex]-\\frac{13\\cdot 3}{40\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factor to simplify.<\/td>\r\n<td>[latex]-\\frac{13}{40}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"230\"]146266[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides two more examples of how to subtract two fractions with unlike denominators.\r\n\r\nhttps:\/\/youtu.be\/aXlkygPPzQ8\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd: [latex]-\\frac{11}{30}+\\frac{23}{42}[\/latex]\r\n[reveal-answer q=\"808641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808641\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466855270\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says negative 11 over 30 plus 23 over 42. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]-\\frac{11}{30}+\\frac{23}{42}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the LCD.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221049\/CNX_BMath_Figure_04_05_033_img-01.png\" alt=\".\" width=\"151\" height=\"82\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\r\n<td>[latex]-\\frac{11\\cdot\\color{red}{7}}{30\\cdot\\color{red}{7}} +\\frac{23\\cdot\\color{red}{5}}{42\\cdot\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each numerator and denominator.<\/td>\r\n<td>[latex]-\\frac{77}{210}+\\frac{115}{210}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\frac{38}{210}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing the common factor of [latex]2[\/latex].<\/td>\r\n<td>[latex]\\frac{19\\cdot 2}{105\\cdot 2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factor to simplify.<\/td>\r\n<td>[latex]\\frac{19}{105}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"230\"]146267[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract: [latex]\\frac{1}{2}-\\left(-\\frac{1}{4}\\right)[\/latex]\r\n[reveal-answer q=\"841760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841760\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466426391\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 1 half minus negative 1 fourth. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{1}{2}-\\left(-\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the LCD of [latex]2[\/latex] and [latex]4[\/latex].<\/td>\r\n<td><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221041\/CNX_BMath_Figure_04_05_030_img-01.png\" alt=\".\" width=\"100\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as equivalent fractions using the LCD [latex]4[\/latex].<\/td>\r\n<td>[latex]\\frac{1\\cdot\\color{red}{2}}{2\\cdot\\color{red}{2}} - (--\\frac{1}{4})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the first fraction.<\/td>\r\n<td>[latex]\\frac{2}{4}-\\left(-\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]\\frac{2-\\left(-1\\right)}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOne of the fractions already had the least common denominator, so we only had to convert the other fraction.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Model fraction addition<\/li>\n<li>Add and subtract fractions with a common denominator<\/li>\n<li>Find equivalent fractions with the common denominator<\/li>\n<li>Add and subtract fractions with different denominators<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Common denominator<\/strong>: a common multiple of all denominators<\/li>\n<li><strong>Least common denominator<\/strong>: the smallest common multiple of all denominators<\/li>\n<\/ul>\n<\/div>\n<h2>Model Fraction Addition<\/h2>\n<p>How many quarters are pictured? One quarter plus [latex]2[\/latex] quarters equals [latex]3[\/latex] quarters.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220949\/CNX_BMath_Figure_04_04_001.png\" alt=\"Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.\" \/><br \/>\nRemember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that<\/p>\n<p style=\"text-align: center;\">[latex]{\\frac{1}{4}}+{\\frac{2}{4}}={\\frac{3}{4}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\text{one quarter }+\\text{ two quarters }=\\text{ three quarters}[\/latex]<\/p>\n<p>Let\u2019s use fraction circles to model the same example, [latex]\\frac{1}{4}+\\frac{2}{4}[\/latex].<\/p>\n<table id=\"eip-id1168467352669\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td>Start with one [latex]\\frac{1}{4}[\/latex] piece.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220951\/CNX_BMath_Figure_04_04_002_img-01.png\" alt=\".\" \/><\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add two more [latex]\\frac{1}{4}[\/latex] pieces.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220953\/CNX_BMath_Figure_04_04_002_img-02.png\" alt=\".\" \/><\/td>\n<td>[latex]+\\frac{2}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The result is [latex]\\frac{3}{4}[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220954\/CNX_BMath_Figure_04_04_002_img-03.png\" alt=\".