{"id":345,"date":"2023-02-17T19:37:11","date_gmt":"2023-02-17T19:37:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=345"},"modified":"2024-10-18T20:52:01","modified_gmt":"2024-10-18T20:52:01","slug":"measurement-background-youll-need-page-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/measurement-background-youll-need-page-2\/","title":{"raw":"Measurement:  Background You'll Need - Page 2","rendered":"Measurement:  Background You&#8217;ll Need &#8211; Page 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Simplify using fractions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Multiply or Divide Fractions<\/h2>\r\n<p>Multiplying and dividing fractions, including mixed numbers, are essential operations in mathematics that extend beyond basic arithmetic into more complex areas like algebra and geometry. Whether you're combining proportions in a recipe or solving for unknowns in equations, understanding how to manipulate these numbers is key. Before we dive into operations with mixed numbers, remember that they need to be converted into improper fractions. This uniform format makes the multiplication or division process straightforward. Here are the steps to ensure you can tackle these operations with confidence.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Multiply or Divide Mixed Numbers<\/strong><\/p>\r\n<ol id=\"eip-id1168468257930\" class=\"stepwise\">\r\n\t<li>Convert the mixed numbers to improper fractions.<\/li>\r\n\t<li>Follow the rules for fraction multiplication or division.<\/li>\r\n\t<li>Simplify if possible.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Multiply the following. Write your answer in simplified form.<center>[latex]2\\Large\\frac{4}{5}\\left(\\normalsize -1\\Large\\frac{7}{8}\\right)[\/latex]<\/center>[reveal-answer q=\"859815\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"859815\"]\r\n\r\n<table id=\"eip-id1168468531210\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]2\\Large\\frac{4}{5}\\left(\\normalsize -1\\Large\\frac{7}{8}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert mixed numbers to improper fractions.<\/td>\r\n<td>[latex]\\Large\\frac{14}{5}\\left(-\\frac{15}{8}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large-\\frac{14\\cdot 15}{5\\cdot 8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{\\color{red}{2} \\cdot 7\\cdot \\color{red}{5} \\cdot 3}{\\color{red}{5} \\cdot \\color{red}{2}\\cdot 4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{7\\cdot 3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large-\\frac{21}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2943[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Divide the following. Write your answer in simplified form.<center>[latex]3\\Large\\frac{4}{7}\\normalsize\\div 5[\/latex]<\/center>[reveal-answer q=\"69025\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"69025\"]\r\n\r\n<table id=\"eip-id1168467104894\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3\\Large\\frac{4}{7}\\normalsize\\div 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert mixed numbers to improper fractions.<\/td>\r\n<td>[latex]\\Large\\frac{25}{7}\\normalsize\\div\\Large\\frac{5}{1}[\/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{25}{7}\\cdot \\frac{1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{25\\cdot 1}{7\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{5} \\cdot 5\\cdot 1}{7\\cdot \\color{red}{5} }[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot 1}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{5}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2944[\/ohm2_question]<\/section>\r\n<h2>Simplify Complex Fractions<\/h2>\r\n<p>Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.<\/p>\r\n<p>Some examples of complex fractions are:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\LARGE\\frac{\\frac{6}{7}}{ 3}, \\frac{\\frac{3}{4}}{\\frac{5}{8}}, \\frac{\\frac{x}{2}}{\\frac{5}{6}}[\/latex]<br \/>\r\nTo simplify a complex fraction, remember that the fraction bar means division. So the complex fraction [latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex] can be written as: [latex]\\Large\\frac{3}{4}\\normalsize\\div\\Large\\frac{5}{8}[\/latex]<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Simplify a Complex Fraction<\/strong><\/p>\r\n<ol id=\"eip-id1168468756479\" class=\"stepwise\">\r\n\t<li>Rewrite the complex fraction as a division problem.<\/li>\r\n\t<li>Follow the rules for dividing fractions.<\/li>\r\n\t<li>Simplify if possible.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]\\LARGE\\frac{-\\frac{6}{7}}{ 3}[\/latex]<br \/>\r\n[reveal-answer q=\"652451\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"652451\"]\r\n\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2945[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify using fractions<\/li>\n<\/ul>\n<\/section>\n<h2>Multiply or Divide Fractions<\/h2>\n<p>Multiplying and dividing fractions, including mixed numbers, are essential operations in mathematics that extend beyond basic arithmetic into more complex areas like algebra and geometry. Whether you&#8217;re combining proportions in a recipe or solving for unknowns in equations, understanding how to manipulate these numbers is key. Before we dive into operations with mixed numbers, remember that they need to be converted into improper fractions. This uniform format makes the multiplication or division process straightforward. Here are the steps to ensure you can tackle these operations with confidence.