{"id":15319,"date":"2021-10-11T23:05:29","date_gmt":"2021-10-11T23:05:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solving-proportions\/"},"modified":"2021-10-17T02:20:39","modified_gmt":"2021-10-17T02:20:39","slug":"solving-proportions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solving-proportions\/","title":{"raw":"Solving Proportions","rendered":"Solving Proportions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve proportions<\/li>\r\n \t<li>Solve applications using proportions<\/li>\r\n<\/ul>\r\n<\/div>\r\nTo solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168468652606\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 17.5 is to 37.5 as 7 is to 15. The cross multiplication shown is 15 times 17.5 = 262.5 and 37.5 times 7 = 262.5.\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\" colspan=\"2\">To isolate [latex]x[\/latex] , multiply both sides by the LCD, [latex]63[\/latex].<\/td>\r\n<td style=\"height: 30px;\">[latex]\\color{red}{63}({\\Large\\frac{x}{63}})=\\color{red}{63}({\\Large\\frac{4}{7}})[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 30px;\">[latex]x={\\Large\\frac{9\\cdot\\color{red}{7}\\cdot4}{\\color{red}{7}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" colspan=\"2\">Divide the common factors.<\/td>\r\n<td style=\"height: 15px;\">[latex]x=36[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Check: To check our answer, we substitute into the original proportion.<\/td>\r\n<td style=\"height: 30px;\"><\/td>\r\n<td style=\"height: 30px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Substitute [latex]x=\\color{red}{36}[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]{\\Large\\frac{\\color{red}{36}}{63}}\\stackrel{?}{=}{\\Large\\frac{4}{7}}[\/latex]<\/td>\r\n<td style=\"height: 30px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Show common factors.<\/td>\r\n<td style=\"height: 30px;\">[latex]{\\Large\\frac{4\\cdot9}{7\\cdot9}}\\stackrel{?}{=}{\\Large\\frac{4}{7}}[\/latex]<\/td>\r\n<td style=\"height: 30px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.875px;\">\r\n<td style=\"height: 15.875px;\">Simplify.<\/td>\r\n<td style=\"height: 15.875px;\">[latex]{\\Large\\frac{4}{7}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\r\n<td style=\"height: 15.875px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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 x.\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<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146811[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show another example of how to solve a proportion equation using the LCD.\r\n\r\nhttps:\/\/youtu.be\/pXvzpSU4DyU\r\n\r\nWhen the variable is in a denominator, we\u2019ll use the fact that the cross products of a proportion are equal to solve the proportions.\r\n\r\nWe can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]{\\Large\\frac{144}{a}}={\\Large\\frac{9}{4}}[\/latex]\r\n[reveal-answer q=\"841326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841326\"]\r\n\r\nSolution\r\nNotice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.\r\n<table id=\"eip-id1168469872045\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to solve the proportion 144 is to a as 9 is to 4 using cross multiplication. It shows how the cross products are found and set equal to one another. The result is 4 times 144 = a times 9. The equation is simplified to 576 = 9 times a. Both sides of the equation is divided by 9. When simplified, the result is 64 = a. The figure shows the steps to checking that a = 64 is the solution to the proportion 144 is to a as 9 is to 4. It shows 64 being substituted for a in the proportion. The proportion becomes 144 is to 64 as 9 is to 4. The common factors in the proportion are shown. The proportion becomes 9 times 16 is to 4 times 16 as 9 is to 4. The proportion is simplified to 9 is to 4 as 9 is to 4. So the solution a = 64 is correct.\">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222140\/CNX_BMath_Figure_06_02_024_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Find the cross products and set them equal.<\/td>\r\n<td>[latex]4\\cdot144=a\\cdot9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]576=9a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Divide both sides by [latex]9[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{576}{9}}={\\Large\\frac{9a}{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]64=a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{144}{a}}={\\Large\\frac{9}{4}}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]a=\\color{red}{64}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{144}{\\color{red}{64}}}\\stackrel{?}{=}{\\Large\\frac{9}{4}}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Show common factors..<\/td>\r\n<td>[latex]{\\Large\\frac{9\\cdot16}{4\\cdot16}}\\stackrel{?}{=}{\\Large\\frac{9}{4}}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{9}{4}}={\\Large\\frac{9}{4}}\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother method to solve this would be to multiply both sides by the LCD, [latex]4a[\/latex]. Try it and verify that you get the same solution.\r\n\r\nThe following video shows an example of how to solve a similar problem by using the LCD.