\" \/><\/td>\n<td>[latex]\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So again, we see that\u00a0[latex]\\frac{1}{4}+\\frac{2}{4}=\\frac{3}{4}[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use a model to find each sum. Show a diagram to illustrate your model.<\/p>\n<p>[latex]\\frac{1}{8}+\\frac{4}{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304582\">Show Solution<\/span><\/p>\n<div id=\"q304582\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{5}{8}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221004\/CNX_BMath_Figure_04_04_004_img.png\" alt=\"A circle divided into 8 sections, 5 of which are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use a model to find each sum. Show a diagram to illustrate your model.<br \/>\n[latex]\\frac{1}{6}+\\frac{4}{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q297291\">Show Solution<\/span><\/p>\n<div id=\"q297291\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{5}{6}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221004\/CNX_BMath_Figure_04_04_005_img.png\" alt=\"A circle divided into 6 sections, 5 of which are shaded.\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146178\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146178&theme=oea&iframe_resize_id=ohm146178&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of how to use models to add fractions with like denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Add Fractions with Like Denominators\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GTkY34kl6Kw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Add Fractions with a Common Denominator<\/h2>\n<p>The example above shows that to add the same-size pieces\u2014meaning that the fractions have the same denominator\u2014we just add the number of pieces.<\/p>\n<div class=\"textbox shaded\">\n<h3>Fraction Addition<\/h3>\n<p>If [latex]a,b,\\text{ and }c[\/latex] are integers where [latex]c\\ne 0[\/latex], then\u00a0[latex]\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}[\/latex]<\/p>\n<p>To add fractions with a common denominator, add the numerators and place the sum over the common denominator.<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum: [latex]\\frac{3}{5}+\\frac{1}{5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q250553\">Show Solution<\/span><\/p>\n<div id=\"q250553\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468585410\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{3}{5}+\\frac{1}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the numerators and place the sum over the common denominator.<\/td>\n<td>[latex]\\frac{3+1}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{4}{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146182\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146182&theme=oea&iframe_resize_id=ohm146182&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Technically, a negative sign on a fraction can be written in the following locations: by the numerator, by the denominator or out in front of the fraction bar.<\/p>\n<p style=\"text-align: center;\">[latex]-\\frac{2}{3}=\\frac{-2}{3}=\\frac{2}{-3}[\/latex]<\/p>\n<p>This is because [latex]\\frac{(-)}{(+)}=(-)[\/latex] and [latex]\\frac{(+)}{(-)}=(-)[\/latex] due to division of integers. Usually, we avoid putting the negative sign on the denominator and keep it either out front or on the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum: [latex]-\\frac{3}{12}+\\left(-\\frac{5}{12}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q855499\">Show Solution<\/span><\/p>\n<div id=\"q855499\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467276148\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-\\frac{3}{12}+\\left(-\\frac{5}{12}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the numerators and place the sum over the common denominator.<\/td>\n<td>[latex]\\frac{-3+\\left(-5\\right)}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\frac{-8}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>[latex]-\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146187\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146187&theme=oea&iframe_resize_id=ohm146187&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<h2>Adding and Subtracting Fractions with Different Denominators<\/h2>\n<p>Can you add one quarter and one dime? You could say there are two coins, but that\u2019s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit\u2014cents. One quarter equals [latex]25[\/latex] cents and one dime equals [latex]10[\/latex] cents, so the sum is [latex]35[\/latex] cents. See the image below.