{"id":4689,"date":"2023-06-20T15:01:52","date_gmt":"2023-06-20T15:01:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4689"},"modified":"2024-10-18T20:52:08","modified_gmt":"2024-10-18T20:52:08","slug":"dimensional-analysis-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/dimensional-analysis-learn-it-1\/","title":{"raw":"Dimensional Analysis of Rates: Learn It 1","rendered":"Dimensional Analysis of Rates: Learn It 1"},"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<p><strong>Proportions<\/strong> are a powerful tool in expressing and solving problems involving <strong>rates<\/strong> and <strong>ratios<\/strong>. By setting two rates or ratios equal to each other, we can find unknown values and understand relationships between different quantities.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>ratios and rates<\/h3>\r\n<ul>\r\n\t<li>A <strong>ratio <\/strong>is a comparison between two quantities or measures. The ratio can be expressed in three ways: [latex]a[\/latex] to [latex]b[\/latex], [latex]a:b[\/latex], or [latex]\\frac{a}{b}[\/latex].<\/li>\r\n\t<li>A <strong>rate<\/strong> is a specific kind of ratio in which two measurements with different units are related to each other. For example, if a car travels [latex]180[\/latex] miles in [latex]3[\/latex] hours, the rate of the car is [latex]60[\/latex] miles per hour.<\/li>\r\n\t<li>A <strong>unit rate<\/strong> is a rate with a denominator of one.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p><strong>Recall Reducing Fractions<\/strong><\/p>\r\n<p>The Equivalent Fractions Property states that:<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <em>\u00a0 If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then:<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex]{\\dfrac{a\\cdot c}{b\\cdot c}}={\\dfrac{a}{b}}[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"567582\"]See Example[\/reveal-answer] [hidden-answer a=\"567582\"]\u00a0[latex]\\dfrac{500}{20}=\\dfrac{25\\cdot 20}{1\\cdot 20}=\\dfrac{25}{1}=25[\/latex][\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Your car can drive [latex]300[\/latex] miles on a tank of [latex]15[\/latex] gallons. Express this as a rate.<br \/>\r\n[reveal-answer q=\"378596\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"378596\"]Expressed as a rate, [latex]\\displaystyle\\frac{300\\text{ miles}}{15\\text{ gallons}}[\/latex]. We can divide to find a unit rate:[latex]\\displaystyle\\frac{20\\text{ miles}}{1\\text{ gallon}}[\/latex], which we could also write as [latex]\\displaystyle{20}\\frac{\\text{miles}}{\\text{gallon}}[\/latex], or just [latex]20[\/latex] miles per gallon.[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8919[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>proportions<\/h3>\r\n<p>A <strong>proportion<\/strong> is an equation showing the equivalence of two rates or ratios.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Proportions have two key properties:<\/p>\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n\t<li><strong>The Cross-Product Property<\/strong>: The cross products of a proportion are always equal. That is, the product of the means equals the product of the extremes.<\/li>\r\n\t<li><strong>The Reciprocal Property<\/strong>: If [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex], then [latex]\\frac{b}{a} = \\frac{d}{c}[\/latex]. This property allows us to write the reciprocal of a proportion.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<p>To express a rate or ratio as a proportion, we write two ratios or rates as fractions that are equal to each other. Here's how to do it:<\/p>\r\n<p>Let's take the example of the car mentioned above traveling at a rate of [latex]60[\/latex] miles per hour. If we want to know how far the car would travel in [latex]5[\/latex] hours at the same rate, we would set up a proportion like this:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{180 \\text{ miles}}{3 \\text{ hours}} = \\frac{x \\text{ miles}}{5 \\text{ hours}}[\/latex]<\/p>\r\n<p>The proportion reads, \"[latex]180[\/latex] miles is to [latex]3[\/latex] hours as [latex]x[\/latex] miles is to [latex]5[\/latex] hours.\" We can solve for [latex]x[\/latex] by cross-multiplying and find that [latex]x[\/latex] equals [latex]300[\/latex] miles. So, the car would travel [latex]300[\/latex] miles in [latex]5[\/latex] hours at the same rate.<\/p>\r\n<section class=\"textbox recall\">\r\n<p><strong>Using Variables to represent unknowns<\/strong><\/p>\r\n<p>Recall that we can use letters we call <strong>variables\u00a0<\/strong>to \"stand in\" for unknown quantities. Then we can use the properties of equality to isolate the variable on one side of the equation. Once we have accomplished that, we say that we have \"solved the equation for the variable.\"<\/p>\r\n<p>You can view the example below to see how a proportion, which is an equality established between two fractions, is resolved to find the unknown value denoted by [latex]x[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"265082\"]See Example[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"265082\"]Solve the proportion [latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/p>\r\n<p>We see that the variable we wish to isolate is being divided by [latex]15[\/latex]. We can reverse that by multiplying on both sides by [latex]15[\/latex].