\r\n\r\nhttps:\/\/youtu.be\/zrgLddU8pFU\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146813[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]{\\Large\\frac{52}{91}}={\\Large\\frac{-4}{y}}[\/latex]\r\n[reveal-answer q=\"761416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"761416\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469612644\" class=\"unnumbered unstyled\" style=\"width: 808px;\" summary=\"The figure shows the steps to solve the proportion 52 is to 91 as -4 is to y using cross multiplication. It shows how the cross products are found and set equal to one another to form the equation y times 52 equals 91 times negative 4. When simplified, the equation becomes 52 times y = negative 364. Both sides of the equation are divided by 52. When simplified, the result is y = negative 7. The figure shows the steps to checking that y = negative 7 is the solution to the proportion 52 to 91 as negative 4 is to y. It shows negative 7 being substituted for y in the proportion. The proportion becomes 52 is to 91 as -4 is to y. The common factors in the proportion are shown. The proportion becomes 13 times 4 is to 13 times 7 as negative 4 is to negative 7. The proportion is simplified to 4 is to 7 as 4 is to 7. So the solution y= negative 7 is correct.\">\r\n<tbody>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px; width: 488px;\" colspan=\"2\">Find the cross products and set them equal.<\/td>\r\n<td style=\"height: 42px; width: 287px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222150\/CNX_BMath_Figure_06_05_026_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px; width: 488px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 23px; width: 287px;\">[latex]y\\cdot52=91(-4)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 488px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 24px; width: 287px;\">[latex]52y=-364[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43px;\">\r\n<td style=\"height: 43px; width: 488px;\" colspan=\"2\">Divide both sides by [latex]52[\/latex].<\/td>\r\n<td style=\"height: 43px; width: 287px;\">[latex]{\\Large\\frac{52y}{52}}={\\Large\\frac{-364}{52}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24.3787px;\">\r\n<td style=\"height: 24.3787px; width: 488px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 24.3787px; width: 287px;\">[latex]y=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 141.75px;\">Check:<\/td>\r\n<td style=\"height: 14px; width: 346.25px;\"><\/td>\r\n<td style=\"height: 14px; width: 287px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 41px;\">\r\n<td style=\"height: 41px; width: 141.75px;\"><\/td>\r\n<td style=\"height: 41px; width: 346.25px;\">[latex]{\\Large\\frac{52}{91}}={\\Large\\frac{-4}{y}}[\/latex]<\/td>\r\n<td style=\"height: 41px; width: 287px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 40px;\">\r\n<td style=\"height: 40px; width: 141.75px;\">Substitute [latex]y=\\color{red}{-7}[\/latex]<\/td>\r\n<td style=\"height: 40px; width: 346.25px;\">[latex]{\\Large\\frac{52}{91}}\\stackrel{?}{=}{\\Large\\frac{-4}{\\color{red}{-7}}}[\/latex]<\/td>\r\n<td style=\"height: 40px; width: 287px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px; width: 141.75px;\">Show common factors.<\/td>\r\n<td style=\"height: 42px; width: 346.25px;\">[latex]{\\Large\\frac{13\\cdot4}{13\\cdot4}}\\stackrel{?}{=}{\\Large\\frac{-4}{\\color{red}{-7}}}[\/latex]<\/td>\r\n<td style=\"height: 42px; width: 287px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px; width: 141.75px;\">Simplify.<\/td>\r\n<td style=\"height: 42px; width: 346.25px;\">[latex]{\\Large\\frac{4}{7}}={\\Large\\frac{4}{7}}\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"height: 42px; width: 287px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146814[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Solve Applications Using Proportions<\/h3>\r\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/solving-applications-with-percent\/\">The strategy for solving applications with percents<\/a>, also works for proportions, since proportions are also equations. When we set up the proportion, we must make sure the units are correct\u2014the units in the numerators match and the units in the denominators match.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhen pediatricians prescribe acetaminophen to children, they prescribe [latex]5[\/latex] milliliters (ml) of acetaminophen for every [latex]25[\/latex] pounds of the child\u2019s weight. If Zoe weighs [latex]80[\/latex] pounds, how many milliliters of acetaminophen will her doctor prescribe?\r\n[reveal-answer q=\"781594\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"781594\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468389749\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Identify what you are asked to find.<\/td>\r\n<td>How many ml of acetaminophen the doctor will prescribe<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choose a variable to represent it.<\/td>\r\n<td>Let [latex]a=[\/latex] ml of acetaminophen.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a sentence that gives the information to find it.