{"id":626,"date":"2021-08-11T15:45:52","date_gmt":"2021-08-11T15:45:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=626"},"modified":"2022-01-27T15:46:59","modified_gmt":"2022-01-27T15:46:59","slug":"operations-of-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/operations-of-fractions\/","title":{"raw":"Operations of Fractions","rendered":"Operations of 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>Multiply fractions<\/li>\r\n \t<li>Divide fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\nProbabilities are often stated as fractions. Sometimes we are interested in probabilities involving several events. The methods used to calculate these probabilities will require addition, subtraction, multiplication and division. This section will remind you how to do these 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\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Add Fractions<\/span>\r\n\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\nSuppose a pizza has been cut into 4 equal slices, and one slice of a pizza has been eaten leaving 3 slices. The fraction [latex]\\frac{3}{4}[\/latex] represents how much of a whole pizza we have left.\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\nSuppose you have another pizza that had been cut into 8 equal parts and 3 of those parts were gone, leaving [latex]\\frac{5}{8}[\/latex]. To describe the total amount of pizza remaining as a single fraction, we need a common denominator. The lowest common denominator is the <strong>least common multiple (LCM)<\/strong> of the two original denominators. Finding the least common multiple of a list of numbers using prime factorizations was described in a previous section. 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\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd [latex]\\frac{5}{12}+\\frac{1}{12}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"230560\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"230560\"]\r\n\r\nEach of the fractions has the same denominator, 12.\u00a0 So we can add the numerators and place the sum over the common denominator.\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{12}+\\frac{1}{12}=\\frac{5+1}{12}=\\frac{6}{12}[\/latex]<\/p>\r\nThe numerator and denominator have a common factor of 6, so we need to simplify our answer.\r\n<p style=\"text-align: center;\">[latex]\\frac{6}{12} = \\frac{6 \\cdot 1}{6 \\cdot 2} = \\frac{1}{2}[\/latex]<\/p>\r\nTherefore, [latex]\\frac{5}{12} + \\frac{1}{12} = \\frac{1}{2}[\/latex]\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\nIn the next example you are shown how to add two fractions with different denominators, then simplify the answer.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd [latex]\\frac{1}{3}+\\frac{1}{2}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"232880\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"232880\"]\r\n\r\nEach of the denominators, 2 and 3, are prime numbers and cannot be factored. Then the least common multiple is [latex]2 \\cdot 3=6[\/latex].\u00a0 We use the Equivalent Fractions Property to write each fraction as an equivalent fraction with our common denominator, 6.\r\n\r\n[latex]\\Large \\frac{1}{3} + \\frac{1}{2} = \\frac{1 \\cdot 2}{3 \\cdot 2} + \\frac{1 \\cdot 3}{2 \\cdot 3} = \\frac{2}{6} + \\frac{3}{6} = \\frac{2+3}{6} = \\frac{5}{6}[\/latex]\r\n\r\nSince 5 and 6 do not have any common factors, our answer is in simplest form.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhile any common multiple of the two denominators can be used, the least common multiple is the easiest.\r\n\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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]624[\/ohm_question]\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. 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 [latex]\\frac{5}{6} - \\frac{2}{9}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"234928\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"234928\"]\r\n\r\nThe lowest common denominator is the least common multiple of 6 and 4.\r\n<p style=\"text-align: center;\">[latex]6=2 \\cdot 3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]9=3 \\cdot 3[\/latex]<\/p>\r\nThe LCM of 6 and 9 is [latex]2 \\cdot 3 \\cdot 3 = 18[\/latex].\r\n\r\nWrite each of the original fractions as an equivalent fraction with the lowest common denominator, 18.\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{6} = \\frac{5 \\cdot 3}{6 \\cdot 3 } = \\frac{15}{18}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{9} = \\frac{2 \\cdot 2}{9 \\cdot 2} = \\frac{4}{18}[\/latex]<\/p>\r\nNow, subtract numerators and place the difference over the common denominator.\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{6} - \\frac{2}{9} = \\frac{15}{18} - \\frac{4}{18} = \\frac{11}{18}[\/latex].<\/p>\r\nSince 11 and 18 have no common factors, our solution is in simplest form.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can perform additions and subtractions on three or more fractions, as seen in the example below.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract [latex]\\frac{1}{6} + \\frac{1}{4} - \\frac{1}{12}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"368982\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"368982\"]\r\n\r\nThe lowest common denominator is the least common multiple of 6 and 4.\r\n<p style=\"text-align: center;\">[latex]6=2 \\cdot 3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]4=2 \\cdot 2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]12 = 2 \\cdot 2 \\cdot 3[\/latex]<\/p>\r\nThe LCM of 6, 4, and 12 is [latex]2 \\cdot 2 \\cdot 3 = 12[\/latex].