{"id":584,"date":"2025-10-09T15:23:38","date_gmt":"2025-10-09T15:23:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=584"},"modified":"2025-11-20T15:35:45","modified_gmt":"2025-11-20T15:35:45","slug":"introduction-to-multivariable-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-multivariable-functions-background-youll-need-2\/","title":{"raw":"Introduction to Multivariable Functions: Background You'll Need 2","rendered":"Introduction to Multivariable Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Simplify complex fractions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Complex Fractions<\/h2>\r\nA <strong>complex fraction<\/strong> is a fraction where the numerator, denominator, or both contain fractions.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\frac{6}{7}}{ 3}, \\dfrac{\\dfrac{3}{4}}{\\frac{5}{8}}, \\dfrac{\\dfrac{x}{2}}{\\dfrac{5}{6}}[\/latex]<\/p>\r\nThe key to simplifying: remember that the fraction bar means division. So [latex]\\dfrac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex] is the same as [latex]\\dfrac{3}{4} \\div \\dfrac{5}{8}[\/latex].\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to Simplify a Complex Fraction<\/strong>\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Rewrite the complex fraction as a division problem<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow the rules for dividing fractions (multiply by the reciprocal)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify if possible<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{-\\frac{6}{7}}{ 3}[\/latex]\r\n[reveal-answer q=\"652451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"652451\"]Solution:\r\n<table id=\"eip-id1168466330347\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\LARGE\\frac{-\\frac{6}{7}}{ 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as division.<\/td>\r\n<td>[latex]\\Large-\\frac{6}{7}\\normalsize\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large-\\frac{6}{7}\\cdot \\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply; the product will be negative.<\/td>\r\n<td>[latex]\\Large-\\frac{6\\cdot 1}{7\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{\\color{red}{3} \\cdot 2\\cdot 1}{7\\cdot \\color{red}{3} }[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors and simplify.<\/td>\r\n<td>[latex]\\Large-\\frac{2}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex]\r\n[reveal-answer q=\"853762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"853762\"]Solution:\r\n<table id=\"eip-id1168468527734\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as division.<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}\\normalsize\\div\\Large\\frac{5}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}\\cdot \\frac{8}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot 8}{4\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot\\color{red}{4}\\cdot 2}{\\color{red}{4} \\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors and simplify.<\/td>\r\n<td>[latex]\\Large\\frac{6}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]146109[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]146110[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{\\frac{x}{2}}{\\frac{xy}{6}}[\/latex]\r\n[reveal-answer q=\"663304\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"663304\"]Solution:\r\n<table id=\"eip-id1168468401740\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\LARGE\\frac{\\frac{x}{2}}{\\frac{xy}{6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as division.<\/td>\r\n<td>[latex]\\Large\\frac{x}{2}\\div \\frac{xy}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\Large\\frac{x}{2}\\cdot \\frac{6}{xy}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{x\\cdot 6}{2\\cdot xy}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{x}\\cdot 3\\cdot \\color{red}{2}}{\\color{red}{2}\\cdot \\color{red}{x}\\cdot y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors and simplify.<\/td>\r\n<td>[latex]\\Large\\frac{3}{y}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]146112[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Simplify complex fractions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Complex Fractions<\/h2>\n<p>A <strong>complex fraction<\/strong> is a fraction where the numerator, denominator, or both contain fractions.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\frac{6}{7}}{ 3}, \\dfrac{\\dfrac{3}{4}}{\\frac{5}{8}}, \\dfrac{\\dfrac{x}{2}}{\\dfrac{5}{6}}[\/latex]<\/p>\n<p>The key to simplifying: remember that the fraction bar means division. So [latex]\\dfrac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex] is the same as [latex]\\dfrac{3}{4} \\div \\dfrac{5}{8}[\/latex].<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to Simplify a Complex Fraction<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Rewrite the complex fraction as a division problem<\/li>\n<li class=\"whitespace-normal break-words\">Follow the rules for dividing fractions (multiply by the reciprocal)<\/li>\n<li class=\"whitespace-normal break-words\">Simplify if possible<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{-\\frac{6}{7}}{ 3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q652451\">Show Solution<\/button><\/p>\n<div id=\"q652451\" class=\"hidden-answer\" style=\"display: none\">Solution:<\/p>\n<table id=\"eip-id1168466330347\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\LARGE\\frac{-\\frac{6}{7}}{ 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as division.<\/td>\n<td>[latex]\\Large-\\frac{6}{7}\\normalsize\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large-\\frac{6}{7}\\cdot \\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply; the product will be negative.<\/td>\n<td>[latex]\\Large-\\frac{6\\cdot 1}{7\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\Large-\\frac{\\color{red}{3} \\cdot 2\\cdot 1}{7\\cdot \\color{red}{3} }[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors and simplify.<\/td>\n<td>[latex]\\Large-\\frac{2}{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q853762\">Show Solution<\/button><\/p>\n<div id=\"q853762\" class=\"hidden-answer\" style=\"display: none\">Solution:<\/p>\n<table id=\"eip-id1168468527734\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as division.<\/td>\n<td>[latex]\\Large\\frac{3}{4}\\normalsize\\div\\Large\\frac{5}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{3}{4}\\cdot \\frac{8}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot 8}{4\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot\\color{red}{4}\\cdot 2}{\\color{red}{4} \\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors and simplify.<\/td>\n<td>[latex]\\Large\\frac{6}{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm146109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146109&theme=lumen&iframe_resize_id=ohm146109&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm146110\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146110&theme=lumen&iframe_resize_id=ohm146110&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox example\" aria-label=\"Example\">Simplify: [latex]\\LARGE\\frac{\\frac{x}{2}}{\\frac{xy}{6}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q663304\">Show Solution<\/button><\/p>\n<div id=\"q663304\" class=\"hidden-answer\" style=\"display: none\">Solution:<\/p>\n<table id=\"eip-id1168468401740\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\LARGE\\frac{\\frac{x}{2}}{\\frac{xy}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as division.<\/td>\n<td>[latex]\\Large\\frac{x}{2}\\div \\frac{xy}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{x}{2}\\cdot \\frac{6}{xy}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{x\\cdot 6}{2\\cdot xy}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{x}\\cdot 3\\cdot \\color{red}{2}}{\\color{red}{2}\\cdot \\color{red}{x}\\cdot y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors and simplify.<\/td>\n<td>[latex]\\Large\\frac{3}{y}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm146112\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146112&theme=lumen&iframe_resize_id=ohm146112&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-584","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/584","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/584\/revisions"}],"predecessor-version":[{"id":2635,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/584\/revisions\/2635"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/584\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=584"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=584"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=584"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=584"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}