{"id":1115,"date":"2023-03-30T16:56:24","date_gmt":"2023-03-30T16:56:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1115"},"modified":"2024-10-18T20:51:04","modified_gmt":"2024-10-18T20:51:04","slug":"percents-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/percents-fresh-take\/","title":{"raw":"Percents: Fresh Take","rendered":"Percents: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Write percents and perform calculations<\/li>\r\n\t<li>Determine unit rate using percentages<\/li>\r\n\t<li>Find both the relative and absolute change<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Percents<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Percentages<\/strong> represent a part of a whole, with the whole being represented as [latex]100\\%[\/latex]. They are a special kind of fraction that are used to indicate proportions out of a hundred.<\/p>\r\n<p>Conversion of percentages to decimals or fractions is straightforward. To convert a percent to a decimal, you simply divide the percentage by [latex]100[\/latex], which effectively means moving the decimal point two places to the left. To convert a percent to a fraction, place the percentage over [latex]100[\/latex] to form a fraction and then simplify if possible. For instance, [latex]45\\%[\/latex] becomes [latex]0.45[\/latex] as a decimal and [latex]\\frac{45}{100}[\/latex] as a fraction, which simplifies to [latex]\\frac{9}{20}[\/latex].<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<div class=\"watchItDiv\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/JeVSmq1Nrpw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+What+Are+Percentages_.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - What Are Percentages?\u201d here (opens in new window).<\/a><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">In a survey, [latex]243[\/latex] out of [latex]400[\/latex] people state that they like dogs. What percent is this?<br \/>\r\n[reveal-answer q=\"987171\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"987171\"][latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex]\r\n\r\n<p>This is [latex]60.75\\%[\/latex]. Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">In the news, you hear \u201ctuition is expected to increase by [latex]7\\%[\/latex] next year.\u201d If tuition this year was [latex]$1200[\/latex] per quarter, what will it be next year?<br \/>\r\n[reveal-answer q=\"475615\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"475615\"]\r\n\r\n<p>The tuition next year will be the current tuition plus an additional [latex]7\\%[\/latex], so it will be [latex]107\\%[\/latex] of this year\u2019s tuition: [latex]$1200(1.07) = $1284[\/latex]. Alternatively, we could have first calculated [latex]7\\%[\/latex] of [latex]$1200[\/latex]: [latex]$1200(0.07) = $84[\/latex]. Notice this is not the expected tuition for next year (we could only wish).<\/p>\r\n<p>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: [latex]$1200 + $84 = $1284[\/latex].<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<h2>Rates<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Rates<\/strong> are a specific kind of ratio, used to compare quantities of different kinds, such as miles per hour or price per pound. A <strong>unit rate<\/strong> is a rate that is simplified so that it has a denominator of one, making it easier to compare different rates directly. For instance, if you can drive [latex]180[\/latex] miles on [latex]10[\/latex] gallons of gas, the unit rate would be [latex]18[\/latex] miles per gallon.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<div class=\"watchItDiv\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/RQ2nYUBVvqI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Ratios+And+Rates.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - Ratios And Rates\u201d here (opens in new window).<\/a><\/p>\r\n<\/div>\r\n<\/section>\r\n<h2>Proportions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Proportions <\/strong>are equations that show two ratios or rates as being equivalent. For example, if you know you can drive [latex]180[\/latex] miles on [latex]10[\/latex] gallons of gas, and you want to know how far you can drive on [latex]15[\/latex] gallons, you would set up a proportion: [latex]\\frac{180}{10} = \\frac{x}{15}[\/latex], and solve for [latex]x[\/latex]. The proportion equation allows us to solve problems by finding missing quantities in equivalent ratios or rates.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<div class=\"watchItDiv\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/USmit5zUGas\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Proportions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - Proportions\u201d here (opens in new window).<\/a><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value [latex]x[\/latex].