<\/td>\r\n<td>If [latex]5[\/latex] ml is prescribed for every [latex]25[\/latex] pounds, how much will be prescribed for [latex]80[\/latex] pounds?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate into a proportion.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222200\/CNX_BMath_Figure_06_05_001_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute given values\u2014be careful of the units.<\/td>\r\n<td>[latex]{\\Large\\frac{5}{25}}={\\Large\\frac{a}{80}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply both sides by [latex]80[\/latex].<\/td>\r\n<td>[latex]80\\cdot{\\Large\\frac{5}{25}}=80\\cdot{\\Large\\frac{a}{80}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and show common factors.<\/td>\r\n<td>[latex]{\\Large\\frac{16\\cdot5\\cdot5}{5\\cdot5}}={\\Large\\frac{80a}{80}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]16=a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check if the answer is reasonable.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Yes. Since [latex]80[\/latex] is about [latex]3[\/latex] times [latex]25[\/latex], the medicine should be about [latex]3[\/latex] times [latex]5[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a complete sentence.<\/td>\r\n<td>The pediatrician would prescribe [latex]16[\/latex] ml of acetaminophen to Zoe.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou could also solve this proportion by setting the cross products equal.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146816[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nOne brand of microwave popcorn has [latex]120[\/latex] calories per serving. A whole bag of this popcorn has [latex]3.5[\/latex] servings. How many calories are in a whole bag of this microwave popcorn?\r\n[reveal-answer q=\"459070\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459070\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466068741\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Identify what you are asked to find.<\/td>\r\n<td>How many calories are in a whole bag of microwave popcorn?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choose a variable to represent it.<\/td>\r\n<td>Let [latex]c=[\/latex] number of calories.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a sentence that gives the information to find it.<\/td>\r\n<td>If there are [latex]120[\/latex] calories per serving, how many calories are in a whole bag with [latex]3.5[\/latex] servings?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate into a proportion.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222206\/CNX_BMath_Figure_06_05_002_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute given values.<\/td>\r\n<td>[latex]{\\Large\\frac{120}{1}}={\\Large\\frac{c}{3.5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply both sides by [latex]3.5[\/latex].<\/td>\r\n<td>[latex](3.5)({\\Large\\frac{120}{1}})=(3.5)({\\Large\\frac{c}{3.5}})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]420=c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check if the answer is reasonable.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Yes. Since [latex]3.5[\/latex] is between [latex]3[\/latex] and [latex]4[\/latex], the total calories should be between [latex]360 (3\u22c5120)[\/latex] and [latex]480 (4\u22c5120)[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a complete sentence.<\/td>\r\n<td>The whole bag of microwave popcorn has [latex]420[\/latex] calories.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146817[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nJosiah went to Mexico for spring break and changed $[latex]325[\/latex] dollars into Mexican pesos. At that time, the exchange rate had $[latex]1[\/latex] U.S. is equal to [latex]12.54[\/latex] Mexican pesos. How many Mexican pesos did he get for his trip?\r\n[reveal-answer q=\"347314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"347314\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469868734\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Identify what you are asked to find.<\/td>\r\n<td>How many Mexican pesos did Josiah get?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choose a variable to represent it.<\/td>\r\n<td>Let [latex]p=[\/latex] number of pesos.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a sentence that gives the information to find it.<\/td>\r\n<td>If [latex]\\text{\\$1}[\/latex] U.S. is equal to [latex]12.54[\/latex] Mexican pesos, then [latex]\\text{\\$325}[\/latex] is how many pesos?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate into a proportion.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222211\/CNX_BMath_Figure_06_05_003_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute given values.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{12.54}}={\\Large\\frac{325}{p}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The variable is in the denominator, so find the cross products and set them equal.<\/td>\r\n<td>[latex]p\\cdot{1}=12.54(325)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]c=4,075.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check if the answer is reasonable.