<\/p>\r\n<center>[latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/center><center>[latex]15\\cdot \\dfrac{7}{3}=x[\/latex],<\/center>\r\n<p>giving [latex]x=35[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Your car can drive [latex]300[\/latex] miles on a tank of [latex]15[\/latex] gallons. How far can it drive on [latex]40[\/latex] gallons?<br \/>\r\n[reveal-answer q=\"526887\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"526887\"]\r\n\r\n<p>We could certainly answer this question using a proportion: [latex]\\displaystyle\\frac{300\\text{ miles}}{15\\text{ gallons}}=\\frac{x\\text{ miles}}{40\\text{ gallons}}[\/latex]. However, we earlier found that [latex]300[\/latex] miles on [latex]15[\/latex] gallons gives a rate of [latex]20[\/latex] miles per gallon. If we multiply the given [latex]40[\/latex] gallon quantity by this rate, the <em>gallons<\/em> unit \u201ccancels\u201d and we\u2019re left with a number of miles:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle40\\text{ gallons}\\cdot\\frac{20\\text{ miles}}{\\text{gallon}}=\\frac{40\\text{ gallons}}{1}\\cdot\\frac{20\\text{ miles}}{\\text{gallons}}=800\\text{ miles}[\/latex]<\/p>\r\n<p>Notice if instead we were asked \u201chow many gallons are needed to drive [latex]50[\/latex] miles?\u201d we could answer this question by inverting the [latex]20[\/latex] mile per gallon rate so that the <em>miles<\/em> unit cancels and we\u2019re left with gallons:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle{50}\\text{ miles}\\cdot\\frac{1\\text{ gallon}}{20\\text{ miles}}=\\frac{50\\text{ miles}}{1}\\cdot\\frac{1\\text{ gallon}}{20\\text{ miles}}=\\frac{50\\text{ gallons}}{20}=2.5\\text{ gallons}[\/latex]<\/p>\r\n<p>A worked example of this last question can be found in the following video.<\/p>\r\n<p>https:\/\/youtu.be\/jYwi3YqP0Wk<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Proportions+using+dimensional+analysis.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProportions using dimensional analysis\u201d here (opens in new window).<\/a><\/p>\r\n<p>[\/hidden-answer]<\/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<p><strong>Proportions<\/strong> are a powerful tool in expressing and solving problems involving <strong>rates<\/strong> and <strong>ratios<\/strong>. By setting two rates or ratios equal to each other, we can find unknown values and understand relationships between different quantities.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>ratios and rates<\/h3>\n<ul>\n<li>A <strong>ratio <\/strong>is a comparison between two quantities or measures. The ratio can be expressed in three ways: [latex]a[\/latex] to [latex]b[\/latex], [latex]a:b[\/latex], or [latex]\\frac{a}{b}[\/latex].<\/li>\n<li>A <strong>rate<\/strong> is a specific kind of ratio in which two measurements with different units are related to each other. For example, if a car travels [latex]180[\/latex] miles in [latex]3[\/latex] hours, the rate of the car is [latex]60[\/latex] miles per hour.<\/li>\n<li>A <strong>unit rate<\/strong> is a rate with a denominator of one.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox recall\">\n<p><strong>Recall Reducing Fractions<\/strong><\/p>\n<p>The Equivalent Fractions Property states that:<\/p>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <em>\u00a0 If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then:<\/em><\/p>\n<p style=\"text-align: center;\">[latex]{\\dfrac{a\\cdot c}{b\\cdot c}}={\\dfrac{a}{b}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q567582\">See Example<\/button> <\/p>\n<div id=\"q567582\" class=\"hidden-answer\" style=\"display: none\">\u00a0[latex]\\dfrac{500}{20}=\\dfrac{25\\cdot 20}{1\\cdot 20}=\\dfrac{25}{1}=25[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Your car can drive [latex]300[\/latex] miles on a tank of [latex]15[\/latex] gallons. Express this as a rate.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q378596\">Show Solution<\/button><\/p>\n<div id=\"q378596\" class=\"hidden-answer\" style=\"display: none\">Expressed as a rate, [latex]\\displaystyle\\frac{300\\text{ miles}}{15\\text{ gallons}}[\/latex]. We can divide to find a unit rate:[latex]\\displaystyle\\frac{20\\text{ miles}}{1\\text{ gallon}}[\/latex], which we could also write as [latex]\\displaystyle{20}\\frac{\\text{miles}}{\\text{gallon}}[\/latex], or just [latex]20[\/latex] miles per gallon.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8919\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8919&theme=lumen&iframe_resize_id=ohm8919&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>proportions<\/h3>\n<p>A <strong>proportion<\/strong> is an equation showing the equivalence of two rates or ratios.<\/p>\n<p>&nbsp;<\/p>\n<p>Proportions have two key properties:<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li><strong>The Cross-Product Property<\/strong>: The cross products of a proportion are always equal. That is, the product of the means equals the product of the extremes.