{"id":129,"date":"2017-06-13T23:19:16","date_gmt":"2017-06-13T23:19:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/?post_type=chapter&#038;p=129"},"modified":"2018-08-14T20:21:08","modified_gmt":"2018-08-14T20:21:08","slug":"proportional-relationships-and-a-bit-of-geometry","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/chapter\/proportional-relationships-and-a-bit-of-geometry\/","title":{"raw":"Proportional Relationships and a Bit of Geometry","rendered":"Proportional Relationships and a Bit of Geometry"},"content":{"raw":"In the 2004 vice-presidential debates, Democratic contender John Edwards claimed that\u00a0US forces have suffered \"90% of the coalition casualties\" in Iraq. Incumbent Vice President Dick Cheney disputed this, saying that in fact Iraqi security forces and coalition allies \"have taken almost 50 percent\" of the casualties.[footnote]<a href=\"http:\/\/www.factcheck.org\/cheney_edwards_mangle_facts.html\" target=\"_blank\">http:\/\/www.factcheck.org\/cheney_edwards_mangle_facts.html<\/a>[\/footnote]\r\n\r\nWho was\u00a0correct? How can we make sense of these numbers?\r\n\r\nIn this section, we will show how the idea of percent is used to describe parts of a whole. \u00a0Percents are prevalent in the media we consume regularly, making it imperative that you understand what they mean and where they come from.\r\n\r\nWe will also show you how to compare different quantities using proportions. \u00a0Proportions can help us understand how things change or relate to each other.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this lesson you will learn how to do the following:\r\n<ul>\r\n \t<li>Given the part and the whole, write a percent<\/li>\r\n \t<li>Calculate both relative\u00a0and absolute change of a quantity<\/li>\r\n \t<li>Calculate tax on a purchase<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Percents<\/h2>\r\nPercent literally means \u201cper 100,\u201d or \u201cparts per hundred.\u201d When we write 40%, this is equivalent to the fraction [latex]\\displaystyle\\frac{40}{100}[\/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\\displaystyle\\frac{80}{200}=\\frac{10}{25}=\\frac{40}{100}[\/latex].\r\n\r\n[caption id=\"attachment_494\" align=\"aligncenter\" width=\"500\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\"><img class=\"wp-image-494\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\" alt=\"Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty.\" width=\"500\" height=\"282\" \/><\/a> A visual depiction of 40%[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>Percent<\/h3>\r\nIf we have a <em>part<\/em> that is some <em>percent<\/em> of a <em>whole<\/em>, then\u00a0[latex]\\displaystyle\\text{percent}=\\frac{\\text{part}}{\\text{whole}}[\/latex], or equivalently, [latex]\\text{part}\\cdot\\text{whole}=\\text{percent}[\/latex].\r\n\r\nTo do the calculations, we write the percent as a decimal.\r\n\r\nFor a refresher on basic percentage rules, using the examples on this page, view the following video.\r\n\r\nhttps:\/\/youtu.be\/Z229RysttR8\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nIn a survey, 243 out of 400 people state that they like dogs. What percent is this?\r\n[reveal-answer q=\"987171\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"987171\"]\r\n\r\n[latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex] This is 60.75%.\r\n\r\nNotice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite each as a percent:\r\n<ol>\r\n \t<li>[latex]\\displaystyle\\frac{1}{4}[\/latex]<\/li>\r\n \t<li>0.02<\/li>\r\n \t<li>2.35<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"660805\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"660805\"]\r\n<ol>\r\n \t<li>[latex]\\displaystyle\\frac{1}{4}=0.25[\/latex] = 25%<\/li>\r\n \t<li>0.02 = 2%<\/li>\r\n \t<li>2.35 = 235%<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\nThroughout this text, you will be given opportunities to answer questions and know immediately whether you answered correctly. To answer the question below, do the calculation on a separate piece of paper and enter your answer in the box. Click on the submit button , and if you are correct, a green box will appear around your answer. \u00a0If you are incorrect, a red box will appear. \u00a0You can click on \"Try Another Version of This Question\" as many times as you like. Practice all you want!\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17441&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn the news, you hear \u201ctuition is expected to increase by 7% next year.\u201d If tuition this year was $1200 per quarter, what will it be next year?\r\n[reveal-answer q=\"475615\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"475615\"]The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year\u2019s tuition:\u00a0$1200(1.07) = $1284.\r\n\r\nAlternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84.\r\n\r\nNotice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase, so to calculate the expected tuition, we\u2019ll need to add this change to the previous year\u2019s tuition:\u00a0$1200 + $84 = $1284.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17447&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?\r\n[reveal-answer q=\"573833\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"573833\"]\r\n\r\nTo compute the percent change, we first need to find the dollar value change: $6800 \u2013\u00a0$7400 = \u2013$600. Often we will take the absolute value of this amount, which is called the <strong>absolute change<\/strong>: |\u2013600| = 600.\r\n\r\nSince we are computing the decrease relative to the starting value, we compute this percent out of $7400:\r\n\r\n[latex]\\displaystyle\\frac{600}{7400}=0.081=[\/latex] 8.1% decrease. This is called a <strong>relative change<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Absolute and Relative Change<\/h3>\r\nGiven two quantities,\r\n\r\nAbsolute change =[latex]\\displaystyle|\\text{ending quantity}-\\text{starting quantity}|[\/latex]\r\n\r\nRelative change: [latex]\\displaystyle\\frac{\\text{absolute change}}{\\text{starting quantity}}[\/latex]\r\n<ul>\r\n \t<li>Absolute change has the same units as the original quantity.<\/li>\r\n \t<li>Relative change gives a percent change.<\/li>\r\n<\/ul>\r\nThe starting quantity is called the <strong>base<\/strong> of the percent change.\r\n\r\nFor a deeper dive on absolute and relative change, using the examples on this page, view the following video.\r\n\r\nhttps:\/\/youtu.be\/QjVeurkg8CQ\r\n\r\n<\/div>\r\nThe base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a \u201cgateway\u201d drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn\u2019t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs.[footnote]<a href=\"http:\/\/tvtropes.org\/pmwiki\/pmwiki.php\/Main\/LiesDamnedLiesAndStatistics\" target=\"_blank\">http:\/\/tvtropes.org\/pmwiki\/pmwiki.php\/Main\/LiesDamnedLiesAndStatistics<\/a>[\/footnote]\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThere are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.\r\n[reveal-answer q=\"933757\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"933757\"]\r\n\r\nWhen we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 \u2013 75 = 140. From this, we could say \u201cAlbertsons has 140 more stores than QFC.\u201d However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:\r\n\r\nUsing QFC as the base, [latex]\\displaystyle\\frac{140}{75}=1.867[\/latex].\r\n\r\nThis tells us Albertsons is 186.7% larger than QFC.\r\n\r\nUsing Albertsons as the base,[latex]\\displaystyle\\frac{140}{215}=0.651[\/latex].\r\n\r\nThis tells us QFC is 65.1% smaller than Albertsons.\r\n\r\nNotice both of these are showing percent <em>differences<\/em>. We could also calculate the size of Albertsons relative to QFC:[latex]\\displaystyle\\frac{215}{75}=2.867[\/latex], which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:[latex]\\displaystyle\\frac{75}{215}=0.349[\/latex], which tells us that QFC is 34.9% of the size of Albertsons.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?\r\n[reveal-answer q=\"568319\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"568319\"]\r\n\r\nTo answer this question, suppose the value started at $100. After one week, the value dropped by 60%:\u00a0$100 \u2013 $100(0.60) = $100 \u2013 $60 = $40.\r\n\r\nIn the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase:\u00a0$40 + $40(0.75) = $40 + $30 = $70.\r\n\r\nIn the end, the stock is still $30 lower, or [latex]\\displaystyle\\frac{\\$30}{100}[\/latex] = 30% lower, valued than it started.\r\n\r\n[\/hidden-answer]\r\n\r\nA video walk-through of this example can be seen here.\r\n\r\nhttps:\/\/youtu.be\/4HNxwYMTNl8\r\n\r\n<\/div>\r\nConsideration of the base of percentages is explored in this video, using the examples on this page.