\r\n\r\nWrite each of the original fractions as an equivalent fraction with the lowest common denominator, 12.\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{6} = \\frac{1 \\cdot 2}{6 \\cdot 2} = \\frac{2}{12}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} = \\frac{1 \\cdot 3}{4 \\cdot 3} = \\frac{3}{12}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{12}[\/latex]<\/p>\r\nTherefore,\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{6} + \\frac{1}{4} - \\frac{1}{12} = \\frac{2}{12} + \\frac{3}{12} - \\frac{1}{12} = \\frac{5}{12} - \\frac{1}{12} = \\frac{4}{12}[\/latex].<\/p>\r\nSince 4 and 12 have a common factor of 4, our answer must be simplified.\r\n<p style=\"text-align: center;\">[latex]\\frac{4}{12} = \\frac{4 \\cdot 1}{4 \\cdot 3} = \\frac{1}{3}[\/latex].<\/p>\r\nOur answer, in simplest form, is [latex]\\frac{1}{3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]629[\/ohm_question]\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\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 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=\"363710\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"363710\"]\r\n\r\nThe product of two fractions is the product of the numerators over the product of the denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{4}{5} = \\frac{2 \\cdot 4}{3 \\cdot 5} = \\frac{8}{15}[\/latex]<\/p>\r\nSince 8 and 15 have no common factors, our answer, [latex]\\frac{8}{15}[\/latex] is in simplest form.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf 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<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{3}{8}[\/latex]\r\n\r\n[reveal-answer q=\"181907\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"181907\"]\r\n\r\nThe product of two fractions is the product of the numerators over the product of the denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{3}{8} = \\frac{2 \\cdot 3}{3 \\cdot 8}[\/latex]<\/p>\r\nBefore performing the multiplications, remember that you can remove any common factors between the numerator and denominator. 2 and 8 each have factors of 2, so write each of 2 and 8 as a product in which one of the factors is 2.\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{3}{8} = \\frac{2 \\cdot 3}{3 \\cdot 8} = \\frac{2 \\cdot 1 \\cdot 3}{3 \\cdot 2 \\cdot 4}[\/latex]<\/p>\r\nRemoving the common factors of 2 and 3 our simplified answer is [latex]\\frac{1}{4}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]533[\/ohm_question]\r\n\r\n<\/div>\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\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<\/ul>\r\nNote that for [latex]a \\neq 0[\/latex] and [latex]b \\neq 0[\/latex], [latex]\\frac{a}{b} \\cdot \\frac{b}{a} = \\frac{a \\cdot b}{b \\cdot a} = \\frac{a \\cdot b}{a \\cdot b} = 1[\/latex].\u00a0The reciprocal of a fraction can be found by interchanging the numerator and denominator.\r\n<p style=\"text-align: center;\">The reciprocal of [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex].<\/p>\r\nDividing fractions requires using the reciprocal of a number or fraction. Here are some examples of reciprocals:\r\n<table style=\"width: 473px; height: 84px;\">\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<th style=\"width: 257.797px; height: 14px;\">Original number<\/th>\r\n<th style=\"width: 165.719px; height: 14px;\">Reciprocal<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 257.797px; height: 14px;\">[latex]\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{4}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 257.797px; height: 14px;\">[latex]\\Large\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{2}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 257.797px; height: 14px;\">[latex] 3=\\Large\\frac{3}{1}[\/latex]<\/td>\r\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 257.797px; height: 28px;\">[latex]2\\Large\\frac{1}{3}=\\Large\\frac{7}{3}[\/latex]<\/td>\r\n<td style=\"width: 165.719px; height: 28px;\">[latex]\\Large\\frac{3}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\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<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\n<span style=\"font-size: 1rem; text-align: initial;\">Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?<\/span>\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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]558[\/ohm_question]\r\n\r\n<\/div>\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>Multiply fractions<\/li>\n<li>Divide fractions<\/li>\n<\/ul>\n<\/div>\n<p>Probabilities are often stated as fractions. Sometimes we are interested in probabilities involving several events. The methods used to calculate these probabilities will require addition, subtraction, multiplication and division. This section will remind you how to do these 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><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Add Fractions<\/span><\/p>\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>Suppose a pizza has been cut into 4 equal slices, and one slice of a pizza has been eaten leaving 3 slices. The fraction [latex]\\frac{3}{4}[\/latex] represents how much of a whole pizza we have left.<\/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>Suppose you have another pizza that had been cut into 8 equal parts and 3 of those parts were gone, leaving [latex]\\frac{5}{8}[\/latex]. To describe the total amount of pizza remaining as a single fraction, we need a common denominator. The lowest common denominator is the <strong>least common multiple (LCM)<\/strong> of the two original denominators. Finding the least common multiple of a list of numbers using prime factorizations was described in a previous section. 