{"id":5588,"date":"2024-06-26T18:01:34","date_gmt":"2024-06-26T18:01:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/?post_type=chapter&#038;p=5588"},"modified":"2024-06-26T18:01:34","modified_gmt":"2024-06-26T18:01:34","slug":"basic-rates-and-proportions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/basic-rates-and-proportions\/","title":{"raw":"Basic Rates and Proportions","rendered":"Basic Rates and Proportions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write a proportion to express a rate or ratio<\/li>\r\n \t<li>Solve a proportion for an unknown<\/li>\r\n<\/ul>\r\n<\/div>\r\nIf you wanted to power the city of Lincoln, Nebraska using wind power, how many wind\u00a0turbines would you need to install? Questions like these can be answered using rates and proportions.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14205401\/wind-364996_1280.jpg\"><img class=\"aligncenter wp-image-497\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14205401\/wind-364996_1280-1024x685.jpg\" alt=\"two wind turbines in a field of flowers and low trees\" width=\"613\" height=\"410\" \/><\/a>\r\n<div class=\"textbox\">\r\n<h2>RATES<\/h2>\r\nA rate is the ratio (fraction) of two quantities.\r\n\r\nA <strong>unit rate<\/strong> is a rate with a denominator of one.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall Reducing Fractions<\/h3>\r\nThe Equivalent Fractions Property states that\r\n\r\nIf [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]{\\dfrac{a\\cdot c}{b\\cdot c}}={\\dfrac{a}{b}}[\/latex].\r\n\r\nEx. [latex]\\dfrac{500}{20}=\\dfrac{25\\cdot 20}{1\\cdot 20}=\\dfrac{25}{1}=25[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nYour car can drive 300 miles on a tank of 15 gallons. Express this as a rate.\r\n[reveal-answer q=\"378596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"378596\"]\r\n\r\nExpressed 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 20 miles per gallon.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h2>Proportion Equation<\/h2>\r\nA proportion equation is an equation showing the equivalence of two rates or ratios.\r\n\r\nAn example of a proportion would be:\u00a0 [latex]\\dfrac{7}{3}=\\dfrac{35}{15}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Using Variables to represent unknowns<\/h3>\r\nRecall 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.\"\r\n\r\nIn the example below, you are asked to solve the proportion (an equality given between two fractions) for the unknown value [latex]x[\/latex].\r\n\r\nEx. Solve the proportion [latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]\r\n<p style=\"padding-left: 30px;\">We see that the variable we wish to isolate is being divided by 15. We can reverse that by multiplying on both sides by 15.<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]15\\cdot \\dfrac{7}{3}=x[\/latex], giving [latex]x=35[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value <em>x<\/em>.\r\n[reveal-answer q=\"737915\"]Show Solution[\/reveal-answer]\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]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write a proportion to express a rate or ratio<\/li>\n<li>Solve a proportion for an unknown<\/li>\n<\/ul>\n<\/div>\n<p>If you wanted to power the city of Lincoln, Nebraska using wind power, how many wind\u00a0turbines would you need to install? Questions like these can be answered using rates and proportions.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14205401\/wind-364996_1280.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-497\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14205401\/wind-364996_1280-1024x685.jpg\" alt=\"two wind turbines in a field of flowers and low trees\" width=\"613\" height=\"410\" \/><\/a><\/p>\n<div class=\"textbox\">\n<h2>RATES<\/h2>\n<p>A rate is the ratio (fraction) of two quantities.<\/p>\n<p>A <strong>unit rate<\/strong> is a rate with a denominator of one.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall Reducing Fractions<\/h3>\n<p>The Equivalent Fractions Property states that<\/p>\n<p>If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]{\\dfrac{a\\cdot c}{b\\cdot c}}={\\dfrac{a}{b}}[\/latex].<\/p>\n<p>Ex. [latex]\\dfrac{500}{20}=\\dfrac{25\\cdot 20}{1\\cdot 20}=\\dfrac{25}{1}=25[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q378596\">Show Solution<\/span><\/p>\n<div id=\"q378596\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 20 miles per gallon.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h2>Proportion Equation<\/h2>\n<p>A proportion equation is an equation showing the equivalence of two rates or ratios.<\/p>\n<p>An example of a proportion would be:\u00a0 [latex]\\dfrac{7}{3}=\\dfrac{35}{15}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Using Variables to represent unknowns<\/h3>\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>In the example below, you are asked to solve the proportion (an equality given between two fractions) for the unknown value [latex]x[\/latex].<\/p>\n<p>Ex. Solve the proportion [latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">We see that the variable we wish to isolate is being divided by 15. We can reverse that by multiplying on both sides by 15.<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\dfrac{7}{3}=\\dfrac{x}{15}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]15\\cdot \\dfrac{7}{3}=x[\/latex], giving [latex]x=35[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value <em>x<\/em>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737915\">Show Solution<\/span><\/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<\/div>\n","protected":false},"author":689068,"menu_order":32,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5588","chapter","type-chapter","status-publish","hentry"],"part":5569,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/5588","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/689068"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/5588\/revisions"}],"predecessor-version":[{"id":5589,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/5588\/revisions\/5589"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/5569"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/5588\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=5588"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=5588"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=5588"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=5588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}