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Multiply or Divide Mixed Numbers<\/strong><\/p>\n<ol id=\"eip-id1168468257930\" class=\"stepwise\">\n<li>Convert the mixed numbers to improper fractions.<\/li>\n<li>Follow the rules for fraction multiplication or division.<\/li>\n<li>Simplify if possible.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Multiply the following. Write your answer in simplified form.<\/p>\n<div style=\"text-align: center;\">[latex]2\\Large\\frac{4}{5}\\left(\\normalsize -1\\Large\\frac{7}{8}\\right)[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q859815\">Show Solution<\/button><\/p>\n<div id=\"q859815\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468531210\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]2\\Large\\frac{4}{5}\\left(\\normalsize -1\\Large\\frac{7}{8}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert mixed numbers to improper fractions.<\/td>\n<td>[latex]\\Large\\frac{14}{5}\\left(-\\frac{15}{8}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large-\\frac{14\\cdot 15}{5\\cdot 8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\Large-\\frac{\\color{red}{2} \\cdot 7\\cdot \\color{red}{5} \\cdot 3}{\\color{red}{5} \\cdot \\color{red}{2}\\cdot 4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large-\\frac{7\\cdot 3}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large-\\frac{21}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2943\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2943&theme=lumen&iframe_resize_id=ohm2943&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Divide the following. Write your answer in simplified form.<\/p>\n<div style=\"text-align: center;\">[latex]3\\Large\\frac{4}{7}\\normalsize\\div 5[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q69025\">Show Solution<\/button><\/p>\n<div id=\"q69025\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467104894\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3\\Large\\frac{4}{7}\\normalsize\\div 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert mixed numbers to improper fractions.<\/td>\n<td>[latex]\\Large\\frac{25}{7}\\normalsize\\div\\Large\\frac{5}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\Large\\frac{25}{7}\\cdot \\frac{1}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{25\\cdot 1}{7\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{5} \\cdot 5\\cdot 1}{7\\cdot \\color{red}{5} }[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot 1}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{5}{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2944\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2944&theme=lumen&iframe_resize_id=ohm2944&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Simplify Complex Fractions<\/h2>\n<p>Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.<\/p>\n<p>Some examples of complex fractions are:<\/p>\n<p style=\"text-align: center;\">[latex]\\LARGE\\frac{\\frac{6}{7}}{ 3}, \\frac{\\frac{3}{4}}{\\frac{5}{8}}, \\frac{\\frac{x}{2}}{\\frac{5}{6}}[\/latex]<br \/>\nTo simplify a complex fraction, remember that the fraction bar means division. So the complex fraction [latex]\\LARGE\\frac{\\frac{3}{4}}{\\frac{5}{8}}[\/latex] can be written as: [latex]\\Large\\frac{3}{4}\\normalsize\\div\\Large\\frac{5}{8}[\/latex]<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Simplify a Complex Fraction<\/strong><\/p>\n<ol id=\"eip-id1168468756479\" class=\"stepwise\">\n<li>Rewrite the complex fraction as a division problem.<\/li>\n<li>Follow the rules for dividing fractions.<\/li>\n<li>Simplify if possible.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox 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\">\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 tryIt\"><iframe loading=\"lazy\" id=\"ohm2945\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2945&theme=lumen&iframe_resize_id=ohm2945&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"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":[],"part":62,"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/345"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions"}],"predecessor-version":[{"id":15277,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions\/15277"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/62"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/345\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=345"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=345"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=345"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}