\r\n\r\nhttps:\/\/youtu.be\/nygw69JqwoQ\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17443&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA <em>Seattle Times<\/em> article on high school graduation rates reported \u201cThe number of schools graduating 60 percent or fewer students in four years\u2014sometimes referred to as 'dropout factories'\u2014decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.\u201d\r\n<ol>\r\n \t<li>Is the \u201cdecreased by 17\u201d number a useful comparison?<\/li>\r\n \t<li>Considering the last sentence, can we conclude that the number of \u201cdropout factories\u201d was originally 34?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"713382\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"713382\"]\r\n<ol>\r\n \t<li>This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of \u201cdropout factories\u201d dropped from 20 to 3, that\u2019d be a very significant change, but if the number dropped from 217 to 200, that\u2019d be less of an improvement.<\/li>\r\n \t<li>The last sentence provides relative change, which helps put the first sentence in perspective. We can estimate that the number of \u201cdropout factories\u201d was probably previously around 34. However, it\u2019s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet's return to the example at the top of this page. In the 2004 vice-presidential debates, Democratic candidate John Edwards claimed that\u00a0US forces have suffered \"90% of the coalition casualties\" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies \"have taken almost 50 percent\" of the casualties. Who is correct?\r\n[reveal-answer q=\"908531\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"908531\"]Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward\u2019s claim was a percent with coalition forces as the base of the percent, while Cheney\u2019s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.[\/hidden-answer]\r\n\r\nA detailed explanation of these examples can be viewed here.\r\n\r\nhttps:\/\/youtu.be\/Svlu2Lurmsc\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Think About It<\/h3>\r\nIn the 2012 presidential elections, one candidate argued that \u201cthe president\u2019s plan will cut $716 billion from Medicare, leading to fewer services for seniors,\u201d while the other candidate rebuts that \u201cour plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.\u201d Are these claims in conflict, in agreement, or not comparable because they\u2019re talking about different things?\r\n<p class=\"p1\"><span class=\"s1\">[practice-area rows=\"8\"][\/practice-area]<\/span><\/p>\r\n\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14201133\/industrial-safety-1492046.png\"><img class=\"alignleft wp-image-492\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14201133\/industrial-safety-1492046.png\" alt=\"Yellow triangle sign of black exclamation mark\" width=\"95\" height=\"85\" \/><\/a>We\u2019ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA politician\u2019s support increases from 40% of voters to 50% of voters. Describe the change.\r\n[reveal-answer q=\"27288\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"27288\"]\r\n\r\nWe could describe this using an absolute change: [latex]|50\\%-40\\%|=10\\%[\/latex]. Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 <strong>percentage points<\/strong>.\r\n\r\nIn contrast, we could compute the percent change:[latex]\\displaystyle\\frac{10\\%}{40\\%}=0.25=25\\%[\/latex] increase. This is the relative change, and we\u2019d say the politician\u2019s support has increased by 25%.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLastly, a caution against averaging percents.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player\u2019s overall field goal percentage.\r\n\r\n[reveal-answer q=\"831091\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"831091\"]It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don\u2019t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then that player\u00a0made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, they player made 110 shots out of 300, for a [latex]\\displaystyle\\frac{110}{300}=0.367=36.7\\%[\/latex] overall field goal percentage.[\/hidden-answer]\r\n\r\n<\/div>\r\nFor more information about these cautionary tales using percentages, view the following.\r\n\r\nhttps:\/\/youtu.be\/vtgEkQUB5F8\r\n<h2>Proportions and Rates<\/h2>\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 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 Answer[\/reveal-answer]\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 20 miles per gallon.[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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\nFor an overview on rates and proportions, using the examples on this page, view the following video.\r\n\r\nhttps:\/\/youtu.be\/aZrio6ztHKE\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 Answer[\/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>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA map scale indicates that \u00bd inch on the map corresponds with 3 real miles. How many miles apart are two cities that are [latex]\\displaystyle{2}\\frac{1}{4}[\/latex] inches apart on the map?\r\n[reveal-answer q=\"439949\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"439949\"]\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, <em>x<\/em>, to represent the unknown quantity\u2014the mile distance between the cities.\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 <em>x\u00a0<\/em>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 3<\/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 2 (or divide by \u00bd)<\/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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nMany proportion problems can also be solved using <strong>dimensional analysis<\/strong>, the process of multiplying a quantity by rates to change the units.\r\n\r\n[\/hidden-answer]\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. How far can it drive on 40 gallons?\r\n[reveal-answer q=\"526887\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"526887\"]\r\n\r\nWe 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].\r\n\r\nHowever, we earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the <em>gallons<\/em> unit \u201ccancels\u201d and we\u2019re left with a number of miles:\r\n\r\n[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]\r\n\r\nNotice if instead we were asked \u201chow many gallons are needed to drive 50 miles?\u201d we could answer this question by inverting the 20 mile per gallon rate so that the <em>miles<\/em> unit cancels and we\u2019re left with gallons:\r\n\r\n[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]\r\n\r\n[\/hidden-answer]\r\n\r\nA worked example of this last question can be found in the following video.\r\n\r\nhttps:\/\/youtu.be\/jYwi3YqP0Wk\r\n\r\n<\/div>\r\nNotice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.\r\n\r\n&nbsp;\r\n\r\nYou have likely encountered distance, rate, and time problems in the past. This is likely because they are easy to visualize and most of us have experienced them first hand. In our next example, we will solve distance, rate and time problems that will require us to change the units that the distance or time is measured in.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?\r\n[reveal-answer q=\"946318\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"946318\"]\r\n\r\nTo answer this question, we need to convert 20 seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don\u2019t, we will need to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might start by converting the 20 seconds into hours:\r\n\r\n[latex]\\displaystyle{20}\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{1}{180}\\text{ hour}[\/latex]\r\n\r\nNow we can multiply by the 15 miles\/hr\r\n\r\n[latex]\\displaystyle\\frac{1}{180}\\text{ hour}\\cdot\\frac{15\\text{ miles}}{1\\text{ hour}}=\\frac{1}{12}\\text{ mile}[\/latex]\r\n\r\nNow we can convert to feet\r\n\r\n[latex]\\displaystyle\\frac{1}{12}\\text{ mile}\\cdot\\frac{5280\\text{ feet}}{1\\text{ mile}}=440\\text{ feet}[\/latex]\r\n\r\nWe could have also done this entire calculation in one long set of products:\r\n\r\n[latex]\\displaystyle20\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{15\\text{ miles}}{1\\text{ miles}}=\\frac{5280\\text{ feet}}{1\\text{ mile}}=\\frac{1}{180}\\text{ hour}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nView the following video to see this problem worked through.\r\n\r\nhttps:\/\/youtu.be\/fyOcLcIVipM\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\nA 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How much will 18 inches of the wire weigh, in ounces?\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17454&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose you\u2019re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?