<br \/>\r\n[reveal-answer q=\"737915\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"737915\"]This proportion is asking us to find a fraction with denominator [latex]6[\/latex] that is equivalent to the fraction[latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]6[\/latex], giving\u00a0[latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">A map scale indicates that [latex]\\frac{1}{2}[\/latex] inch on the map corresponds with [latex]3[\/latex] real miles. How many miles apart are two cities that are [latex]\\displaystyle{2}\\frac{1}{4}[\/latex] inches apart on the map?<br \/>\r\n[reveal-answer q=\"439949\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"439949\"]<br \/>\r\nWe can set up a proportion by setting equal two [latex]\\displaystyle\\frac{\\text{map inches}}{\\text{real miles}}[\/latex]\u00a0rates, and introducing a variable, [latex]x[\/latex], to represent the unknown quantity\u2014the mile distance between the cities.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}\\text{map inch}}{3\\text{ miles}}=\\frac{2\\frac{1}{4}\\text{map inches}}{x\\text{ miles}}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]x[\/latex] and rewriting the mixed number<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}}{3}\\cdot{x}=\\frac{9}{4}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle\\frac{1}{2}x=\\frac{27}{4}[\/latex]<\/td>\r\n<td>Multiply both sides by [latex]2[\/latex] (or divide by [latex]\u00bd[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\displaystyle{x}=\\frac{27}{2}=13\\frac{1}{2}\\text{ miles}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\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>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose you\u2019re tiling the floor of a [latex]10[\/latex] ft by [latex]10[\/latex] ft room, and find that [latex]100[\/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[\/latex] ft by [latex]20[\/latex] ft room?<br \/>\r\n[reveal-answer q=\"815477\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"815477\"]In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, [latex]400[\/latex] tiles will be needed. We could find this using a proportion based on the areas of the rooms:[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Absolute and Relative Change<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Absolute change and relative change are two ways of quantifying the difference between two values.<\/p>\r\n<p><strong>Absolute change<\/strong> refers to the simple difference between the initial value and the final value. For example, if a stock price goes from [latex]$10[\/latex] to [latex]$15[\/latex], the absolute change is [latex]$5[\/latex].<\/p>\r\n<p><strong>Relative change<\/strong>, on the other hand, expresses the absolute change as a percentage of the original value. This provides a sense of the scale or significance of the change in relation to the starting point. In the above example, the relative change would be [latex]50\\%[\/latex], as the stock price increased by half of its original value. This is calculated by dividing the absolute change ([latex]$5[\/latex]) by the initial value ([latex]$10[\/latex]), and then multiplying the result by [latex]100[\/latex] to get a percentage.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<div class=\"watchItDiv\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/TGFTUibUquQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Change%2C+Absolute+Change%2C+and+Relative+Change.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Change, Absolute Change, and Relative Change\u201d here (opens in new window).<\/a><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose a stock drops in value by [latex]60\\%[\/latex] one week, then increases in value the next week by [latex]75\\%[\/latex]. Is the value higher or lower than where it started?<br \/>\r\n[reveal-answer q=\"568319\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"568319\"]\r\n\r\n<p>To answer this question, suppose the value started at [latex]$100[\/latex]. After one week, the value dropped by [latex]60\\%: $100 \u2013 $100(0.60) = $100 \u2013 $60 = $40[\/latex]. In the next week, notice that base of the percent has changed to the new value, [latex]$40[\/latex]. Computing the [latex]75\\%[\/latex] increase: [latex]$40 + $40(0.75) = $40 + $30 = $70[\/latex].<\/p>\r\n<p>In the end, the stock is still [latex]$30[\/latex] lower, or [latex]\\displaystyle\\frac{\\$30}{100} = 30\\%[\/latex] lower, valued than it started. A video walk-through of this example can be seen below:<\/p>\r\n<br \/>\r\n<br \/>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4HNxwYMTNl8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Combining+percents.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCombining percents\u201d here (opens in new window).<\/a><\/p>\r\n\r\n[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Write percents and perform calculations<\/li>\n<li>Determine unit rate using percentages<\/li>\n<li>Find both the relative and absolute change<\/li>\n<\/ul>\n<\/section>\n<h2>Percents<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Percentages<\/strong> represent a part of a whole, with the whole being represented as [latex]100\\%[\/latex]. They are a special kind of fraction that are used to indicate proportions out of a hundred.