{"id":621,"date":"2021-08-11T14:57:05","date_gmt":"2021-08-11T14:57:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=621"},"modified":"2022-01-24T21:23:50","modified_gmt":"2022-01-24T21:23:50","slug":"simplify-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/simplify-fractions\/","title":{"raw":"Simplify Fractions","rendered":"Simplify Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify fractions by finding common factors between the numerator and denominator<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplify Fractions<\/h2>\r\n<img class=\"alignright wp-image-622 size-medium\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/08\/11145443\/pexels-sarah-chai-7262788-200x300.jpg\" alt=\"There are a red and a orange pepper in a sack and lemons in a second sack on a couch.\" width=\"200\" height=\"300\" \/>\r\n\r\nThere are many ways to write fractions that have the same value, or represent the same part of the whole. For example, suppose a bag contains 8 peppers, of which 3 are yellow, 4 are green, and 1 is red. Without looking, we reach in and randomly select a pepper. To find the chance we select a green pepper we need the fraction of green peppers in the bag.\u00a0 Since 4 out of the 3+4+1=8 peppers are green, the fraction of green peppers is [latex]\\frac{4}{8}[\/latex]. But 4 is half of 8, so we could also say the fraction of green peppers is [latex]\\frac{1}{2}[\/latex].\r\n\r\nHow do you know which one to use? Often, we\u2019ll use the fraction that is in simplified form.\r\n\r\nA fraction is considered <strong>simplified<\/strong> if there are no common factors, other than 1, in the <strong>numerator<\/strong> and <strong>denominator<\/strong>. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplified Fraction<\/h3>\r\nA fraction is considered simplified if there are no common factors in the numerator and denominator.\r\n\r\n<\/div>\r\nFor example,\r\n<ul id=\"fs-id1302300\">\r\n \t<li>[latex]\\Large\\frac{2}{3}[\/latex] is simplified because there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\r\n \t<li>[latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\r\n<\/ul>\r\nThe process of simplifying a fraction is often called <strong>reducing the fraction<\/strong><em>.<\/em>\u00a0We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex].\r\n\r\n<\/div>\r\nNotice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.\u00a0Sometimes we say we cancel the common factor, [latex]c[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Simplify a fraction<\/h3>\r\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\r\n \t<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\r\n \t<li>Simplify, using the equivalent fractions property, by removing common factors.<\/li>\r\n \t<li>Multiply any remaining factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]\\Large\\frac{10}{15}[\/latex]\r\n\r\nSolution:\r\nTo simplify the fraction, we look for any common factors in the numerator and the denominator.\r\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{10}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the numerator and denominator.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factors.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{2}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question height=\"400\"]146014[\/ohm_question]\r\n\r\n<\/div>\r\nTo simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]\\Large-\\frac{18}{24}[\/latex]\r\n[reveal-answer q=\"270732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"270732\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\r\n<tbody>\r\n<tr>\r\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\r\n<td>[latex]\\Large-\\frac{18}{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large-\\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"400\"]146015[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to simplify a fraction.\r\n\r\nhttps:\/\/youtu.be\/_2Wk7jXf3Ok\r\n\r\nAfter simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify fractions by finding common factors between the numerator and denominator<\/li>\n<\/ul>\n<\/div>\n<h2>Simplify Fractions<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-622 size-medium\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/08\/11145443\/pexels-sarah-chai-7262788-200x300.jpg\" alt=\"There are a red and a orange pepper in a sack and lemons in a second sack on a couch.\" width=\"200\" height=\"300\" \/><\/p>\n<p>There are many ways to write fractions that have the same value, or represent the same part of the whole. For example, suppose a bag contains 8 peppers, of which 3 are yellow, 4 are green, and 1 is red. Without looking, we reach in and randomly select a pepper. To find the chance we select a green pepper we need the fraction of green peppers in the bag.\u00a0 Since 4 out of the 3+4+1=8 peppers are green, the fraction of green peppers is [latex]\\frac{4}{8}[\/latex]. But 4 is half of 8, so we could also say the fraction of green peppers is [latex]\\frac{1}{2}[\/latex].<\/p>\n<p>How do you know which one to use? Often, we\u2019ll use the fraction that is in simplified form.<\/p>\n<p>A fraction is considered <strong>simplified<\/strong> if there are no common factors, other than 1, in the <strong>numerator<\/strong> and <strong>denominator<\/strong>. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplified Fraction<\/h3>\n<p>A fraction is considered simplified if there are no common factors in the numerator and denominator.<\/p>\n<\/div>\n<p>For example,<\/p>\n<ul id=\"fs-id1302300\">\n<li>[latex]\\Large\\frac{2}{3}[\/latex] is simplified because there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\n<li>[latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\n<\/ul>\n<p>The process of simplifying a fraction is often called <strong>reducing the fraction<\/strong><em>.<\/em>\u00a0We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex].<\/p>\n<\/div>\n<p>Notice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.\u00a0Sometimes we say we cancel the common factor, [latex]c[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplify a fraction<\/h3>\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\n<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\n<li>Simplify, using the equivalent fractions property, by removing common factors.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\Large\\frac{10}{15}[\/latex]<\/p>\n<p>Solution:<br \/>\nTo simplify the fraction, we look for any common factors in the numerator and the denominator.<\/p>\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\n<td>[latex]\\Large\\frac{10}{15}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the numerator and denominator.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factors.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146014\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146014&theme=oea&iframe_resize_id=ohm146014&show_question_numbers\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\Large-\\frac{18}{24}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q270732\">Show Solution<\/span><\/p>\n<div id=\"q270732\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\n<tbody>\n<tr>\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\n<td>[latex]\\Large-\\frac{18}{24}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large-\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146015\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146015&theme=oea&iframe_resize_id=ohm146015&show_question_numbers\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see another example of how to simplify a fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Simplify Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_2Wk7jXf3Ok?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.<\/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-621\">\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>Question ID: 146014, 146015. <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>Ex 1: Simplify Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_2Wk7jXf3Ok\">https:\/\/youtu.be\/_2Wk7jXf3Ok<\/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>Prealgebra. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">Public domain content<\/div><ul class=\"citation-list\"><li>Lemons and bell peppers on couch. <strong>Authored by<\/strong>: Sarah Chai. <strong>Provided by<\/strong>: Pexels. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.pexels.com\/photo\/lemons-and-bell-peppers-on-couch-7262788\/\">https:\/\/www.pexels.com\/photo\/lemons-and-bell-peppers-on-couch-7262788\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/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":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"pd\",\"description\":\"Lemons and bell peppers on couch\",\"author\":\"Sarah 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