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Yes, [latex]\\text{\\$100}[\/latex] would be [latex]\\text{\\$1,254}[\/latex] pesos. [latex]\\text{\\$325}[\/latex] is a little more than [latex]3[\/latex] times this amount.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a complete sentence.<\/td>\r\n<td>Josiah has [latex]4075.5[\/latex] pesos for his spring break trip.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146819[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to solve an application that involves proportions.\r\n\r\nhttps:\/\/youtu.be\/vnB1mh5X5cA\r\n\r\nSome quantities though don\u2019t scale proportionally at all.\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\n&nbsp;\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\" rel=\"noopener\">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\" rel=\"noopener\">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><\/h2>\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1EkP8lv94ZuYQedqKM0k5i7pKg4orTjBaTgXZi5bsDXE\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve proportions<\/li>\n<li>Solve applications using proportions<\/li>\n<\/ul>\n<\/div>\n<p>To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468652606\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to finding the cross products of the proportion 17.5 is to 37.5 as 7 is to 15. The cross multiplication shown is 15 times 17.5 = 262.5 and 37.5 times 7 = 262.5.\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\" colspan=\"2\">To isolate [latex]x[\/latex] , multiply both sides by the LCD, [latex]63[\/latex].<\/td>\n<td style=\"height: 30px;\">[latex]\\color{red}{63}({\\Large\\frac{x}{63}})=\\color{red}{63}({\\Large\\frac{4}{7}})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 30px;\">[latex]x={\\Large\\frac{9\\cdot\\color{red}{7}\\cdot4}{\\color{red}{7}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" colspan=\"2\">Divide the common factors.<\/td>\n<td style=\"height: 15px;\">[latex]x=36[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Check: To check our answer, we substitute into the original proportion.<\/td>\n<td style=\"height: 30px;\"><\/td>\n<td style=\"height: 30px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{x}{63}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Substitute [latex]x=\\color{red}{36}[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]{\\Large\\frac{\\color{red}{36}}{63}}\\stackrel{?}{=}{\\Large\\frac{4}{7}}[\/latex]<\/td>\n<td style=\"height: 30px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Show common factors.<\/td>\n<td style=\"height: 30px;\">[latex]{\\Large\\frac{4\\cdot9}{7\\cdot9}}\\stackrel{?}{=}{\\Large\\frac{4}{7}}[\/latex]<\/td>\n<td style=\"height: 30px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15.875px;\">\n<td style=\"height: 15.875px;\">Simplify.<\/td>\n<td style=\"height: 15.875px;\">[latex]{\\Large\\frac{4}{7}}={\\Large\\frac{4}{7}}[\/latex]<\/td>\n<td style=\"height: 15.875px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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 x.<\/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-1\" title=\"Proportions using dimensional analysis\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jYwi3YqP0Wk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/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<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146811\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146811&theme=oea&iframe_resize_id=ohm146811&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show another example of how to solve a proportion equation using the LCD.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Solve a Proportion by Clearing Fractions (x\/a=b\/c, Whole Num  Solution)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/pXvzpSU4DyU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When the variable is in a denominator, we\u2019ll use the fact that the cross products of a proportion are equal to solve the proportions.<\/p>\n<p>We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]{\\Large\\frac{144}{a}}={\\Large\\frac{9}{4}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841326\">Show Solution<\/span><\/p>\n<div id=\"q841326\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nNotice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.<\/p>\n<table id=\"eip-id1168469872045\" class=\"unnumbered unstyled\" summary=\"The figure shows the steps to solve the proportion 144 is to a as 9 is to 4 using cross multiplication. It shows how the cross products are found and set equal to one another. The result is 4 times 144 = a times 9. The equation is simplified to 576 = 9 times a. Both sides of the equation is divided by 9. When simplified, the result is 64 = a. The figure shows the steps to checking that a = 64 is the solution to the proportion 144 is to a as 9 is to 4. It shows 64 being substituted for a in the proportion. The proportion becomes 144 is to 64 as 9 is to 4. The common factors in the proportion are shown. The proportion becomes 9 times 16 is to 4 times 16 as 9 is to 4. The proportion is simplified to 9 is to 4 as 9 is to 4. So the solution a = 64 is correct.