<\/p>\n<p>Conversion of percentages to decimals or fractions is straightforward. To convert a percent to a decimal, you simply divide the percentage by [latex]100[\/latex], which effectively means moving the decimal point two places to the left. To convert a percent to a fraction, place the percentage over [latex]100[\/latex] to form a fraction and then simplify if possible. For instance, [latex]45\\%[\/latex] becomes [latex]0.45[\/latex] as a decimal and [latex]\\frac{45}{100}[\/latex] as a fraction, which simplifies to [latex]\\frac{9}{20}[\/latex].<\/p>\n<\/div>\n<section class=\"textbox watchIt\">\n<div class=\"watchItDiv\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/JeVSmq1Nrpw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+What+Are+Percentages_.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; What Are Percentages?\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">In a survey, [latex]243[\/latex] out of [latex]400[\/latex] people state that they like dogs. What percent is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q987171\">Show Solution<\/button><\/p>\n<div id=\"q987171\" class=\"hidden-answer\" style=\"display: none\">[latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex]<\/p>\n<p>This is [latex]60.75\\%[\/latex]. 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<\/section>\n<section class=\"textbox example\">In the news, you hear \u201ctuition is expected to increase by [latex]7\\%[\/latex] next year.\u201d If tuition this year was [latex]$1200[\/latex] per quarter, what will it be next year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q475615\">Show Solution<\/button><\/p>\n<div id=\"q475615\" class=\"hidden-answer\" style=\"display: none\">\n<p>The tuition next year will be the current tuition plus an additional [latex]7\\%[\/latex], so it will be [latex]107\\%[\/latex] of this year\u2019s tuition: [latex]$1200(1.07) = $1284[\/latex]. Alternatively, we could have first calculated [latex]7\\%[\/latex] of [latex]$1200[\/latex]: [latex]$1200(0.07) = $84[\/latex]. Notice this is not the expected tuition for next year (we could only wish).<\/p>\n<p>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: [latex]$1200 + $84 = $1284[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Rates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Rates<\/strong> are a specific kind of ratio, used to compare quantities of different kinds, such as miles per hour or price per pound. A <strong>unit rate<\/strong> is a rate that is simplified so that it has a denominator of one, making it easier to compare different rates directly. For instance, if you can drive [latex]180[\/latex] miles on [latex]10[\/latex] gallons of gas, the unit rate would be [latex]18[\/latex] miles per gallon.<\/p>\n<\/div>\n<section class=\"textbox watchIt\">\n<div class=\"watchItDiv\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/RQ2nYUBVvqI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Ratios+And+Rates.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; Ratios And Rates\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/section>\n<h2>Proportions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Proportions <\/strong>are equations that show two ratios or rates as being equivalent. For example, if you know you can drive [latex]180[\/latex] miles on [latex]10[\/latex] gallons of gas, and you want to know how far you can drive on [latex]15[\/latex] gallons, you would set up a proportion: [latex]\\frac{180}{10} = \\frac{x}{15}[\/latex], and solve for [latex]x[\/latex]. The proportion equation allows us to solve problems by finding missing quantities in equivalent ratios or rates.<\/p>\n<\/div>\n<section class=\"textbox watchIt\">\n<div class=\"watchItDiv\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/USmit5zUGas\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Proportions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; Proportions\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve the proportion [latex]\\displaystyle\\frac{5}{3}=\\frac{x}{6}[\/latex] for the unknown value [latex]x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q737915\">Show Solution<\/button><\/p>\n<div id=\"q737915\" class=\"hidden-answer\" style=\"display: none\">This proportion is asking us to find a fraction with denominator [latex]6[\/latex] that is equivalent to the fraction[latex]\\displaystyle\\frac{5}{3}[\/latex]. We can solve this by multiplying both sides of the equation by [latex]6[\/latex], giving\u00a0[latex]\\displaystyle{x}=\\frac{5}{3}\\cdot6=10[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A map scale indicates that [latex]\\frac{1}{2}[\/latex] inch on the map corresponds with [latex]3[\/latex] real miles. How many miles apart are two cities that are [latex]\\displaystyle{2}\\frac{1}{4}[\/latex] inches apart on the map?