<\/p>\n<p>Together, a quarter and a dime are worth [latex]35[\/latex] cents, or [latex]{\\Large\\frac{35}{100}}[\/latex] of a dollar.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221013\/CNX_BMath_Figure_04_05_002_img.png\" alt=\"A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.\" \/><br \/>\nSimilarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is [latex]100[\/latex]. Since there are [latex]100[\/latex] cents in one dollar, [latex]25[\/latex] cents is [latex]\\frac{25}{100}[\/latex] and [latex]10[\/latex] cents is [latex]\\frac{10}{100}[\/latex]. So we add [latex]\\frac{25}{100}+\\frac{10}{100}[\/latex] to get [latex]\\frac{35}{100}[\/latex], which is [latex]35[\/latex] cents.<\/p>\n<p>We have practiced adding and subtracting fractions with common denominators. Now let\u2019s see what we need to do with fractions that have different denominators.<\/p>\n<p>To add fractions, they must have the same denominator. If they have different denominators, we must build equivalent fractions so that they have the same denominator. The new denominator must be a multiple of each of the fractions denominators.<\/p>\n<p>For example, [latex]\\frac{5}{6}+\\frac{3}{4}[\/latex] has denominators of [latex]6[\/latex] and [latex]4[\/latex]. We need a new denominator that has factors of both\u00a0[latex]6[\/latex] and [latex]4[\/latex]. One way to achieve this is to simply multiply the two denominators to get [latex]24[\/latex]. \u00a0[latex]6[\/latex] and [latex]4[\/latex] both divide exactly into\u00a0[latex]24[\/latex]. Now we can build equivalent fractions with denominators of\u00a0[latex]24[\/latex].<\/p>\n<p>[latex]\\frac{5}{6}=\\frac{5 \\color{red}{\\cdot 4}}{6 \\color{red}{\\cdot 4}}=\\frac{20}{24}[\/latex]<\/p>\n<p>[latex]\\frac{3}{4}=\\frac{3 \\color{red}{\\cdot 6}}{4 \\color{red}{\\cdot 6}}=\\frac{18}{24}[\/latex]<\/p>\n<p>Then we can add: \u00a0\u00a0[latex]\\frac{5}{6}+\\frac{3}{4}=\\frac{20}{24}+\\frac{18}{24}=\\frac{20+18}{24}=\\frac{38}{24}[\/latex]<\/p>\n<p>Finally, we simplify the sum: \u00a0 [latex]\\large\\frac{38}{24}=\\frac{\\color{red}{\\cancel 2}\\cdot 19}{\\color{red}{\\cancel 2}\\cdot 12}=\\frac{19}{12}[\/latex]<\/p>\n<p>Notice that although multiplying the denominators yields a common denominator, it is not the smallest number that could be used. Any common multiple of the denominators will work, but the least common multiple (LCM) is typically used for efficiency.<\/p>\n<p>let&#8217;s take another look at\u00a0[latex]\\frac{5}{6}+\\frac{3}{4}[\/latex]. The LCM of the denominators [latex]6[\/latex] and [latex]4[\/latex] is [latex]12[\/latex].<\/p>\n<p>We build equivalent fractions with denominator of\u00a0[latex]12[\/latex]:<\/p>\n<p>[latex]\\frac{5}{6}=\\frac{5 \\color{red}{\\cdot 2}}{6 \\color{red}{\\cdot 2}}=\\frac{10}{12}[\/latex]<\/p>\n<p>[latex]\\frac{3}{4}=\\frac{3 \\color{red}{\\cdot 3}}{4 \\color{red}{\\cdot 3}}=\\frac{9}{12}[\/latex]<\/p>\n<p>Then we add: [latex]\\frac{5}{6}+\\frac{3}{4}=\\frac{10}{12}+\\frac{9}{12}=\\frac{10+9}{12}=\\frac{19}{12}[\/latex]<\/p>\n<p>Either way we get the same answer, but we are using smaller integer values using the LCM.<\/p>\n<div class=\"textbox shaded\">\n<h3>Least Common Denominator<\/h3>\n<p>The lowest common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146252\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146252&theme=oea&iframe_resize_id=ohm146252&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox shaded\">\n<h3>Convert two fractions to equivalent fractions with their LCD as the common denominator<\/h3>\n<ol id=\"eip-id1168468227692\" class=\"stepwise\">\n<li>Find the LCD.<\/li>\n<li>For each fraction, determine the natural number needed to multiply the denominator to get the LCD.<\/li>\n<li>Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.<\/li>\n<li>Simplify the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146254\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146254&theme=oea&iframe_resize_id=ohm146254&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146255\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146255&theme=oea&iframe_resize_id=ohm146255&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.<\/p>\n<div class=\"textbox shaded\">\n<h3>Add or subtract fractions with different denominators<\/h3>\n<ol id=\"eip-id1168468303196\" class=\"stepwise\">\n<li>Find the LCD.<\/li>\n<li>Convert each fraction to an equivalent form with the LCD as the denominator.<\/li>\n<li>Add or subtract the fractions.<\/li>\n<li>Write the result in simplified form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add: [latex]\\frac{1}{2}+\\frac{1}{3}[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466144855\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 1 half plus 1 third. