{"id":4482,"date":"2020-04-13T14:27:31","date_gmt":"2020-04-13T14:27:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4482"},"modified":"2021-02-06T00:04:04","modified_gmt":"2021-02-06T00:04:04","slug":"writing-proportions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/writing-proportions\/","title":{"raw":"Writing Proportions","rendered":"Writing Proportions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given a statement, write a proportion<\/li>\r\n \t<li>Given an equation, determine whether it is a proportion<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Proportion<\/h3>\r\nA proportion is an equation of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex].\r\nThe proportion states two ratios or rates are equal. The proportion is read [latex]\\text{\"}a[\/latex] is to [latex]b[\/latex], as [latex]c[\/latex] is to [latex]d\\text{\".}[\/latex]\r\n\r\n<\/div>\r\nThe equation [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] is a proportion because the two fractions are equal. The proportion [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] is read \"[latex]1[\/latex] is to [latex]2[\/latex] as [latex]4[\/latex] is to [latex]8[\/latex]\".\r\n\r\nIf we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion [latex]{\\Large\\frac{\\text{20 students}}{\\text{1 teacher}}}={\\Large\\frac{\\text{60 students}}{\\text{3 teachers}}}[\/latex] we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite each sentence as a proportion:\r\n\r\n1. [latex]3[\/latex] is to [latex]7[\/latex] as [latex]15[\/latex] is to [latex]35[\/latex].\r\n2. [latex]5[\/latex] hits in [latex]8[\/latex] at bats is the same as [latex]30[\/latex] hits in [latex]48[\/latex] at-bats.\r\n3. [latex]\\text{\\$1.50}[\/latex] for [latex]6[\/latex] ounces is equivalent to [latex]\\text{\\$2.25}[\/latex] for [latex]9[\/latex] ounces.\r\n\r\nSolution\r\n<table id=\"eip-id1168468496922\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex] is to [latex]7[\/latex] as [latex]15[\/latex] is to [latex]35[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a proportion.<\/td>\r\n<td>[latex]{\\Large\\frac{3}{7}}={\\Large\\frac{15}{35}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047457950\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex] hits in [latex]8[\/latex] at-bats is the same as [latex]30[\/latex] hits in [latex]48[\/latex] at-bats.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write each fraction to compare hits to at-bats.<\/td>\r\n<td>[latex]{\\Large\\frac{\\text{hits}}{\\text{at-bats}}}={\\Large\\frac{\\text{hits}}{\\text{at-bats}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a proportion.<\/td>\r\n<td>[latex]{\\Large\\frac{5}{8}}={\\Large\\frac{30}{48}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047661662\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{\\$1.50}[\/latex] for [latex]6[\/latex] ounces is equivalent to [latex]\\text{\\$2.25}[\/latex] for [latex]9[\/latex] ounces.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write each fraction to compare dollars to ounces.<\/td>\r\n<td>[latex]{\\Large\\frac{$}{\\text{ounces}}}={\\Large\\frac{$}{\\text{ounces}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a proportion.<\/td>\r\n<td>[latex]{\\Large\\frac{1.50}{6}}={\\Large\\frac{2.25}{9}}[\/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&nbsp;\r\n\r\n[ohm_question]146798[\/ohm_question]\r\n\r\n[ohm_question]146799[\/ohm_question]\r\n\r\n<\/div>\r\nLook at the proportions [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] and [latex]{\\Large\\frac{2}{3}}={\\Large\\frac{6}{9}}[\/latex]. From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?\r\n\r\nTo determine if a proportion is true, we find the <strong>cross products<\/strong> of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222123\/CNX_BMath_Figure_06_05_028_img.png\" alt=\"The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 \u00b7 1 = 8 and 2 \u00b7 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 \u00b7 2 = 18 and 3 \u00b7 6 = 18.\" \/>\r\n\r\nCross Products of a Proportion\r\n\r\nFor any proportion of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex], its cross products are equal.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222125\/CNX_BMath_Figure_06_05_003_img.png\" alt=\"No Alt Text\" \/>\r\n\r\nCross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether each equation is a proportion:\r\n\r\n1.\u00a0 [latex]{\\Large\\frac{4}{9}}={\\Large\\frac{12}{28}}[\/latex]\r\n2.