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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add [latex]\\frac{5}{12}+\\frac{1}{12}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q230560\">Show Answer<\/span><\/p>\n<div id=\"q230560\" class=\"hidden-answer\" style=\"display: none\">\n<p>Each of the fractions has the same denominator, 12.\u00a0 So we can add the numerators and place the sum over the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{12}+\\frac{1}{12}=\\frac{5+1}{12}=\\frac{6}{12}[\/latex]<\/p>\n<p>The numerator and denominator have a common factor of 6, so we need to simplify our answer.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{6}{12} = \\frac{6 \\cdot 1}{6 \\cdot 2} = \\frac{1}{2}[\/latex]<\/p>\n<p>Therefore, [latex]\\frac{5}{12} + \\frac{1}{12} = \\frac{1}{2}[\/latex]<\/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<p>In the next example you are shown how to add two fractions with different denominators, then simplify the answer.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add [latex]\\frac{1}{3}+\\frac{1}{2}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232880\">Show Answer<\/span><\/p>\n<div id=\"q232880\" class=\"hidden-answer\" style=\"display: none\">\n<p>Each of the denominators, 2 and 3, are prime numbers and cannot be factored. Then the least common multiple is [latex]2 \\cdot 3=6[\/latex].\u00a0 We use the Equivalent Fractions Property to write each fraction as an equivalent fraction with our common denominator, 6.<\/p>\n<p>[latex]\\Large \\frac{1}{3} + \\frac{1}{2} = \\frac{1 \\cdot 2}{3 \\cdot 2} + \\frac{1 \\cdot 3}{2 \\cdot 3} = \\frac{2}{6} + \\frac{3}{6} = \\frac{2+3}{6} = \\frac{5}{6}[\/latex]<\/p>\n<p>Since 5 and 6 do not have any common factors, our answer is in simplest form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>While any common multiple of the two denominators can be used, the least common multiple is the easiest.<\/p>\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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm624\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=624&theme=oea&iframe_resize_id=ohm624&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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. 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 [latex]\\frac{5}{6} - \\frac{2}{9}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q234928\">Show Answer<\/span><\/p>\n<div id=\"q234928\" class=\"hidden-answer\" style=\"display: none\">\n<p>The lowest common denominator is the least common multiple of 6 and 4.<\/p>\n<p style=\"text-align: center;\">[latex]6=2 \\cdot 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]9=3 \\cdot 3[\/latex]<\/p>\n<p>The LCM of 6 and 9 is [latex]2 \\cdot 3 \\cdot 3 = 18[\/latex].<\/p>\n<p>Write each of the original fractions as an equivalent fraction with the lowest common denominator, 18.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{6} = \\frac{5 \\cdot 3}{6 \\cdot 3 } = \\frac{15}{18}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{9} = \\frac{2 \\cdot 2}{9 \\cdot 2} = \\frac{4}{18}[\/latex]<\/p>\n<p>Now, subtract numerators and place the difference over the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{6} - \\frac{2}{9} = \\frac{15}{18} - \\frac{4}{18} = \\frac{11}{18}[\/latex].<\/p>\n<p>Since 11 and 18 have no common factors, our solution is in simplest form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can perform additions and subtractions on three or more fractions, as seen in the example below.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract [latex]\\frac{1}{6} + \\frac{1}{4} - \\frac{1}{12}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368982\">Show Answer<\/span><\/p>\n<div id=\"q368982\" class=\"hidden-answer\" style=\"display: none\">\n<p>The lowest common denominator is the least common multiple of 6 and 4.<\/p>\n<p style=\"text-align: center;\">[latex]6=2 \\cdot 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]4=2 \\cdot 2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]12 = 2 \\cdot 2 \\cdot 3[\/latex]<\/p>\n<p>The LCM of 6, 4, and 12 is [latex]2 \\cdot 2 \\cdot 3 = 12[\/latex].<\/p>\n<p>Write each of the original fractions as an equivalent fraction with the lowest common denominator, 12.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{6} = \\frac{1 \\cdot 2}{6 \\cdot 2} = \\frac{2}{12}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} = \\frac{1 \\cdot 3}{4 \\cdot 3} = \\frac{3}{12}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{12}[\/latex]<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{6} + \\frac{1}{4} - \\frac{1}{12} = \\frac{2}{12} + \\frac{3}{12} - \\frac{1}{12} = \\frac{5}{12} - \\frac{1}{12} = \\frac{4}{12}[\/latex].<\/p>\n<p>Since 4 and 12 have a common factor of 4, our answer must be simplified.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4}{12} = \\frac{4 \\cdot 1}{4 \\cdot 3} = \\frac{1}{3}[\/latex].