\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222140\/CNX_BMath_Figure_06_02_024_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Find the cross products and set them equal.<\/td>\n<td>[latex]4\\cdot144=a\\cdot9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]576=9a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Divide both sides by [latex]9[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{576}{9}}={\\Large\\frac{9a}{9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]64=a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer.<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{144}{a}}={\\Large\\frac{9}{4}}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]a=\\color{red}{64}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{144}{\\color{red}{64}}}\\stackrel{?}{=}{\\Large\\frac{9}{4}}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Show common factors..<\/td>\n<td>[latex]{\\Large\\frac{9\\cdot16}{4\\cdot16}}\\stackrel{?}{=}{\\Large\\frac{9}{4}}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{9}{4}}={\\Large\\frac{9}{4}}\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Another method to solve this would be to multiply both sides by the LCD, [latex]4a[\/latex]. Try it and verify that you get the same solution.<\/p>\n<p>The following video shows an example of how to solve a similar problem by using the LCD.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Solve a Proportion by Clearing Fractions ((a\/x=b\/c, Fraction Solution)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/zrgLddU8pFU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146813\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146813&theme=oea&iframe_resize_id=ohm146813&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]{\\Large\\frac{52}{91}}={\\Large\\frac{-4}{y}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q761416\">Show Solution<\/span><\/p>\n<div id=\"q761416\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469612644\" class=\"unnumbered unstyled\" style=\"width: 808px;\" summary=\"The figure shows the steps to solve the proportion 52 is to 91 as -4 is to y using cross multiplication. It shows how the cross products are found and set equal to one another to form the equation y times 52 equals 91 times negative 4. When simplified, the equation becomes 52 times y = negative 364. Both sides of the equation are divided by 52. When simplified, the result is y = negative 7. The figure shows the steps to checking that y = negative 7 is the solution to the proportion 52 to 91 as negative 4 is to y. It shows negative 7 being substituted for y in the proportion. The proportion becomes 52 is to 91 as -4 is to y. The common factors in the proportion are shown. The proportion becomes 13 times 4 is to 13 times 7 as negative 4 is to negative 7. The proportion is simplified to 4 is to 7 as 4 is to 7. So the solution y= negative 7 is correct.\">\n<tbody>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px; width: 488px;\" colspan=\"2\">Find the cross products and set them equal.<\/td>\n<td style=\"height: 42px; width: 287px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222150\/CNX_BMath_Figure_06_05_026_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px; width: 488px;\" colspan=\"2\"><\/td>\n<td style=\"height: 23px; width: 287px;\">[latex]y\\cdot52=91(-4)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 488px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 24px; width: 287px;\">[latex]52y=-364[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px; width: 488px;\" colspan=\"2\">Divide both sides by [latex]52[\/latex].<\/td>\n<td style=\"height: 43px; width: 287px;\">[latex]{\\Large\\frac{52y}{52}}={\\Large\\frac{-364}{52}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24.3787px;\">\n<td style=\"height: 24.3787px; width: 488px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 24.3787px; width: 287px;\">[latex]y=-7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 141.75px;\">Check:<\/td>\n<td style=\"height: 14px; width: 346.25px;\"><\/td>\n<td style=\"height: 14px; width: 287px;\"><\/td>\n<\/tr>\n<tr style=\"height: 41px;\">\n<td style=\"height: 41px; width: 141.75px;\"><\/td>\n<td style=\"height: 41px; width: 346.25px;\">[latex]{\\Large\\frac{52}{91}}={\\Large\\frac{-4}{y}}[\/latex]<\/td>\n<td style=\"height: 41px; width: 287px;\"><\/td>\n<\/tr>\n<tr style=\"height: 40px;\">\n<td style=\"height: 40px; width: 141.75px;\">Substitute [latex]y=\\color{red}{-7}[\/latex]<\/td>\n<td style=\"height: 40px; width: 346.25px;\">[latex]{\\Large\\frac{52}{91}}\\stackrel{?}{=}{\\Large\\frac{-4}{\\color{red}{-7}}}[\/latex]<\/td>\n<td style=\"height: 40px; width: 287px;\"><\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px; width: 141.75px;\">Show common factors.<\/td>\n<td style=\"height: 42px; width: 346.25px;\">[latex]{\\Large\\frac{13\\cdot4}{13\\cdot4}}\\stackrel{?}{=}{\\Large\\frac{-4}{\\color{red}{-7}}}[\/latex]<\/td>\n<td style=\"height: 42px; width: 287px;\"><\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px; width: 141.75px;\">Simplify.