\r\n[reveal-answer q=\"815477\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"815477\"]\r\n\r\nIn 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, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:\r\n\r\n[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nOther quantities just don\u2019t scale proportionally at all.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000?\r\n[reveal-answer q=\"597027\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"597027\"]While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.[\/hidden-answer]\r\n\r\nMatters of scale in this example and the previous one are explained in more detail here.\r\n\r\nhttps:\/\/youtu.be\/-e2typcrhLE\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nThe 2010 U.S. military budget was $683.7 billion. To gain perspective on how much money this is, answer the following questions.\r\n<ol>\r\n \t<li>What would the salary of each of the 1.4 million Walmart employees in the US be if the military budget were distributed evenly amongst them?<\/li>\r\n \t<li>If you distributed the military budget of 2010 evenly amongst the 300 million people who live in the US, how much money would you give to each person?<\/li>\r\n \t<li>If you converted the US budget into $100 bills, how long would it take you to count it out - assume it takes one second to count one $100 bill.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"447493\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"447493\"]\r\n\r\nHere we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.\r\n<ol>\r\n \t<li>If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000.<\/li>\r\n \t<li>There are about 300 million people in the U.S. The military budget is about $2,200 per person.<\/li>\r\n \t<li>If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCompare the electricity consumption per capita in China to the rate in Japan.\r\n[reveal-answer q=\"924187\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"924187\"]\r\n\r\nTo address this question, we will first need data. From the CIA[footnote]<a href=\"https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html\" target=\"_blank\">https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html<\/a>[\/footnote]\u00a0website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries.\u00a0\u00a0 From the World Bank,[footnote]<a href=\"http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL\" target=\"_blank\">http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL<\/a>[\/footnote] we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million.\r\n\r\nComputing the consumption per capita for each country:\r\n\r\nChina: [latex]\\displaystyle\\frac{4,693,000,000,000\\text{KWH}}{1,344,130,000\\text{ people}}[\/latex] \u2248 3491.5 KWH per person\r\n\r\nJapan: [latex]\\displaystyle\\frac{859,700,000,000\\text{KWH}}{127,817,277\\text{ people}}[\/latex]\u00a0\u2248 6726 KWH per person\r\n\r\nWhile China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.\r\n\r\n[\/hidden-answer]\r\n\r\nWorking with large numbers is examined in more detail in this video.\r\n\r\nhttps:\/\/youtu.be\/rCLh8ZvSQr8\r\n\r\n<\/div>\r\n<h2>A Bit of Geometry<\/h2>\r\nGeometric shapes, as well as area and volumes, can often be important in problem solving.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14214054\/10131639494_43480a0a1f_z.jpg\"><img class=\"aligncenter size-full wp-image-501\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14214054\/10131639494_43480a0a1f_z.jpg\" alt=\"terraced hillside of rice paddies\" width=\"640\" height=\"427\" \/><\/a>\r\n\r\nLet's start things off with an example, rather than trying to explain geometric concepts to you.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<div>\r\n\r\nYou are curious how tall a tree is, but don\u2019t have any way to climb it. Describe a method for determining the height.\r\n[reveal-answer q=\"924148\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"924148\"]\r\n\r\nThere are several approaches we could take. We\u2019ll use one based on triangles, which requires that it\u2019s a sunny day. Suppose the tree is casting a shadow, say 15 ft long. I can then have a friend help me measure my own shadow. Suppose I am 6 ft tall, and cast a 1.5 ft shadow. Since the triangle formed by the tree and its shadow has the same angles as the triangle formed by me and my shadow, these triangles are called <strong>similar triangles<\/strong> and their sides will scale proportionally. In other words, the ratio of height to width will be the same in both triangles. Using this, we can find the height of the tree, which we\u2019ll denote by <em>h<\/em>:\r\n\r\n[latex]\\frac{6\\text{ft. tall}}{1.5\\text{ft. shadow}}=frac{h\\text{ft. tall}}{15\\text{ft. shadow}}\r\n<div>\r\n\r\nMultiplying both sides by 15, we get <em>h<\/em> = 60. The tree is about 60 ft tall.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Similar Triangles<\/h2>\r\nWe introduced the idea of similar triangles in the previous example. One property of geometric shapes that we have learned is a helpful problem-solving tool is that of similarity. \u00a0If two triangles are the same, meaning the angles between the sides are all the same, we can find an unknown length or height as in the last example. This idea of similarity holds for other geometric shapes as well.\r\n<div class=\"textbox exercises\">\r\n<h3>Guided Example<\/h3>\r\nMary was out in the yard one day and had her two daughters with her. She\u00a0was doing some renovations and wanted to know how tall the house was.\u00a0She noticed a shadow 3 feet long when her daughter was standing 12 feet\u00a0from the house and used it to set up figure 1.\r\n\r\n[caption id=\"attachment_447\" align=\"alignnone\" width=\"358\"]<img class=\"wp-image-447 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155003\/Fig2_3_1.png\" alt=\"Fig2_3_1\" width=\"358\" height=\"339\" \/> Figure 1.[\/caption]\r\n\r\nWe can take that drawing and separate the two triangles as follows allowing us to\u00a0focus on the numbers and the shapes.\r\n\r\nThese triangles are what are called <strong>similar triangles<\/strong>. They have the\u00a0same angles and sides in <em>proportion<\/em>\u00a0to each other. We can use that\u00a0information to determine the height\u00a0of the house as seen in figure 2.\r\n\r\n[caption id=\"attachment_448\" align=\"alignnone\" width=\"419\"]<img class=\"size-full wp-image-448\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155004\/Fig2_3_2.png\" alt=\"Figure 2.\" width=\"419\" height=\"268\" \/> Figure 2.[\/caption]\r\n\r\nTo determine the height of the house, we set up the following proportion:\r\n\r\n[latex]\\displaystyle\\frac{x}{15}=\\frac{5}{3}\\\\[\/latex]\r\n\r\nThen, we solve for the unknown <em>x<\/em> by using cross products as we have done before:\r\n\r\n[latex]\\displaystyle{x}=\\frac{5\\times{15}}{3}=\\frac{75}{3}=25\\\\[\/latex]\r\n\r\nTherefore, we can conclude that the house is 25 feet high.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=42493&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"460\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIt may be helpful to recall some formulas for areas and volumes of a few basic shapes:\r\n<div class=\"textbox\">\r\n<h3>Areas<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Rectangle<\/strong><\/td>\r\n<td><strong>Circle<\/strong>, radius <em>r<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Area: [latex]L\\times{W}[\/latex]<\/td>\r\n<td>Area: [latex]\\pi{r^2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Perimeter: [latex]2l+2W[\/latex]<\/td>\r\n<td>\u00a0Circumference[latex]2\\pi{r}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10191723\/Screen-Shot-2017-02-10-at-11.17.08-AM.png\">\r\n<img class=\" wp-image-1502 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10191723\/Screen-Shot-2017-02-10-at-11.17.08-AM-300x82.png\" alt=\"\" width=\"531\" height=\"146\" \/><\/a>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Rectangular Box<\/strong><\/td>\r\n<td>Cylinder<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Volume: [latex]L\\times{W}\\times{H}[\/latex]<em>\u00a0 \u00a0<\/em><\/td>\r\n<td>\u00a0<em>\u00a0<\/em>Volume: [latex]\\pi{r^2}h[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<em>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10192318\/Screen-Shot-2017-02-10-at-11.23.02-AM.png\"><img class=\" wp-image-1503 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10192318\/Screen-Shot-2017-02-10-at-11.23.02-AM-300x82.png\" alt=\"\" width=\"512\" height=\"140\" \/><\/a><\/em>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn our next two examples, we will combine the ideas we have explored about ratios with the geometry of some basic shapes to answer questions. \u00a0In the first example, we will predict how much dough will be needed for a pizza that is 16 inches in diameter given that we know how much dough it takes for a pizza with a diameter of 12 inches. The second example uses the volume of a cylinder to determine the number of calories in a marshmallow.