<\/p>\n<p>Our answer, in simplest form, is [latex]\\frac{1}{3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm629\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=629&theme=oea&iframe_resize_id=ohm629&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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>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 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=\"q363710\">Show Answer<\/span><\/p>\n<div id=\"q363710\" class=\"hidden-answer\" style=\"display: none\">\n<p>The product of two fractions is the product of the numerators over the product of the denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{4}{5} = \\frac{2 \\cdot 4}{3 \\cdot 5} = \\frac{8}{15}[\/latex]<\/p>\n<p>Since 8 and 15 have no common factors, our answer, [latex]\\frac{8}{15}[\/latex] is in simplest form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{3}{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q181907\">Show Answer<\/span><\/p>\n<div id=\"q181907\" class=\"hidden-answer\" style=\"display: none\">\n<p>The product of two fractions is the product of the numerators over the product of the denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{3}{8} = \\frac{2 \\cdot 3}{3 \\cdot 8}[\/latex]<\/p>\n<p>Before performing the multiplications, remember that you can remove any common factors between the numerator and denominator. 2 and 8 each have factors of 2, so write each of 2 and 8 as a product in which one of the factors is 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\cdot \\frac{3}{8} = \\frac{2 \\cdot 3}{3 \\cdot 8} = \\frac{2 \\cdot 1 \\cdot 3}{3 \\cdot 2 \\cdot 4}[\/latex]<\/p>\n<p>Removing the common factors of 2 and 3 our simplified answer is [latex]\\frac{1}{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm533\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=533&theme=oea&iframe_resize_id=ohm533&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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<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<\/ul>\n<p>Note that for [latex]a \\neq 0[\/latex] and [latex]b \\neq 0[\/latex], [latex]\\frac{a}{b} \\cdot \\frac{b}{a} = \\frac{a \\cdot b}{b \\cdot a} = \\frac{a \\cdot b}{a \\cdot b} = 1[\/latex].\u00a0The reciprocal of a fraction can be found by interchanging the numerator and denominator.<\/p>\n<p style=\"text-align: center;\">The reciprocal of [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex].<\/p>\n<p>Dividing fractions requires using the reciprocal of a number or fraction. Here are some examples of reciprocals:<\/p>\n<table style=\"width: 473px; height: 84px;\">\n<thead>\n<tr style=\"height: 14px;\">\n<th style=\"width: 257.797px; height: 14px;\">Original number<\/th>\n<th style=\"width: 165.719px; height: 14px;\">Reciprocal<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 257.797px; height: 14px;\">[latex]\\Large\\frac{3}{4}[\/latex]<\/td>\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{4}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 257.797px; height: 14px;\">[latex]\\Large\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{2}{1}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 257.797px; height: 14px;\">[latex]3=\\Large\\frac{3}{1}[\/latex]<\/td>\n<td style=\"width: 165.719px; height: 14px;\">[latex]\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 257.797px; height: 28px;\">[latex]2\\Large\\frac{1}{3}=\\Large\\frac{7}{3}[\/latex]<\/td>\n<td style=\"width: 165.719px; height: 28px;\">[latex]\\Large\\frac{3}{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/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<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><span style=\"font-size: 1rem; text-align: initial;\">Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?<\/span><\/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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm558\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=558&theme=oea&iframe_resize_id=ohm558&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-626\">\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><li>Question ID: 624, 629, 533, 558. <strong>Authored by<\/strong>: Lippman, D. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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>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><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> 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":169134,"menu_order":5,"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 Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaption\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Add Fractions with Unlike Denominators (Basic with Model)\",\"author\":\"James Sousa (Mathispower4u)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/zV4q7j1-89I\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\" Ex 1: Multiply Fractions (Basic)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/f_L-EFC8Z7c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Diviide Fractions (Basic)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/F5YSNLel3n8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 624, 629, 533, 558\",\"author\":\"Lippman, D\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-626","chapter","type-chapter","status-publish","hentry"],"part":43,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/626","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/626\/revisions"}],"predecessor-version":[{"id":3508,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/626\/revisions\/3508"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/43"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/626\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=626"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=626"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=626"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=626"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}