<\/td>\n<td style=\"height: 42px; width: 346.25px;\">[latex]{\\Large\\frac{4}{7}}={\\Large\\frac{4}{7}}\\quad\\checkmark[\/latex]<\/td>\n<td style=\"height: 42px; width: 287px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146814\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146814&theme=oea&iframe_resize_id=ohm146814&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Solve Applications Using Proportions<\/h3>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/solving-applications-with-percent\/\">The strategy for solving applications with percents<\/a>, also works for proportions, since proportions are also equations. When we set up the proportion, we must make sure the units are correct\u2014the units in the numerators match and the units in the denominators match.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>When pediatricians prescribe acetaminophen to children, they prescribe [latex]5[\/latex] milliliters (ml) of acetaminophen for every [latex]25[\/latex] pounds of the child\u2019s weight. If Zoe weighs [latex]80[\/latex] pounds, how many milliliters of acetaminophen will her doctor prescribe?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q781594\">Show Solution<\/span><\/p>\n<div id=\"q781594\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468389749\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Identify what you are asked to find.<\/td>\n<td>How many ml of acetaminophen the doctor will prescribe<\/td>\n<\/tr>\n<tr>\n<td>Choose a variable to represent it.<\/td>\n<td>Let [latex]a=[\/latex] ml of acetaminophen.<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence that gives the information to find it.<\/td>\n<td>If [latex]5[\/latex] ml is prescribed for every [latex]25[\/latex] pounds, how much will be prescribed for [latex]80[\/latex] pounds?<\/td>\n<\/tr>\n<tr>\n<td>Translate into a proportion.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222200\/CNX_BMath_Figure_06_05_001_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Substitute given values\u2014be careful of the units.<\/td>\n<td>[latex]{\\Large\\frac{5}{25}}={\\Large\\frac{a}{80}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply both sides by [latex]80[\/latex].<\/td>\n<td>[latex]80\\cdot{\\Large\\frac{5}{25}}=80\\cdot{\\Large\\frac{a}{80}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply and show common factors.<\/td>\n<td>[latex]{\\Large\\frac{16\\cdot5\\cdot5}{5\\cdot5}}={\\Large\\frac{80a}{80}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]16=a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check if the answer is reasonable.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Yes. Since [latex]80[\/latex] is about [latex]3[\/latex] times [latex]25[\/latex], the medicine should be about [latex]3[\/latex] times [latex]5[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write a complete sentence.<\/td>\n<td>The pediatrician would prescribe [latex]16[\/latex] ml of acetaminophen to Zoe.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>You could also solve this proportion by setting the cross products equal.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146816\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146816&theme=oea&iframe_resize_id=ohm146816&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>One brand of microwave popcorn has [latex]120[\/latex] calories per serving. A whole bag of this popcorn has [latex]3.5[\/latex] servings. How many calories are in a whole bag of this microwave popcorn?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459070\">Show Solution<\/span><\/p>\n<div id=\"q459070\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466068741\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Identify what you are asked to find.<\/td>\n<td>How many calories are in a whole bag of microwave popcorn?<\/td>\n<\/tr>\n<tr>\n<td>Choose a variable to represent it.<\/td>\n<td>Let [latex]c=[\/latex] number of calories.<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence that gives the information to find it.<\/td>\n<td>If there are [latex]120[\/latex] calories per serving, how many calories are in a whole bag with [latex]3.5[\/latex] servings?<\/td>\n<\/tr>\n<tr>\n<td>Translate into a proportion.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222206\/CNX_BMath_Figure_06_05_002_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Substitute given values.<\/td>\n<td>[latex]{\\Large\\frac{120}{1}}={\\Large\\frac{c}{3.5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply both sides by [latex]3.5[\/latex].<\/td>\n<td>[latex](3.5)({\\Large\\frac{120}{1}})=(3.5)({\\Large\\frac{c}{3.5}})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]420=c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check if the answer is reasonable.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Yes. Since [latex]3.5[\/latex] is between [latex]3[\/latex] and [latex]4[\/latex], the total calories should be between [latex]360 (3\u22c5120)[\/latex] and [latex]480 (4\u22c5120)[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write a complete sentence.<\/td>\n<td>The whole bag of microwave popcorn has [latex]420[\/latex] calories.