{"id":4691,"date":"2023-06-20T15:02:39","date_gmt":"2023-06-20T15:02:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4691"},"modified":"2024-10-18T20:52:10","modified_gmt":"2024-10-18T20:52:10","slug":"dimensional-analysis-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/dimensional-analysis-fresh-take\/","title":{"raw":"Dimensional Analysis of Rates: Fresh Take","rendered":"Dimensional Analysis of Rates: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Write a proportion to express a rate or ratio<\/li>\r\n\t<li>Use units to determine which conversion factors are needed for dimensional analysis<\/li>\r\n\t<li>Use a conversion factor to convert a rate<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Proportions and Rates<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Ratio:<\/strong> A ratio is a way of comparing or relating one quantity to another. It is a mathematical expression that presents the relationship between two numbers indicating how many times the first number contains the second.<\/p>\r\n<p><strong>Rate:<\/strong> A rate is a special kind of ratio, used to compare quantities of different kinds. Rates are often used in everyday life, such as when we talk about speed (miles per hour), cost (dollars per item), or pay (dollars per hour). For example, if a car travels [latex]60[\/latex] miles in [latex]2[\/latex] hours, the rate is [latex]60[\/latex] miles per [latex]2[\/latex] hours, or [latex]30[\/latex] miles per hour.<\/p>\r\n<p><strong>Unit Rate:<\/strong> A unit rate is a rate in which the second quantity in the comparison is one unit. In other words, it's the rate for just one of something. For example, if a car travels [latex]60[\/latex] miles in [latex]2[\/latex] hours, the unit rate is [latex]30[\/latex] miles per [latex]1[\/latex] hour, which is often simply expressed as 30 miles per hour. Unit rates are useful in understanding the relationship between two units and are commonly used in everyday life, such as when comparing prices at the grocery store (e.g., dollars per pound).<\/p>\r\n<p>A <strong>proportion <\/strong>is a statement that two ratios or rates are equivalent. It is an equation that equates two ratios.<\/p>\r\n<\/div>\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].<br \/>\r\n[reveal-answer q=\"737915\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"737915\"]This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction[latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by 6, giving\u00a0[latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">A map scale indicates that [latex]\\frac{1}{2}[\/latex] inch on the map corresponds with [latex]3[\/latex] real miles. How many miles apart are two cities that are [latex]\\displaystyle{2}\\frac{1}{4}[\/latex] inches apart on the map?<br \/>\r\n[reveal-answer q=\"439949\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"439949\"]<br \/>\r\nWe can set up a proportion by setting equal two [latex]\\displaystyle\\frac{\\text{map inches}}{\\text{real miles}}[\/latex]\u00a0rates, and introducing a variable, [latex]x[\/latex], to represent the unknown quantity\u2014the mile distance between the cities.\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}\\text{map inch}}{3\\text{ miles}}=\\frac{2\\frac{1}{4}\\text{map inches}}{x\\text{ miles}}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]x[\/latex] and rewriting the mixed number<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}}{3}\\cdot{x}=\\frac{9}{4}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{1}{2}x=\\frac{27}{4}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]2[\/latex] (or divide by [latex]\u00bd[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle{x}=\\frac{27}{2}=13\\frac{1}{2}\\text{ miles}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose you\u2019re tiling the floor of a [latex]10[\/latex] ft by [latex]10[\/latex] ft room, and find that [latex]100[\/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[\/latex] ft by [latex]20[\/latex] ft room?<br \/>\r\n[reveal-answer q=\"815477\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"815477\"]In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, [latex]400[\/latex] tiles will be needed. We could find this using a proportion based on the areas of the rooms:[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex][\/hidden-answer]<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10305248&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=GnIJQkzYWWs&amp;video_target=tpm-plugin-dpj7ltya-GnIJQkzYWWs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/An+Intro+to+Ratios+%7C+What+is+a+Ratio%3F+%7C+Understanding+Ratios+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAn Intro to Ratios | What is a Ratio? | Understanding Ratios | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10305249&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OMTFweN0TJQ&amp;video_target=tpm-plugin-m9nnhsvo-OMTFweN0TJQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Solve+Proportions+Using+Cross+Multiplication+%7C+Solving+Proportions+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Solve Proportions Using Cross Multiplication | Solving Proportions | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Dimensional Analysis of Rates<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong>Converting rates involves transforming a given rate into a different unit of rate. This is achieved by applying the appropriate conversion factors in a process known as dimensional analysis.<\/p>\r\n<p><strong>Dimensional Analysis<\/strong>: A mathematical technique used to convert one unit of measurement to another by using conversion factors.<\/p>\r\n<p><strong>Conversion Factors<\/strong>: These are ratios that express the relationship between two different units of measurement.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_IsVUpqtMAQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/1.2.3+-+Converting+Rates+Using+Dimensional+Analysis.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c1.2.3 - Converting Rates Using Dimensional Analysis\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10305250&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=c4Q1PHhI4d8&amp;video_target=tpm-plugin-wtescq8z-c4Q1PHhI4d8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Dimensional+Analysis+and+Rates.