\u00a0 [latex]{\\Large\\frac{17.5}{37.5}}={\\Large\\frac{7}{15}}[\/latex]\r\n[reveal-answer q=\"586373\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"586373\"]\r\n\r\nSolution\r\nTo determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.\r\n<table id=\"eip-id1168468703669\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 4 is to 9 as 12 is to 28. The cross multiplication shown is 28 times 4 = 112 and 9 times 12 = 28.\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{\\Large\\frac{4}{9}}={\\Large\\frac{12}{28}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the cross products.<\/td>\r\n<td>[latex]28\\cdot 4=1129\\cdot 12=108[\/latex]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222127\/CNX_BMath_Figure_06_05_020_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the cross products are not equal, [latex]28\\cdot 4\\ne 9\\cdot 12[\/latex], the equation is not a proportion.\r\n<table id=\"eip-id1168469699712\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 17.5 is to 37.5 as 7 is to 15. The cross multiplication shown is 15 times 17.5 = 262.5 and 37.5 times 7 = 262.5.\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{\\Large\\frac{17.5}{37.5}}={\\Large\\frac{7}{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the cross products.<\/td>\r\n<td>[latex]15\\cdot 17.5=262.5[\/latex]\r\n\r\n[latex]37.5\\cdot 7=262.5[\/latex]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222130\/CNX_BMath_Figure_06_05_021_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the cross products are equal, [latex]15\\cdot 17.5=37.5\\cdot 7[\/latex], the equation is a proportion.\r\n\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]146809[\/ohm_question]\r\n\r\n<\/div>\r\n<p><span style=\"color: #ff0000\">\u00a0<\/span><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Given a statement, write a proportion<\/li>\n<li>Given an equation, determine whether it is a proportion<\/li>\n<\/ul>\n<\/div>\n<p>In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.<\/p>\n<div class=\"textbox shaded\">\n<h3>Proportion<\/h3>\n<p>A proportion is an equation of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex].<br \/>\nThe proportion states two ratios or rates are equal. The proportion is read [latex]\\text{\"}a[\/latex] is to [latex]b[\/latex], as [latex]c[\/latex] is to [latex]d\\text{\".}[\/latex]<\/p>\n<\/div>\n<p>The equation [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] is a proportion because the two fractions are equal. The proportion [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] is read &#8220;[latex]1[\/latex] is to [latex]2[\/latex] as [latex]4[\/latex] is to [latex]8[\/latex]&#8220;.<\/p>\n<p>If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion [latex]{\\Large\\frac{\\text{20 students}}{\\text{1 teacher}}}={\\Large\\frac{\\text{60 students}}{\\text{3 teachers}}}[\/latex] we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write each sentence as a proportion:<\/p>\n<p>1. [latex]3[\/latex] is to [latex]7[\/latex] as [latex]15[\/latex] is to [latex]35[\/latex].<br \/>\n2. [latex]5[\/latex] hits in [latex]8[\/latex] at bats is the same as [latex]30[\/latex] hits in [latex]48[\/latex] at-bats.<br \/>\n3. [latex]\\text{\\$1.50}[\/latex] for [latex]6[\/latex] ounces is equivalent to [latex]\\text{\\$2.25}[\/latex] for [latex]9[\/latex] ounces.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468496922\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex] is to [latex]7[\/latex] as [latex]15[\/latex] is to [latex]35[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Write as a proportion.<\/td>\n<td>[latex]{\\Large\\frac{3}{7}}={\\Large\\frac{15}{35}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047457950\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex] hits in [latex]8[\/latex] at-bats is the same as [latex]30[\/latex] hits in [latex]48[\/latex] at-bats.<\/td>\n<\/tr>\n<tr>\n<td>Write each fraction to compare hits to at-bats.<\/td>\n<td>[latex]{\\Large\\frac{\\text{hits}}{\\text{at-bats}}}={\\Large\\frac{\\text{hits}}{\\text{at-bats}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a proportion.<\/td>\n<td>[latex]{\\Large\\frac{5}{8}}={\\Large\\frac{30}{48}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047661662\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{\\$1.50}[\/latex] for [latex]6[\/latex] ounces is equivalent to [latex]\\text{\\$2.25}[\/latex] for [latex]9[\/latex] ounces.<\/td>\n<\/tr>\n<tr>\n<td>Write each fraction to compare dollars to ounces.<\/td>\n<td>[latex]{\\Large\\frac{$}{\\text{ounces}}}={\\Large\\frac{$}{\\text{ounces}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a proportion.