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146817\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146817&theme=oea&iframe_resize_id=ohm146817&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Josiah went to Mexico for spring break and changed $[latex]325[\/latex] dollars into Mexican pesos. At that time, the exchange rate had $[latex]1[\/latex] U.S. is equal to [latex]12.54[\/latex] Mexican pesos. How many Mexican pesos did he get for his trip?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q347314\">Show Solution<\/span><\/p>\n<div id=\"q347314\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469868734\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Identify what you are asked to find.<\/td>\n<td>How many Mexican pesos did Josiah get?<\/td>\n<\/tr>\n<tr>\n<td>Choose a variable to represent it.<\/td>\n<td>Let [latex]p=[\/latex] number of pesos.<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence that gives the information to find it.<\/td>\n<td>If [latex]\\text{\\$1}[\/latex] U.S. is equal to [latex]12.54[\/latex] Mexican pesos, then [latex]\\text{\\$325}[\/latex] is how many pesos?<\/td>\n<\/tr>\n<tr>\n<td>Translate into a proportion.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222211\/CNX_BMath_Figure_06_05_003_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Substitute given values.<\/td>\n<td>[latex]{\\Large\\frac{1}{12.54}}={\\Large\\frac{325}{p}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The variable is in the denominator, so find the cross products and set them equal.<\/td>\n<td>[latex]p\\cdot{1}=12.54(325)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]c=4,075.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check if the answer is reasonable.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Yes, [latex]\\text{\\$100}[\/latex] would be [latex]\\text{\\$1,254}[\/latex] pesos. [latex]\\text{\\$325}[\/latex] is a little more than [latex]3[\/latex] times this amount.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write a complete sentence.<\/td>\n<td>Josiah has [latex]4075.5[\/latex] pesos for his spring break trip.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146819\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146819&theme=oea&iframe_resize_id=ohm146819&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to solve an application that involves proportions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Examples:  Applications Using Proportions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vnB1mh5X5cA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Some quantities though don\u2019t scale proportionally at all.<\/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>&nbsp;<\/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-5\" 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-15319-1\" href=\"#footnote-15319-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/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-15319-2\" href=\"#footnote-15319-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/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-6\" 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><\/h2>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1EkP8lv94ZuYQedqKM0k5i7pKg4orTjBaTgXZi5bsDXE\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/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-15319\">\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>Question ID 146819, 146818, 146817. <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>Ex: Solve a Proportion by Clearing Fractions (x\/a=b\/c, Whole Num Solution). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pXvzpSU4DyU\">https:\/\/youtu.be\/pXvzpSU4DyU<\/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>Ex: Solve a Proportion by Clearing Fractions ((a\/x=b\/c, Fraction Solution). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zrgLddU8pFU\">https:\/\/youtu.be\/zrgLddU8pFU<\/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>Examples: Applications Using Proportions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vnB1mh5X5cA\">https:\/\/youtu.be\/vnB1mh5X5cA<\/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>Problem Solving. <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>Project<\/strong>: Math in Society. <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>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>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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-15319-1\"><a href=\"https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html\" target=\"_blank\" rel=\"noopener\">https:\/\/www.cia.gov\/library\/publications\/the-world-factbook\/rankorder\/2042rank.html<\/a> <a href=\"#return-footnote-15319-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-15319-2\"><a href=\"http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL\" target=\"_blank\" rel=\"noopener\">http:\/\/data.worldbank.org\/indicator\/SP.POP.TOTL<\/a> <a href=\"#return-footnote-15319-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":167848,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146819, 146818, 146817\",\"author\":\"Lumen 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Solving\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"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\":\"Considering how\/if things 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