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{1}{2}+\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the LCD of [latex]2[\/latex], [latex]3[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221040\/CNX_BMath_Figure_04_05_029_img-01.png\" alt=\".\" width=\"81\" height=\"110\" \/><\/td>\n<\/tr>\n<tr>\n<td>Change into equivalent fractions with the LCD [latex]6[\/latex].<\/td>\n<td>[latex]\\frac{1\\cdot\\color{red}{3}}{2\\cdot\\color{red}{3}} +\\frac{1\\cdot\\color{red}{2}}{3\\cdot\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerators and denominators.<\/td>\n<td>[latex]\\frac{3}{6}+\\frac{2}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Remember, always check to see if the answer can be simplified. Since [latex]5[\/latex] and [latex]6[\/latex] have no common factors (other than 1), the fraction [latex]\\frac{5}{6}[\/latex] cannot be simplified.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146262\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146262&theme=oea&iframe_resize_id=ohm146262&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples and explanation about how to add two fractions with unlike denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146264\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146264&theme=oea&iframe_resize_id=ohm146264&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add: [latex]\\frac{7}{12}+\\frac{5}{18}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q826911\">Show Solution<\/span><\/p>\n<div id=\"q826911\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468285390\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 7 over 12 plus 5 over 18. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{7}{12}+\\frac{5}{18}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the LCD of [latex]12[\/latex] and [latex]18[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221043\/CNX_BMath_Figure_04_05_031_img-01.png\" alt=\".\" width=\"146\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\n<td>[latex]\\frac{7\\cdot\\color{red}{3}}{12\\cdot\\color{red}{3}} +\\frac{5\\cdot\\color{red}{2}}{18\\cdot\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerators and denominators.<\/td>\n<td>[latex]\\frac{21}{36}+\\frac{10}{36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\frac{31}{36}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Because [latex]31[\/latex] is a prime number, it has no factors in common with [latex]36[\/latex]. The answer is simplified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146265\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146265&theme=oea&iframe_resize_id=ohm146265&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\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]\\frac{3}{4}+\\frac{1}{6}+\\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 4, 6, and 8.<\/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 24.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3}{4}\\cdot\\frac{6}{6}=\\frac{18}{24}\\\\\\\\\\frac{1}{6}\\cdot\\frac{4}{4}=\\frac{4}{24}\\\\\\\\\\frac{5}{8}\\cdot\\frac{3}{3}=\\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]\\frac{18}{24}+\\frac{4}{24}+\\frac{15}{24}=\\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]\\frac{37}{24}=1\\,\\,\\frac{13}{24}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{3}{4}+\\frac{1}{6}+\\frac{5}{8}=1\\frac{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=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract: [latex]\\frac{7}{15}-\\frac{19}{24}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169999\">Show Solution<\/span><\/p>\n<div id=\"q169999\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466319808\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 7 over 15 minus 19 over 24. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{7}{15}-\\frac{19}{24}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the LCD.<\/p>\n<p>&nbsp;<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221046\/CNX_BMath_Figure_04_05_032_img-01.png\" alt=\".\" width=\"170\" height=\"80\" \/><\/p>\n<p>[latex]15[\/latex] is &#8216;missing&#8217; three factors of [latex]2[\/latex]<\/p>\n<p>[latex]24[\/latex] is &#8216;missing&#8217; a factor of [latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\n<td>[latex]\\frac{7\\cdot\\color{red}{8}}{15\\cdot\\color{red}{8}} -\\frac{19\\cdot\\color{red}{5}}{24\\cdot\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each numerator and denominator.<\/td>\n<td>[latex]\\frac{56}{120}-\\frac{95}{120}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-\\frac{39}{120}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing the common factor of [latex]3[\/latex].