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q439949\">Show Solution<\/button><\/p>\n<div id=\"q439949\" class=\"hidden-answer\" style=\"display: none\">\nWe can set up a proportion by setting equal two [latex]\\displaystyle\\frac{\\text{map inches}}{\\text{real miles}}[\/latex]\u00a0rates, and introducing a variable, [latex]x[\/latex], to represent the unknown quantity\u2014the mile distance between the cities.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}\\text{map inch}}{3\\text{ miles}}=\\frac{2\\frac{1}{4}\\text{map inches}}{x\\text{ miles}}[\/latex]<\/td>\n<td>Multiply both sides by [latex]x[\/latex] and rewriting the mixed number<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle\\frac{\\frac{1}{2}}{3}\\cdot{x}=\\frac{9}{4}[\/latex]<\/td>\n<td>Multiply both sides by [latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle\\frac{1}{2}x=\\frac{27}{4}[\/latex]<\/td>\n<td>Multiply both sides by [latex]2[\/latex] (or divide by [latex]\u00bd[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle{x}=\\frac{27}{2}=13\\frac{1}{2}\\text{ miles}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\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<\/section>\n<section class=\"textbox example\">Suppose you\u2019re tiling the floor of a [latex]10[\/latex] ft by [latex]10[\/latex] ft room, and find that [latex]100[\/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[\/latex] ft by [latex]20[\/latex] ft room?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q815477\">Show Solution<\/button><\/p>\n<div id=\"q815477\" class=\"hidden-answer\" style=\"display: none\">In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, [latex]400[\/latex] tiles will be needed. We could find this using a proportion based on the areas of the rooms:[latex]\\displaystyle\\frac{100\\text{ tiles}}{100\\text{ft}^2}=\\frac{n\\text{ tiles}}{400\\text{ft}^2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Absolute and Relative Change<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Absolute change and relative change are two ways of quantifying the difference between two values.<\/p>\n<p><strong>Absolute change<\/strong> refers to the simple difference between the initial value and the final value. For example, if a stock price goes from [latex]$10[\/latex] to [latex]$15[\/latex], the absolute change is [latex]$5[\/latex].<\/p>\n<p><strong>Relative change<\/strong>, on the other hand, expresses the absolute change as a percentage of the original value. This provides a sense of the scale or significance of the change in relation to the starting point. In the above example, the relative change would be [latex]50\\%[\/latex], as the stock price increased by half of its original value. This is calculated by dividing the absolute change ([latex]$5[\/latex]) by the initial value ([latex]$10[\/latex]), and then multiplying the result by [latex]100[\/latex] to get a percentage.<\/p>\n<\/div>\n<section class=\"textbox watchIt\">\n<div class=\"watchItDiv\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/TGFTUibUquQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Change%2C+Absolute+Change%2C+and+Relative+Change.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Change, Absolute Change, and Relative Change\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Suppose a stock drops in value by [latex]60\\%[\/latex] one week, then increases in value the next week by [latex]75\\%[\/latex]. Is the value higher or lower than where it started?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q568319\">Show Solution<\/button><\/p>\n<div id=\"q568319\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer this question, suppose the value started at [latex]$100[\/latex]. After one week, the value dropped by [latex]60\\%: $100 \u2013 $100(0.60) = $100 \u2013 $60 = $40[\/latex]. In the next week, notice that base of the percent has changed to the new value, [latex]$40[\/latex]. Computing the [latex]75\\%[\/latex] increase: [latex]$40 + $40(0.75) = $40 + $30 = $70[\/latex].<\/p>\n<p>In the end, the stock is still [latex]$30[\/latex] lower, or [latex]\\displaystyle\\frac{\\$30}{100} = 30\\%[\/latex] lower, valued than it started. A video walk-through of this example can be seen below:<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4HNxwYMTNl8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Combining+percents.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCombining percents\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1115"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":43,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1115\/revisions"}],"predecessor-version":[{"id":15214,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1115\/revisions\/15214"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1115\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1115"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1115"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1115"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}