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nIf a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?\r\n[reveal-answer q=\"967082\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"967082\"]\r\n\r\nTo answer this question, we need to consider how the weight of the dough will scale. The weight will be based on the volume of the dough. However, since both pizzas will be about the same thickness, the weight will scale with the area of the top of the pizza. We can find the area of each pizza using the formula for area of a circle, [latex]A=\\pi{r}^2[\/latex]:\r\n\r\nA 12\" pizza has radius 6 inches, so the area will be [latex]\\pi6^2[\/latex]\u00a0= about 113 square inches.\r\n\r\nA 16\" pizza has radius 8 inches, so the area will be [latex]\\pi8^2[\/latex]\u00a0= about 201 square inches.\r\n\r\nNotice that if both pizzas were 1 inch thick, the volumes would be 113 in<sup>3<\/sup> and 201 in<sup>3<\/sup> respectively, which are at the same ratio as the areas. As mentioned earlier, since the thickness is the same for both pizzas, we can safely ignore it.\r\n\r\nWe can now set up a proportion to find the weight of the dough for a 16\" pizza:\r\n\r\n[latex]\\displaystyle\\frac{10\\text{ ounces}}{113\\text{in}^2}=\\frac{x\\text{ ounces}}{201\\text{in}^2}[\/latex]\r\n\r\nMultiply both sides by 201\r\n\r\n[latex]\\displaystyle{x}=201\\cdot\\frac{10}{113}[\/latex] = about 17.8 ounces of dough for a 16\"\u00a0pizza.\r\n\r\nIt is interesting to note that while the diameter is [latex]\\displaystyle\\frac{16}{12}[\/latex] = 1.33 times larger, the dough required, which scales with area, is 1.33<sup>2<\/sup> = 1.78 times larger.\r\n\r\n[\/hidden-answer]\r\n\r\nThe following video illustrates how to solve this problem.\r\n\r\nhttps:\/\/youtu.be\/e75bk1qCsUE\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA company makes regular and jumbo marshmallows. The regular marshmallow has 25 calories. How many calories will the jumbo marshmallow have?\r\n[reveal-answer q=\"353804\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"353804\"]\r\n\r\nWe would expect the calories to scale with volume. Since the marshmallows have cylindrical shapes, we can use that formula to find the volume. From the grid in the image, we can estimate the radius and height of each marshmallow.\r\n\r\nThe regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units. The volume is about \u03c0(1.75)<sup>2<\/sup>(3.5) = 33.7 units<sup>3<\/sup>.\r\n\r\nThe jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units. The volume is about \u03c0(2.75)<sup>2<\/sup>(5) = 118.8\u00a0units<sup>3<\/sup>.\r\n\r\nWe could now set up a proportion, or use rates. The regular marshmallow has 25 calories for 33.7 cubic units of volume. The jumbo marshmallow will have:\r\n\r\n[latex]\\displaystyle{118.8}\\text{ units}^3\\cdot\\frac{25\\text{ calories}}{33.7\\text{ units}^3}=88.1\\text{ calories}[\/latex]\r\n\r\nIt is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about 1.5<sup>3<\/sup> = 3.375 times larger.\r\n\r\n[\/hidden-answer]\r\n\r\nFor more about the marshmallow example, watch this video.\r\n\r\nhttps:\/\/youtu.be\/QJgGpxzRt6Y\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\nA website says that you\u2019ll need 48 fifty-pound bags of sand to fill a sandbox that measure 8ft by 8ft by 1ft. How many bags would you need for a sandbox 6ft by 4ft by 1ft?\r\n[reveal-answer q=\"953701\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"953701\"]\r\n<div>\r\n\r\nThe original sandbox has volume [latex]64\\text{ft}^3[\/latex]. The smaller sandbox has volume [latex]24\\text{ft}^3[\/latex].\r\n\r\n[latex]\\displaystyle\\frac{48\\text{bags}}{64\\text{ft}^2}=\\frac{x\\text{ bags}}{24\\text{in}^3}[\/latex] results in <em>x<\/em> = 18 bags.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\nMary (from the application that started this topic), decides to use what she knows about the height of the roof to measure the height of her second daughter. If her second daughter casts a shadow that is 1.5 feet long when she is 13.5 feet from the house, what is the height of the second daughter? Draw an accurate diagram and use similar triangles to solve.\r\n[reveal-answer q=\"434062\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"434062\"]2.5 ft[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17456&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nIn the next section, we will explore the process of combining different types of information to answer questions.","rendered":"<p>In the 2004 vice-presidential debates, Democratic contender John Edwards claimed that\u00a0US forces have suffered &#8220;90% of the coalition casualties&#8221; in Iraq. Incumbent Vice President Dick Cheney disputed this, saying that in fact Iraqi security forces and coalition allies &#8220;have taken almost 50 percent&#8221; of the casualties.<a class=\"footnote\" title=\"http:\/\/www.factcheck.org\/cheney_edwards_mangle_facts.html\" id=\"return-footnote-129-1\" href=\"#footnote-129-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>Who was\u00a0correct? How can we make sense of these numbers?<\/p>\n<p>In this section, we will show how the idea of percent is used to describe parts of a whole. \u00a0Percents are prevalent in the media we consume regularly, making it imperative that you understand what they mean and where they come from.<\/p>\n<p>We will also show you how to compare different quantities using proportions. \u00a0Proportions can help us understand how things change or relate to each other.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this lesson you will learn how to do the following:<\/p>\n<ul>\n<li>Given the part and the whole, write a percent<\/li>\n<li>Calculate both relative\u00a0and absolute change of a quantity<\/li>\n<li>Calculate tax on a purchase<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Percents<\/h2>\n<p>Percent literally means \u201cper 100,\u201d or \u201cparts per hundred.\u201d When we write 40%, this is equivalent to the fraction [latex]\\displaystyle\\frac{40}{100}[\/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\\displaystyle\\frac{80}{200}=\\frac{10}{25}=\\frac{40}{100}[\/latex].<\/p>\n<div id=\"attachment_494\" style=\"width: 510px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-494\" class=\"wp-image-494\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\" alt=\"Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty.\" width=\"500\" height=\"282\" \/><\/a><\/p>\n<p id=\"caption-attachment-494\" class=\"wp-caption-text\">A visual depiction of 40%<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Percent<\/h3>\n<p>If we have a <em>part<\/em> that is some <em>percent<\/em> of a <em>whole<\/em>, then\u00a0[latex]\\displaystyle\\text{percent}=\\frac{\\text{part}}{\\text{whole}}[\/latex], or equivalently, [latex]\\text{part}\\cdot\\text{whole}=\\text{percent}[\/latex].<\/p>\n<p>To do the calculations, we write the percent as a decimal.<\/p>\n<p>For a refresher on basic percentage rules, using the examples on this page, view the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Review of basic percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Z229RysttR8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>In a survey, 243 out of 400 people state that they like dogs. What percent is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987171\">Show Answer<\/span><\/p>\n<div id=\"q987171\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex] This is 60.75%.<\/p>\n<p>Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write each as a percent:<\/p>\n<ol>\n<li>[latex]\\displaystyle\\frac{1}{4}[\/latex]<\/li>\n<li>0.02<\/li>\n<li>2.35<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660805\">Show Answer<\/span><\/p>\n<div id=\"q660805\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\displaystyle\\frac{1}{4}=0.25[\/latex] = 25%<\/li>\n<li>0.02 = 2%<\/li>\n<li>2.35 = 235%<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p>Throughout this text, you will be given opportunities to answer questions and know immediately whether you answered correctly. To answer the question below, do the calculation on a separate piece of paper and enter your answer in the box. Click on the submit button , and if you are correct, a green box will appear around your answer. \u00a0If you are incorrect, a red box will appear. \u00a0You can click on &#8220;Try Another Version of This Question&#8221; as many times as you like. Practice all you want!<\/p>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17441&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In the news, you hear \u201ctuition is expected to increase by 7% next year.\u201d If tuition this year was $1200 per quarter, what will it be next year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q475615\">Show Answer<\/span><\/p>\n<div id=\"q475615\" class=\"hidden-answer\" style=\"display: none\">The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year\u2019s tuition:\u00a0$1200(1.07) = $1284.