<\/td>\n<td>[latex]-\\frac{13\\cdot 3}{40\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factor to simplify.<\/td>\n<td>[latex]-\\frac{13}{40}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146266\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146266&theme=oea&iframe_resize_id=ohm146266&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides two more examples of how to subtract two fractions with unlike denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Example:  Subtract Fractions with Unlike Denominators\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aXlkygPPzQ8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add: [latex]-\\frac{11}{30}+\\frac{23}{42}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q808641\">Show Solution<\/span><\/p>\n<div id=\"q808641\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466855270\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says negative 11 over 30 plus 23 over 42. The next line says,\">\n<tbody>\n<tr>\n<td>[latex]-\\frac{11}{30}+\\frac{23}{42}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the LCD.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221049\/CNX_BMath_Figure_04_05_033_img-01.png\" alt=\".\" width=\"151\" height=\"82\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD.<\/td>\n<td>[latex]-\\frac{11\\cdot\\color{red}{7}}{30\\cdot\\color{red}{7}} +\\frac{23\\cdot\\color{red}{5}}{42\\cdot\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each numerator and denominator.<\/td>\n<td>[latex]-\\frac{77}{210}+\\frac{115}{210}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\frac{38}{210}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing the common factor of [latex]2[\/latex].<\/td>\n<td>[latex]\\frac{19\\cdot 2}{105\\cdot 2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factor to simplify.<\/td>\n<td>[latex]\\frac{19}{105}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146267\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146267&theme=oea&iframe_resize_id=ohm146267&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract: [latex]\\frac{1}{2}-\\left(-\\frac{1}{4}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841760\">Show Solution<\/span><\/p>\n<div id=\"q841760\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466426391\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 1 half minus negative 1 fourth. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{1}{2}-\\left(-\\frac{1}{4}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the LCD of [latex]2[\/latex] and [latex]4[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221041\/CNX_BMath_Figure_04_05_030_img-01.png\" alt=\".\" width=\"100\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions using the LCD [latex]4[\/latex].<\/td>\n<td>[latex]\\frac{1\\cdot\\color{red}{2}}{2\\cdot\\color{red}{2}} - (--\\frac{1}{4})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the first fraction.<\/td>\n<td>[latex]\\frac{2}{4}-\\left(-\\frac{1}{4}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]\\frac{2-\\left(-1\\right)}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>One of the fractions already had the least common denominator, so we only had to convert the other fraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1410\">\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>Add Fractions with Variables and Common Denominators. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/V6N0ZYB6Pu8\">https:\/\/youtu.be\/V6N0ZYB6Pu8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 146178, 146182, 146183, 146185, 146186, 146187, 146252, 146251, 146255146262, 146264, 146265, 146266, 146267, 146268. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Determine the Least Common Denominator of Two Fractions (Column Method). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JsHF9CW_SUM\">https:\/\/youtu.be\/JsHF9CW_SUM<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Add Fractions with Like Denominators. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GTkY34kl6Kw\">https:\/\/youtu.be\/GTkY34kl6Kw<\/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>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/i5XHfiQ2Fe0\">https:\/\/youtu.be\/i5XHfiQ2Fe0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Ex: Add Fractions with Unlike Denominators (Basic with Model). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <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>Example: Subtract Fractions with Unlike Denominators. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aXlkygPPzQ8\">https:\/\/youtu.be\/aXlkygPPzQ8<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Adapted and revised: Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t 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