<\/li>\n<li><strong>The Reciprocal Property<\/strong>: If [latex]\\frac{a}{b} = \\frac{c}{d}[\/latex], then [latex]\\frac{b}{a} = \\frac{d}{c}[\/latex]. This property allows us to write the reciprocal of a proportion.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>To express a rate or ratio as a proportion, we write two ratios or rates as fractions that are equal to each other. Here&#8217;s how to do it:<\/p>\n<p>Let&#8217;s take the example of the car mentioned above traveling at a rate of [latex]60[\/latex] miles per hour. If we want to know how far the car would travel in [latex]5[\/latex] hours at the same rate, we would set up a proportion like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{180 \\text{ miles}}{3 \\text{ hours}} = \\frac{x \\text{ miles}}{5 \\text{ hours}}[\/latex]<\/p>\n<p>The proportion reads, &#8220;[latex]180[\/latex] miles is to [latex]3[\/latex] hours as [latex]x[\/latex] miles is to [latex]5[\/latex] hours.&#8221; We can solve for [latex]x[\/latex] by cross-multiplying and find that [latex]x[\/latex] equals [latex]300[\/latex] miles. So, the car would travel [latex]300[\/latex] miles in [latex]5[\/latex] hours at the same rate.<\/p>\n<section class=\"textbox recall\">\n<p><strong>Using Variables to represent unknowns<\/strong><\/p>\n<p>Recall that we can use letters we call <strong>variables\u00a0<\/strong>to &#8220;stand in&#8221; for unknown quantities. Then we can use the properties of equality to isolate the variable on one side of the equation. Once we have accomplished that, we say that we have &#8220;solved the equation for the variable.&#8221;<\/p>\n<p>You can view the example below to see how a proportion, which is an equality established between two fractions, is resolved to find the unknown value denoted by [latex]x[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q265082\">See Example<\/button><\/p>\n<div id=\"q265082\" class=\"hidden-answer\" style=\"display: none\">Solve the proportion [latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/p>\n<p>We see that the variable we wish to isolate is being divided by [latex]15[\/latex]. We can reverse that by multiplying on both sides by [latex]15[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]15\\cdot \\dfrac{7}{3}=x[\/latex],<\/div>\n<p>giving [latex]x=35[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Your car can drive [latex]300[\/latex] miles on a tank of [latex]15[\/latex] gallons. How far can it drive on [latex]40[\/latex] gallons?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q526887\">Show Solution<\/button><\/p>\n<div id=\"q526887\" class=\"hidden-answer\" style=\"display: none\">\n<p>We could certainly answer this question using a proportion: [latex]\\displaystyle\\frac{300\\text{ miles}}{15\\text{ gallons}}=\\frac{x\\text{ miles}}{40\\text{ gallons}}[\/latex]. However, we earlier found that [latex]300[\/latex] miles on [latex]15[\/latex] gallons gives a rate of [latex]20[\/latex] miles per gallon. If we multiply the given [latex]40[\/latex] gallon quantity by this rate, the <em>gallons<\/em> unit \u201ccancels\u201d and we\u2019re left with a number of miles:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle40\\text{ gallons}\\cdot\\frac{20\\text{ miles}}{\\text{gallon}}=\\frac{40\\text{ gallons}}{1}\\cdot\\frac{20\\text{ miles}}{\\text{gallons}}=800\\text{ miles}[\/latex]<\/p>\n<p>Notice if instead we were asked \u201chow many gallons are needed to drive [latex]50[\/latex] miles?\u201d we could answer this question by inverting the [latex]20[\/latex] mile per gallon rate so that the <em>miles<\/em> unit cancels and we\u2019re left with gallons:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle{50}\\text{ miles}\\cdot\\frac{1\\text{ gallon}}{20\\text{ miles}}=\\frac{50\\text{ miles}}{1}\\cdot\\frac{1\\text{ gallon}}{20\\text{ miles}}=\\frac{50\\text{ gallons}}{20}=2.5\\text{ gallons}[\/latex]<\/p>\n<p>A worked example of this last question can be found in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Proportions using dimensional analysis\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jYwi3YqP0Wk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Proportions+using+dimensional+analysis.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProportions using dimensional analysis\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":20,"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":"learn_it","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\/4689"}],"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":48,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4689\/revisions"}],"predecessor-version":[{"id":15310,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4689\/revisions\/15310"}],"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\/4689\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4689"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4689"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4689"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4689"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}