<\/p>\n<p>Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84.<\/p>\n<p>Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase, so to calculate the expected tuition, we\u2019ll need to add this change to the previous year\u2019s tuition:\u00a0$1200 + $84 = $1284.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17447&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q573833\">Show Answer<\/span><\/p>\n<div id=\"q573833\" class=\"hidden-answer\" style=\"display: none\">\n<p>To compute the percent change, we first need to find the dollar value change: $6800 \u2013\u00a0$7400 = \u2013$600. Often we will take the absolute value of this amount, which is called the <strong>absolute change<\/strong>: |\u2013600| = 600.<\/p>\n<p>Since we are computing the decrease relative to the starting value, we compute this percent out of $7400:<\/p>\n<p>[latex]\\displaystyle\\frac{600}{7400}=0.081=[\/latex] 8.1% decrease. This is called a <strong>relative change<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Absolute and Relative Change<\/h3>\n<p>Given two quantities,<\/p>\n<p>Absolute change =[latex]\\displaystyle|\\text{ending quantity}-\\text{starting quantity}|[\/latex]<\/p>\n<p>Relative change: [latex]\\displaystyle\\frac{\\text{absolute change}}{\\text{starting quantity}}[\/latex]<\/p>\n<ul>\n<li>Absolute change has the same units as the original quantity.<\/li>\n<li>Relative change gives a percent change.<\/li>\n<\/ul>\n<p>The starting quantity is called the <strong>base<\/strong> of the percent change.<\/p>\n<p>For a deeper dive on absolute and relative change, using the examples on this page, view the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Absolute and Relative Differences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QjVeurkg8CQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a \u201cgateway\u201d drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn\u2019t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs.<a class=\"footnote\" title=\"http:\/\/tvtropes.org\/pmwiki\/pmwiki.php\/Main\/LiesDamnedLiesAndStatistics\" id=\"return-footnote-129-2\" href=\"#footnote-129-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933757\">Show Answer<\/span><\/p>\n<div id=\"q933757\" class=\"hidden-answer\" style=\"display: none\">\n<p>When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 \u2013 75 = 140. From this, we could say \u201cAlbertsons has 140 more stores than QFC.\u201d However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:<\/p>\n<p>Using QFC as the base, [latex]\\displaystyle\\frac{140}{75}=1.867[\/latex].<\/p>\n<p>This tells us Albertsons is 186.7% larger than QFC.<\/p>\n<p>Using Albertsons as the base,[latex]\\displaystyle\\frac{140}{215}=0.651[\/latex].<\/p>\n<p>This tells us QFC is 65.1% smaller than Albertsons.<\/p>\n<p>Notice both of these are showing percent <em>differences<\/em>. We could also calculate the size of Albertsons relative to QFC:[latex]\\displaystyle\\frac{215}{75}=2.867[\/latex], which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:[latex]\\displaystyle\\frac{75}{215}=0.349[\/latex], which tells us that QFC is 34.9% of the size of Albertsons.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568319\">Show Answer<\/span><\/p>\n<div id=\"q568319\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer this question, suppose the value started at $100. After one week, the value dropped by 60%:\u00a0$100 \u2013 $100(0.60) = $100 \u2013 $60 = $40.<\/p>\n<p>In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase:\u00a0$40 + $40(0.75) = $40 + $30 = $70.<\/p>\n<p>In the end, the stock is still $30 lower, or [latex]\\displaystyle\\frac{\\$30}{100}[\/latex] = 30% lower, valued than it started.<\/p>\n<\/div>\n<\/div>\n<p>A video walk-through of this example can be seen here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Combining percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4HNxwYMTNl8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>Consideration of the base of percentages is explored in this video, using the examples on this page.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Importance of base in percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nygw69JqwoQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17443&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A <em>Seattle Times<\/em> article on high school graduation rates reported \u201cThe number of schools graduating 60 percent or fewer students in four years\u2014sometimes referred to as &#8216;dropout factories&#8217;\u2014decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.\u201d<\/p>\n<ol>\n<li>Is the \u201cdecreased by 17\u201d number a useful comparison?<\/li>\n<li>Considering the last sentence, can we conclude that the number of \u201cdropout factories\u201d was originally 34?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713382\">Show Answer<\/span><\/p>\n<div id=\"q713382\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of \u201cdropout factories\u201d dropped from 20 to 3, that\u2019d be a very significant change, but if the number dropped from 217 to 200, that\u2019d be less of an improvement.<\/li>\n<li>The last sentence provides relative change, which helps put the first sentence in perspective. We can estimate that the number of \u201cdropout factories\u201d was probably previously around 34. However, it\u2019s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let&#8217;s return to the example at the top of this page. In the 2004 vice-presidential debates, Democratic candidate John Edwards claimed that\u00a0US forces have suffered &#8220;90% of the coalition casualties&#8221; in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies &#8220;have taken almost 50 percent&#8221; of the casualties. Who is correct?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q908531\">Show Answer<\/span><\/p>\n<div id=\"q908531\" class=\"hidden-answer\" style=\"display: none\">Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward\u2019s claim was a percent with coalition forces as the base of the percent, while Cheney\u2019s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.<\/div>\n<\/div>\n<p>A detailed explanation of these examples can be viewed here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Evaluating claims involving percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Svlu2Lurmsc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Think About It<\/h3>\n<p>In the 2012 presidential elections, one candidate argued that \u201cthe president\u2019s plan will cut $716 billion from Medicare, leading to fewer services for seniors,\u201d while the other candidate rebuts that \u201cour plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.\u201d Are these claims in conflict, in agreement, or not comparable because they\u2019re talking about different things?<\/p>\n<p class=\"p1\"><span class=\"s1\"><textarea aria-label=\"Your Answer\" rows=\"8\"><\/textarea><\/span><\/p>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14201133\/industrial-safety-1492046.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-492\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14201133\/industrial-safety-1492046.png\" alt=\"Yellow triangle sign of black exclamation mark\" width=\"95\" height=\"85\" \/><\/a>We\u2019ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A politician\u2019s support increases from 40% of voters to 50% of voters. Describe the change.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q27288\">Show Answer<\/span><\/p>\n<div id=\"q27288\" class=\"hidden-answer\" style=\"display: none\">\n<p>We could describe this using an absolute change: [latex]|50\\%-40\\%|=10\\%[\/latex]. Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 <strong>percentage points<\/strong>.<\/p>\n<p>In contrast, we could compute the percent change:[latex]\\displaystyle\\frac{10\\%}{40\\%}=0.25=25\\%[\/latex] increase. This is the relative change, and we\u2019d say the politician\u2019s support has increased by 25%.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Lastly, a caution against averaging percents.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player\u2019s overall field goal percentage.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q831091\">Show Answer<\/span><\/p>\n<div id=\"q831091\" class=\"hidden-answer\" style=\"display: none\">It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don\u2019t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then that player\u00a0made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, they player made 110 shots out of 300, for a [latex]\\displaystyle\\frac{110}{300}=0.367=36.7\\%[\/latex] overall field goal percentage.<\/div>\n<\/div>\n<\/div>\n<p>For more information about these cautionary tales using percentages, view the following.