{"id":573,"date":"2023-03-02T20:06:35","date_gmt":"2023-03-02T20:06:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=573"},"modified":"2024-10-18T20:51:29","modified_gmt":"2024-10-18T20:51:29","slug":"geometry-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/geometry-background-youll-need-2\/","title":{"raw":"Geometry:  Background You'll Need 2","rendered":"Geometry:  Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve a proportion for an unknown&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve a proportion for an unknown<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Proportions<\/h2>\r\n<p>Proportions are the heart of understanding how different quantities relate to one another. They allow us to compare two rates and determine how one quantity changes in relation to another. Mastering the art of solving proportions is not just a mathematical skill; it's a critical thinking skill that applies to real-world situations such as comparing prices, calculating speed, or adjusting recipes.<\/p>\r\n<p>A proportion equation sets two rates or ratios equal to each other, providing a clear pathway to finding unknown values. By cross-multiplying or using equivalent ratios, we can unlock these unknowns and gain insights into the relationship between the quantities at hand. Let\u2019s explore how this powerful tool works.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>rates<\/h3>\r\n<p>A <strong>rate<\/strong> is the ratio (fraction) of two quantities.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A\u00a0<strong>unit rate<\/strong> is a rate with a denominator of one.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>proportion equation<\/h3>\r\n<p>A <strong>proportion equation<\/strong> is an equation showing the equivalence of two rates or ratios.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Solve Proportions for an Unknown Value<\/strong><\/p>\r\n<p>When faced with a proportion, such as [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex], where you are solving for an unknown variable, follow these steps:<\/p>\r\n<ul>\r\n\t<li><strong>Identify the Unknown:<\/strong> Determine which part of the proportion is unknown. This will be the value you are solving for.<\/li>\r\n\t<li><strong>Cross-Multiply<\/strong>: Create an equation by cross-multiplying the terms of the proportion. Multiply the numerator of one ratio by the denominator of the other ratio, and set the two products equal to each other.<\/li>\r\n\t<li><strong>Set Up the Equation:<\/strong> After cross-multiplying, you will have an equation with one unknown. It will look like [latex]a\u00d7d=b\u00d7c[\/latex] for the proportion above.<\/li>\r\n\t<li><strong>Solve for the Unknown:<\/strong> Isolate the unknown variable on one side of the equation. If necessary, use algebraic operations such as division or multiplication to solve for the variable.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value [latex]x[\/latex].[reveal-answer q=\"259261\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"259261\"]\r\n\r\n<p>This proportion is asking us to find a fraction with denominator [latex]6[\/latex] that is equivalent to the fraction [latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]6[\/latex], giving [latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3133[\/ohm2_question]<\/section>\r\n<section>\r\n<section>\r\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{2}{7}=\\frac{4}{x}[\/latex] for the unknown value [latex]x[\/latex].[reveal-answer q=\"760210\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"760210\"]\r\n\r\n<p>This proportion is asking us to find a fraction with numerator [latex]4[\/latex] that is equivalent to the fraction [latex]\\displaystyle\\frac{2}{7}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]7[\/latex] and [latex]x[\/latex], giving...<\/p>\r\n<p style=\"text-align: center;\">[latex]2 \\cdot x = 4 \\cdot 7[\/latex].<\/p>\r\n<p>We then divide both sides by [latex]2[\/latex] to get [latex]x[\/latex] by itself.<\/p>\r\n<p style=\"text-align: center;\">[latex]x = \\frac{4 \\cdot 7}{2} = \\frac{28}{4} =14[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=14[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3136[\/ohm2_question]<\/section>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve a proportion for an unknown&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve a proportion for an unknown<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Solving Proportions<\/h2>\n<p>Proportions are the heart of understanding how different quantities relate to one another. They allow us to compare two rates and determine how one quantity changes in relation to another. Mastering the art of solving proportions is not just a mathematical skill; it&#8217;s a critical thinking skill that applies to real-world situations such as comparing prices, calculating speed, or adjusting recipes.<\/p>\n<p>A proportion equation sets two rates or ratios equal to each other, providing a clear pathway to finding unknown values. By cross-multiplying or using equivalent ratios, we can unlock these unknowns and gain insights into the relationship between the quantities at hand. Let\u2019s explore how this powerful tool works.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>rates<\/h3>\n<p>A <strong>rate<\/strong> is the ratio (fraction) of two quantities.<\/p>\n<p>&nbsp;<\/p>\n<p>A\u00a0<strong>unit rate<\/strong> is a rate with a denominator of one.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>proportion equation<\/h3>\n<p>A <strong>proportion equation<\/strong> is an equation showing the equivalence of two rates or ratios.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Solve Proportions for an Unknown Value<\/strong><\/p>\n<p>When faced with a proportion, such as [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex], where you are solving for an unknown variable, follow these steps:<\/p>\n<ul>\n<li><strong>Identify the Unknown:<\/strong> Determine which part of the proportion is unknown. This will be the value you are solving for.<\/li>\n<li><strong>Cross-Multiply<\/strong>: Create an equation by cross-multiplying the terms of the proportion. Multiply the numerator of one ratio by the denominator of the other ratio, and set the two products equal to each other.<\/li>\n<li><strong>Set Up the Equation:<\/strong> After cross-multiplying, you will have an equation with one unknown. It will look like [latex]a\u00d7d=b\u00d7c[\/latex] for the proportion above.<\/li>\n<li><strong>Solve for the Unknown:<\/strong> Isolate the unknown variable on one side of the equation. If necessary, use algebraic operations such as division or multiplication to solve for the variable.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value [latex]x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q259261\">Show Answer<\/button><\/p>\n<div id=\"q259261\" class=\"hidden-answer\" style=\"display: none\">\n<p>This proportion is asking us to find a fraction with denominator [latex]6[\/latex] that is equivalent to the fraction [latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]6[\/latex], giving [latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3133\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3133&theme=lumen&iframe_resize_id=ohm3133&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section>\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{2}{7}=\\frac{4}{x}[\/latex] for the unknown value [latex]x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q760210\">Show Answer<\/button><\/p>\n<div id=\"q760210\" class=\"hidden-answer\" style=\"display: none\">\n<p>This proportion is asking us to find a fraction with numerator [latex]4[\/latex] that is equivalent to the fraction [latex]\\displaystyle\\frac{2}{7}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]7[\/latex] and [latex]x[\/latex], giving&#8230;<\/p>\n<p style=\"text-align: center;\">[latex]2 \\cdot x = 4 \\cdot 7[\/latex].<\/p>\n<p>We then divide both sides by [latex]2[\/latex] to get [latex]x[\/latex] by itself.<\/p>\n<p style=\"text-align: center;\">[latex]x = \\frac{4 \\cdot 7}{2} = \\frac{28}{4} =14[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=14[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3136\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3136&theme=lumen&iframe_resize_id=ohm3136&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Problem Solving\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"background_you_need","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Problem Solving","author":"David Lippman","organization":"","url":"http:\/\/www.opentextbookstore.com\/mathinsociety\/","project":"Math in Society","license":"cc-by-sa","license_terms":""}],"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\/573"}],"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":23,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/573\/revisions"}],"predecessor-version":[{"id":13838,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/573\/revisions\/13838"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/573\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=573"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=573"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=573"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}