<\/td>\n<td>[latex]{\\Large\\frac{1.50}{6}}={\\Large\\frac{2.25}{9}}[\/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>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146798\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146798&theme=oea&iframe_resize_id=ohm146798&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146799\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146799&theme=oea&iframe_resize_id=ohm146799&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Look at the proportions [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{4}{8}}[\/latex] and [latex]{\\Large\\frac{2}{3}}={\\Large\\frac{6}{9}}[\/latex]. From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?<\/p>\n<p>To determine if a proportion is true, we find the <strong>cross products<\/strong> of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.<\/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\/24222123\/CNX_BMath_Figure_06_05_028_img.png\" alt=\"The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 \u00b7 1 = 8 and 2 \u00b7 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 \u00b7 2 = 18 and 3 \u00b7 6 = 18.\" \/><\/p>\n<p>Cross Products of a Proportion<\/p>\n<p>For any proportion of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex], its cross products are equal.<\/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\/24222125\/CNX_BMath_Figure_06_05_003_img.png\" alt=\"No Alt Text\" \/><\/p>\n<p>Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine whether each equation is a proportion:<\/p>\n<p>1.\u00a0 [latex]{\\Large\\frac{4}{9}}={\\Large\\frac{12}{28}}[\/latex]<br \/>\n2.\u00a0 [latex]{\\Large\\frac{17.5}{37.5}}={\\Large\\frac{7}{15}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q586373\">Show Solution<\/span><\/p>\n<div id=\"q586373\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.<\/p>\n<table id=\"eip-id1168468703669\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 4 is to 9 as 12 is to 28. The cross multiplication shown is 28 times 4 = 112 and 9 times 12 = 28.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]{\\Large\\frac{4}{9}}={\\Large\\frac{12}{28}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the cross products.<\/td>\n<td>[latex]28\\cdot 4=1129\\cdot 12=108[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222127\/CNX_BMath_Figure_06_05_020_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the cross products are not equal, [latex]28\\cdot 4\\ne 9\\cdot 12[\/latex], the equation is not a proportion.<\/p>\n<table id=\"eip-id1168469699712\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 17.5 is to 37.5 as 7 is to 15. The cross multiplication shown is 15 times 17.5 = 262.5 and 37.5 times 7 = 262.5.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td>[latex]{\\Large\\frac{17.5}{37.5}}={\\Large\\frac{7}{15}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the cross products.<\/td>\n<td>[latex]15\\cdot 17.5=262.5[\/latex]<\/p>\n<p>[latex]37.5\\cdot 7=262.5[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222130\/CNX_BMath_Figure_06_05_021_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the cross products are equal, [latex]15\\cdot 17.5=37.5\\cdot 7[\/latex], the equation is a proportion.<\/p>\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=\"ohm146809\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146809&theme=oea&iframe_resize_id=ohm146809&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #ff0000\">\u00a0<\/span><\/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-4482\">\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 146809, 146808, 146807. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>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 <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146809, 146808, 146807\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4482","chapter","type-chapter","status-web-only","hentry"],"part":3698,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4482","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4482\/revisions"}],"predecessor-version":[{"id":4483,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4482\/revisions\/4483"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/3698"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4482\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4482"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4482"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4482"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}