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Percentage points and averaging percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vtgEkQUB5F8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Proportions and Rates<\/h2>\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 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 Answer<\/span><\/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 20 miles per gallon.<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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>For an overview on rates and proportions, using the examples on this page, view the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Basic rates and proportions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aZrio6ztHKE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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 Answer<\/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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A map scale indicates that \u00bd inch on the map corresponds with 3 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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q439949\">Show Answer<\/span><\/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, <em>x<\/em>, 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 <em>x\u00a0<\/em>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 3<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle\\frac{1}{2}x=\\frac{27}{4}[\/latex]<\/td>\n<td>Multiply both sides by 2 (or divide by \u00bd)<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle{x}=\\frac{27}{2}=13\\frac{1}{2}\\text{ miles}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Many proportion problems can also be solved using <strong>dimensional analysis<\/strong>, the process of multiplying a quantity by rates to change the units.<\/p>\n<\/div>\n<\/div>\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. How far can it drive on 40 gallons?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q526887\">Show Answer<\/span><\/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].<\/p>\n<p>However, we earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the <em>gallons<\/em> unit \u201ccancels\u201d and we\u2019re left with a number of miles:<\/p>\n<p>[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 50 miles?\u201d we could answer this question by inverting the 20 mile per gallon rate so that the <em>miles<\/em> unit cancels and we\u2019re left with gallons:<\/p>\n<p>[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<\/div>\n<\/div>\n<p>A worked example of this last question can be found in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" 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<\/div>\n<p>Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.<\/p>\n<p>&nbsp;<\/p>\n<p>You have likely encountered distance, rate, and time problems in the past. This is likely because they are easy to visualize and most of us have experienced them first hand. In our next example, we will solve distance, rate and time problems that will require us to change the units that the distance or time is measured in.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q946318\">Show Answer<\/span><\/p>\n<div id=\"q946318\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer this question, we need to convert 20 seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don\u2019t, we will need to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might start by converting the 20 seconds into hours:<\/p>\n<p>[latex]\\displaystyle{20}\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{1}{180}\\text{ hour}[\/latex]<\/p>\n<p>Now we can multiply by the 15 miles\/hr<\/p>\n<p>[latex]\\displaystyle\\frac{1}{180}\\text{ hour}\\cdot\\frac{15\\text{ miles}}{1\\text{ hour}}=\\frac{1}{12}\\text{ mile}[\/latex]<\/p>\n<p>Now we can convert to feet<\/p>\n<p>[latex]\\displaystyle\\frac{1}{12}\\text{ mile}\\cdot\\frac{5280\\text{ feet}}{1\\text{ mile}}=440\\text{ feet}[\/latex]<\/p>\n<p>We could have also done this entire calculation in one long set of products:<\/p>\n<p>[latex]\\displaystyle20\\text{ seconds}\\cdot\\frac{1\\text{ minute}}{60\\text{ seconds}}\\cdot\\frac{1\\text{ hour}}{60\\text{ minutes}}=\\frac{15\\text{ miles}}{1\\text{ miles}}=\\frac{5280\\text{ feet}}{1\\text{ mile}}=\\frac{1}{180}\\text{ hour}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>View the following video to see this problem worked through.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Proportions with unit conversion\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fyOcLcIVipM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p>A 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How much will 18 inches of the wire weigh, in ounces?<br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17454&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose you\u2019re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815477\">Show Answer<\/span><\/p>\n<div id=\"q815477\" class=\"hidden-answer\" style=\"display: none\">\n<p>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, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:<\/p>\n<p>[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Other quantities just don\u2019t scale proportionally at all.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q597027\">Show Answer<\/span><\/p>\n<div id=\"q597027\" class=\"hidden-answer\" style=\"display: none\">While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.<\/div>\n<\/div>\n<p>Matters of scale in this example and the previous one are explained in more detail here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Considering how\/if things scale\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-e2typcrhLE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>The 2010 U.S. military budget was $683.7 billion. To gain perspective on how much money this is, answer the following questions.<\/p>\n<ol>\n<li>What would the salary of each of the 1.4 million Walmart employees in the US be if the military budget were distributed evenly amongst them?<\/li>\n<li>If you distributed the military budget of 2010 evenly amongst the 300 million people who live in the US, how much money would you give to each person?<\/li>\n<li>If you converted the US budget into $100 bills, how long would it take you to count it out &#8211; assume it takes one second to count one $100 bill.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q447493\">Show Answer<\/span><\/p>\n<div id=\"q447493\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.<\/p>\n<ol>\n<li>If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000.<\/li>\n<li>There are about 300 million people in the U.S. The military budget is about $2,200 per person.<\/li>\n<li>If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Compare the electricity consumption per capita in China to the rate in Japan.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924187\">Show Answer<\/span><\/p>\n<div id=\"q924187\" class=\"hidden-answer\" style=\"display: none\">\n<p>To address this question, we will first need data. From the CIA<a class=\"footnote\" title=\"https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html\" id=\"return-footnote-129-3\" href=\"#footnote-129-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>\u00a0website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries.\u00a0\u00a0 From the World Bank,<a class=\"footnote\" title=\"http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL\" id=\"return-footnote-129-4\" href=\"#footnote-129-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a> we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million.<\/p>\n<p>Computing the consumption per capita for each country:<\/p>\n<p>China: [latex]\\displaystyle\\frac{4,693,000,000,000\\text{KWH}}{1,344,130,000\\text{ people}}[\/latex] \u2248 3491.5 KWH per person<\/p>\n<p>Japan: [latex]\\displaystyle\\frac{859,700,000,000\\text{KWH}}{127,817,277\\text{ people}}[\/latex]\u00a0\u2248 6726 KWH per person<\/p>\n<p>While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.<\/p>\n<\/div>\n<\/div>\n<p>Working with large numbers is examined in more detail in this video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Comparing quantities involving large numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rCLh8ZvSQr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<h2>A Bit of Geometry<\/h2>\n<p>Geometric shapes, as well as area and volumes, can often be important in problem solving.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14214054\/10131639494_43480a0a1f_z.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-501\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14214054\/10131639494_43480a0a1f_z.jpg\" alt=\"terraced hillside of rice paddies\" width=\"640\" height=\"427\" \/><\/a><\/p>\n<p>Let&#8217;s start things off with an example, rather than trying to explain geometric concepts to you.