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDimensional Analysis and Rates\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Write a proportion to express a rate or ratio<\/li>\n<li>Use units to determine which conversion factors are needed for dimensional analysis<\/li>\n<li>Use a conversion factor to convert a rate<\/li>\n<\/ul>\n<\/section>\n<h2>Proportions and Rates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Ratio:<\/strong> A ratio is a way of comparing or relating one quantity to another. It is a mathematical expression that presents the relationship between two numbers indicating how many times the first number contains the second.<\/p>\n<p><strong>Rate:<\/strong> A rate is a special kind of ratio, used to compare quantities of different kinds. Rates are often used in everyday life, such as when we talk about speed (miles per hour), cost (dollars per item), or pay (dollars per hour). For example, if a car travels [latex]60[\/latex] miles in [latex]2[\/latex] hours, the rate is [latex]60[\/latex] miles per [latex]2[\/latex] hours, or [latex]30[\/latex] miles per hour.<\/p>\n<p><strong>Unit Rate:<\/strong> A unit rate is a rate in which the second quantity in the comparison is one unit. In other words, it&#8217;s the rate for just one of something. For example, if a car travels [latex]60[\/latex] miles in [latex]2[\/latex] hours, the unit rate is [latex]30[\/latex] miles per [latex]1[\/latex] hour, which is often simply expressed as 30 miles per hour. Unit rates are useful in understanding the relationship between two units and are commonly used in everyday life, such as when comparing prices at the grocery store (e.g., dollars per pound).<\/p>\n<p>A <strong>proportion <\/strong>is a statement that two ratios or rates are equivalent. It is an equation that equates two ratios.<\/p>\n<\/div>\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=\"q737915\">Show Solution<\/button><\/p>\n<div id=\"q737915\" class=\"hidden-answer\" style=\"display: none\">This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction[latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by 6, giving\u00a0[latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A map scale indicates that [latex]\\frac{1}{2}[\/latex] inch on the map corresponds with [latex]3[\/latex] real miles. How many miles apart are two cities that are [latex]\\displaystyle{2}\\frac{1}{4}[\/latex] inches apart on the map?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q439949\">Show Solution<\/button><\/p>\n<div id=\"q439949\" class=\"hidden-answer\" style=\"display: none\">\nWe can set up a proportion by setting equal two [latex]\\displaystyle\\frac{\\text{map inches}}{\\text{real miles}}[\/latex]\u00a0rates, and introducing a variable, [latex]x[\/latex], to represent the unknown quantity\u2014the mile distance between the cities.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}\\text{map inch}}{3\\text{ miles}}=\\frac{2\\frac{1}{4}\\text{map inches}}{x\\text{ miles}}[\/latex]<\/td>\n<td>Multiply both sides by [latex]x[\/latex] and rewriting the mixed number<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}}{3}\\cdot{x}=\\frac{9}{4}[\/latex]<\/td>\n<td>Multiply both sides by [latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle\\frac{1}{2}x=\\frac{27}{4}[\/latex]<\/td>\n<td>Multiply both sides by [latex]2[\/latex] (or divide by [latex]\u00bd[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle{x}=\\frac{27}{2}=13\\frac{1}{2}\\text{ miles}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Suppose you\u2019re tiling the floor of a [latex]10[\/latex] ft by [latex]10[\/latex] ft room, and find that [latex]100[\/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[\/latex] ft by [latex]20[\/latex] ft room?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q815477\">Show Solution<\/button><\/p>\n<div id=\"q815477\" class=\"hidden-answer\" style=\"display: none\">In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, [latex]400[\/latex] tiles will be needed. We could find this using a proportion based on the areas of the rooms:[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10305248&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=GnIJQkzYWWs&amp;video_target=tpm-plugin-dpj7ltya-GnIJQkzYWWs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/An+Intro+to+Ratios+%7C+What+is+a+Ratio%3F+%7C+Understanding+Ratios+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAn Intro to Ratios | What is a Ratio? | Understanding Ratios | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10305249&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OMTFweN0TJQ&amp;video_target=tpm-plugin-m9nnhsvo-OMTFweN0TJQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Solve+Proportions+Using+Cross+Multiplication+%7C+Solving+Proportions+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Solve Proportions Using Cross Multiplication | Solving Proportions | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Dimensional Analysis of Rates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong>Converting rates involves transforming a given rate into a different unit of rate. This is achieved by applying the appropriate conversion factors in a process known as dimensional analysis.<\/p>\n<p><strong>Dimensional Analysis<\/strong>: A mathematical technique used to convert one unit of measurement to another by using conversion factors.<\/p>\n<p><strong>Conversion Factors<\/strong>: These are ratios that express the relationship between two different units of measurement.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_IsVUpqtMAQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/1.2.3+-+Converting+Rates+Using+Dimensional+Analysis.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c1.2.3 &#8211; Converting Rates Using Dimensional Analysis\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10305250&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=c4Q1PHhI4d8&amp;video_target=tpm-plugin-wtescq8z-c4Q1PHhI4d8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Dimensional+Analysis+and+Rates.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDimensional Analysis and Rates\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":24,"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":"fresh_take","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\/4691"}],"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\/15"}],"version-history":[{"count":16,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4691\/revisions"}],"predecessor-version":[{"id":15313,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4691\/revisions\/15313"}],"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\/4691\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4691"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4691"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4691"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4691"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}