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<div>\n<p>You are curious how tall a tree is, but don\u2019t have any way to climb it. Describe a method for determining the height.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924148\">Show Answer<\/span><\/p>\n<div id=\"q924148\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are several approaches we could take. We\u2019ll use one based on triangles, which requires that it\u2019s a sunny day. Suppose the tree is casting a shadow, say 15 ft long. I can then have a friend help me measure my own shadow. Suppose I am 6 ft tall, and cast a 1.5 ft shadow. Since the triangle formed by the tree and its shadow has the same angles as the triangle formed by me and my shadow, these triangles are called <strong>similar triangles<\/strong> and their sides will scale proportionally. In other words, the ratio of height to width will be the same in both triangles. Using this, we can find the height of the tree, which we\u2019ll denote by <em>h<\/em>:<\/p>\n<p>[latex][\/latex]\\frac{6\\text{ft. tall}}{1.5\\text{ft. shadow}}=frac{h\\text{ft. tall}}{15\\text{ft. shadow}}<\/p>\n<div>\n<p>Multiplying both sides by 15, we get <em>h<\/em> = 60. The tree is about 60 ft tall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Similar Triangles<\/h2>\n<p>We introduced the idea of similar triangles in the previous example. One property of geometric shapes that we have learned is a helpful problem-solving tool is that of similarity. \u00a0If two triangles are the same, meaning the angles between the sides are all the same, we can find an unknown length or height as in the last example. This idea of similarity holds for other geometric shapes as well.<\/p>\n<div class=\"textbox exercises\">\n<h3>Guided Example<\/h3>\n<p>Mary was out in the yard one day and had her two daughters with her. She\u00a0was doing some renovations and wanted to know how tall the house was.\u00a0She noticed a shadow 3 feet long when her daughter was standing 12 feet\u00a0from the house and used it to set up figure 1.<\/p>\n<div id=\"attachment_447\" style=\"width: 368px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-447\" class=\"wp-image-447 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155003\/Fig2_3_1.png\" alt=\"Fig2_3_1\" width=\"358\" height=\"339\" \/><\/p>\n<p id=\"caption-attachment-447\" class=\"wp-caption-text\">Figure 1.<\/p>\n<\/div>\n<p>We can take that drawing and separate the two triangles as follows allowing us to\u00a0focus on the numbers and the shapes.<\/p>\n<p>These triangles are what are called <strong>similar triangles<\/strong>. They have the\u00a0same angles and sides in <em>proportion<\/em>\u00a0to each other. We can use that\u00a0information to determine the height\u00a0of the house as seen in figure 2.<\/p>\n<div id=\"attachment_448\" style=\"width: 429px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-448\" class=\"size-full wp-image-448\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155004\/Fig2_3_2.png\" alt=\"Figure 2.\" width=\"419\" height=\"268\" \/><\/p>\n<p id=\"caption-attachment-448\" class=\"wp-caption-text\">Figure 2.<\/p>\n<\/div>\n<p>To determine the height of the house, we set up the following proportion:<\/p>\n<p>[latex]\\displaystyle\\frac{x}{15}=\\frac{5}{3}\\\\[\/latex]<\/p>\n<p>Then, we solve for the unknown <em>x<\/em> by using cross products as we have done before:<\/p>\n<p>[latex]\\displaystyle{x}=\\frac{5\\times{15}}{3}=\\frac{75}{3}=25\\\\[\/latex]<\/p>\n<p>Therefore, we can conclude that the house is 25 feet high.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=42493&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"460\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>It may be helpful to recall some formulas for areas and volumes of a few basic shapes:<\/p>\n<div class=\"textbox\">\n<h3>Areas<\/h3>\n<table>\n<tbody>\n<tr>\n<td><strong>Rectangle<\/strong><\/td>\n<td><strong>Circle<\/strong>, radius <em>r<\/em><\/td>\n<\/tr>\n<tr>\n<td>Area: [latex]L\\times{W}[\/latex]<\/td>\n<td>Area: [latex]\\pi{r^2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Perimeter: [latex]2l+2W[\/latex]<\/td>\n<td>\u00a0Circumference[latex]2\\pi{r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10191723\/Screen-Shot-2017-02-10-at-11.17.08-AM.png\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1502 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10191723\/Screen-Shot-2017-02-10-at-11.17.08-AM-300x82.png\" alt=\"\" width=\"531\" height=\"146\" \/><\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<table>\n<tbody>\n<tr>\n<td><strong>Rectangular Box<\/strong><\/td>\n<td>Cylinder<\/td>\n<\/tr>\n<tr>\n<td>Volume: [latex]L\\times{W}\\times{H}[\/latex]<em>\u00a0 \u00a0<\/em><\/td>\n<td>\u00a0<em>\u00a0<\/em>Volume: [latex]\\pi{r^2}h[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><em>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10192318\/Screen-Shot-2017-02-10-at-11.23.02-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1503 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/10192318\/Screen-Shot-2017-02-10-at-11.23.02-AM-300x82.png\" alt=\"\" width=\"512\" height=\"140\" \/><\/a><\/em><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In our next two examples, we will combine the ideas we have explored about ratios with the geometry of some basic shapes to answer questions. \u00a0In the first example, we will predict how much dough will be needed for a pizza that is 16 inches in diameter given that we know how much dough it takes for a pizza with a diameter of 12 inches. The second example uses the volume of a cylinder to determine the number of calories in a marshmallow.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967082\">Show Answer<\/span><\/p>\n<div id=\"q967082\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer this question, we need to consider how the weight of the dough will scale. The weight will be based on the volume of the dough. However, since both pizzas will be about the same thickness, the weight will scale with the area of the top of the pizza. We can find the area of each pizza using the formula for area of a circle, [latex]A=\\pi{r}^2[\/latex]:<\/p>\n<p>A 12\" pizza has radius 6 inches, so the area will be [latex]\\pi6^2[\/latex]\u00a0= about 113 square inches.<\/p>\n<p>A 16\" pizza has radius 8 inches, so the area will be [latex]\\pi8^2[\/latex]\u00a0= about 201 square inches.<\/p>\n<p>Notice that if both pizzas were 1 inch thick, the volumes would be 113 in<sup>3<\/sup> and 201 in<sup>3<\/sup> respectively, which are at the same ratio as the areas. As mentioned earlier, since the thickness is the same for both pizzas, we can safely ignore it.<\/p>\n<p>We can now set up a proportion to find the weight of the dough for a 16\" pizza:<\/p>\n<p>[latex]\\displaystyle\\frac{10\\text{ ounces}}{113\\text{in}^2}=\\frac{x\\text{ ounces}}{201\\text{in}^2}[\/latex]<\/p>\n<p>Multiply both sides by 201<\/p>\n<p>[latex]\\displaystyle{x}=201\\cdot\\frac{10}{113}[\/latex] = about 17.8 ounces of dough for a 16\"\u00a0pizza.<\/p>\n<p>It is interesting to note that while the diameter is [latex]\\displaystyle\\frac{16}{12}[\/latex] = 1.33 times larger, the dough required, which scales with area, is 1.33<sup>2<\/sup> = 1.78 times larger.<\/p>\n<\/div>\n<\/div>\n<p>The following video illustrates how to solve this problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-12\" title=\"Scaling with area\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/e75bk1qCsUE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A company makes regular and jumbo marshmallows. The regular marshmallow has 25 calories. How many calories will the jumbo marshmallow have?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q353804\">Show Answer<\/span><\/p>\n<div id=\"q353804\" class=\"hidden-answer\" style=\"display: none\">\n<p>We would expect the calories to scale with volume. Since the marshmallows have cylindrical shapes, we can use that formula to find the volume. From the grid in the image, we can estimate the radius and height of each marshmallow.<\/p>\n<p>The regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units. The volume is about \u03c0(1.75)<sup>2<\/sup>(3.5) = 33.7 units<sup>3<\/sup>.<\/p>\n<p>The jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units. The volume is about \u03c0(2.75)<sup>2<\/sup>(5) = 118.8\u00a0units<sup>3<\/sup>.<\/p>\n<p>We could now set up a proportion, or use rates. The regular marshmallow has 25 calories for 33.7 cubic units of volume. The jumbo marshmallow will have:<\/p>\n<p>[latex]\\displaystyle{118.8}\\text{ units}^3\\cdot\\frac{25\\text{ calories}}{33.7\\text{ units}^3}=88.1\\text{ calories}[\/latex]<\/p>\n<p>It is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about 1.5<sup>3<\/sup> = 3.375 times larger.<\/p>\n<\/div>\n<\/div>\n<p>For more about the marshmallow example, watch this video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-13\" title=\"Scaling with volume\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QJgGpxzRt6Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p>A website says that you\u2019ll need 48 fifty-pound bags of sand to fill a sandbox that measure 8ft by 8ft by 1ft. How many bags would you need for a sandbox 6ft by 4ft by 1ft?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q953701\">Show Answer<\/span><\/p>\n<div id=\"q953701\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>The original sandbox has volume [latex]64\\text{ft}^3[\/latex]. The smaller sandbox has volume [latex]24\\text{ft}^3[\/latex].<\/p>\n<p>[latex]\\displaystyle\\frac{48\\text{bags}}{64\\text{ft}^2}=\\frac{x\\text{ bags}}{24\\text{in}^3}[\/latex] results in <em>x<\/em> = 18 bags.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Mary (from the application that started this topic), decides to use what she knows about the height of the roof to measure the height of her second daughter. If her second daughter casts a shadow that is 1.5 feet long when she is 13.5 feet from the house, what is the height of the second daughter? Draw an accurate diagram and use similar triangles to solve.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q434062\">Show Answer<\/span><\/p>\n<div id=\"q434062\" class=\"hidden-answer\" style=\"display: none\">2.5 ft<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17456&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In the next section, we will explore the process of combining different types of information to answer questions.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-129\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Introduction: Proportional Relationships and a Bit of Geometry. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Caution sign. <strong>Authored by<\/strong>: JDDesign. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/\">https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>40% shaded rectangle. <strong>Authored by<\/strong>: Clker-Free-Vector-Images. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/percent-40-bar-progress-meter-40844\/\">https:\/\/pixabay.com\/en\/percent-40-bar-progress-meter-40844\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Review of basic percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Z229RysttR8\">https:\/\/youtu.be\/Z229RysttR8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Absolute and Relative Differences. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QjVeurkg8CQ\">https:\/\/youtu.be\/QjVeurkg8CQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Importance of base in percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nygw69JqwoQ\">https:\/\/youtu.be\/nygw69JqwoQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Combining percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/4HNxwYMTNl8\">https:\/\/youtu.be\/4HNxwYMTNl8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluating claims involving percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Svlu2Lurmsc\">https:\/\/youtu.be\/Svlu2Lurmsc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Percentage points and averaging percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vtgEkQUB5F8\">https:\/\/youtu.be\/vtgEkQUB5F8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 17441, 17447, 17443. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>wind-364996_1280. <strong>Authored by<\/strong>: Stevebidmead. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/wind-turbines-farmland-364996\/\">https:\/\/pixabay.com\/en\/wind-turbines-farmland-364996\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Basic rates and proportions. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aZrio6ztHKE\">https:\/\/youtu.be\/aZrio6ztHKE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Proportions using dimensional analysis. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jYwi3YqP0Wk\">https:\/\/youtu.be\/jYwi3YqP0Wk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Proportions with unit conversion. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/fyOcLcIVipM\">https:\/\/youtu.be\/fyOcLcIVipM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Considering how\/if things scale. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-e2typcrhLE\">https:\/\/youtu.be\/-e2typcrhLE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Comparing quantities involving large numbers. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rCLh8ZvSQr8\">https:\/\/youtu.be\/rCLh8ZvSQr8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 17454. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-129-1\"><a href=\"http:\/\/www.factcheck.org\/cheney_edwards_mangle_facts.html\" target=\"_blank\">http:\/\/www.factcheck.org\/cheney_edwards_mangle_facts.html<\/a> <a href=\"#return-footnote-129-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-129-2\"><a href=\"http:\/\/tvtropes.org\/pmwiki\/pmwiki.php\/Main\/LiesDamnedLiesAndStatistics\" target=\"_blank\">http:\/\/tvtropes.org\/pmwiki\/pmwiki.php\/Main\/LiesDamnedLiesAndStatistics<\/a> <a href=\"#return-footnote-129-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-129-3\"><a href=\"https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html\" target=\"_blank\">https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html<\/a> <a href=\"#return-footnote-129-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-129-4\"><a href=\"http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL\" target=\"_blank\">http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL<\/a> <a href=\"#return-footnote-129-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Introduction: Proportional Relationships and a Bit of Geometry\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Caution sign\",\"author\":\"JDDesign\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"40% shaded rectangle\",\"author\":\"Clker-Free-Vector-Images\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/percent-40-bar-progress-meter-40844\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Review of basic percents\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Z229RysttR8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Absolute and Relative Differences\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QjVeurkg8CQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Importance of base in percents\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/nygw69JqwoQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Combining percents\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/4HNxwYMTNl8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Evaluating claims involving percents\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Svlu2Lurmsc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Percentage points and averaging percents\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vtgEkQUB5F8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 17441, 17447, 17443\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"wind-364996_1280\",\"author\":\"Stevebidmead\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/wind-turbines-farmland-364996\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Basic rates and proportions\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/aZrio6ztHKE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Proportions using dimensional analysis\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/jYwi3YqP0Wk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Proportions with unit conversion\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/fyOcLcIVipM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Considering how\/if things scale\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-e2typcrhLE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Comparing quantities involving large numbers\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/rCLh8ZvSQr8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 17454\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-129","chapter","type-chapter","status-publish","hentry"],"part":119,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapters\/129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapters\/129\/revisions"}],"predecessor-version":[{"id":1131,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapters\/129\/revisions\/1131"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/parts\/119"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapters\/129\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/wp\/v2\/media?parent=129"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/pressbooks\/v2\/chapter-type?post=129"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/wp\/v2\/contributor?post=129"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/wp-json\/wp\/v2\/license?post=129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}