{"id":83,"date":"2023-06-05T15:29:45","date_gmt":"2023-06-05T15:29:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/putting-it-together-general-problem-solving\/"},"modified":"2023-07-11T15:10:03","modified_gmt":"2023-07-11T15:10:03","slug":"putting-it-together-general-problem-solving","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/putting-it-together-general-problem-solving\/","title":{"raw":"Taxes","rendered":"Taxes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate sales and income taxes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Calculate sales and income taxes<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/27192407\/5120304358_f64b93a2ec_o.jpg\"><img class=\" wp-image-2212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/27192407\/5120304358_f64b93a2ec_o-300x212.jpg\" alt=\"magnifying lens over the word &quot;tax&quot;\" width=\"442\" height=\"312\" \/><\/a>\r\n\r\nGovernments collect taxes to pay for the services they provide. In the United States, federal income taxes help fund the military, the environmental protection agency, and thousands of other programs. Property taxes help fund schools. Gasoline taxes help pay for road improvements. While very few people enjoy paying taxes, they are necessary to pay for the services we all depend upon.\r\n\r\nTaxes can be computed in a variety of ways, but are typically computed as a percentage of a sale, of one\u2019s income, or of one\u2019s assets.\r\n<div class=\"textbox examples\">\r\n<h3>recall calculations with percents<\/h3>\r\nTo make calculations using percents, first convert the percent to a decimal by dropping the % symbol and moving the decimal two places to the left (adding more zeros to the left as needed).\u00a0 When you move the decimal two places to the left, this is the same as dividing by 100.\u00a0 Recall the \"percent\" means \"per 100\" and the word \"per\" in math is often used to refer to division.\r\n\r\nEx. Write 2.7% as a decimal\r\n\r\nWrite 2.7 then move the decimal two places left to divide by 100 (adding more zeros to the left as needed).\r\n\r\n0.027 is the decimal representation of 2.7%.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Sales Tax<\/h3>\r\nThe sales tax rate in a city is 9.3%. How much sales tax will you pay on a $140 purchase?\r\n[reveal-answer q=\"620729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620729\"]\r\n\r\nThe sales tax will be 9.3% of $140. To compute this, we multiply $140 by the percent written as a decimal: $140(0.093) = $13.02.\r\n\r\n[\/hidden-answer][ohm_question]80109[\/ohm_question]\r\n\r\n<\/div>\r\nWhen taxes are not given as a fixed percentage rate, sometimes it is necessary to calculate the <strong>effective tax rate<\/strong>:<strong>\u00a0<\/strong>the equivalent percent rate of the tax paid out of the dollar amount the tax is based on.\r\n<div class=\"textbox examples\">\r\n<h3>Write a ratio as a percent<\/h3>\r\nTo obtain a percent from a ratio, divide the part by the whole and multiply by 100.\r\n\r\nEx. If 87 out of 932 people polled claim to dislike pizza, the percent of people polled who stated they don't like pizza is:\r\n\r\n[latex]\\dfrac{\\text{ part }}{\\text{ whole }}=\\dfrac{87}{932} \\approx .0934[\/latex]\r\n\r\n[latex].0934 \\cdot 100 = 9.34 \\%[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Property Tax<\/h3>\r\nJaquim\u00a0paid $3,200 in property taxes on his\u00a0house valued at $215,000 last year. What is the effective tax rate?\r\n[reveal-answer q=\"696377\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"696377\"]\r\n\r\nWe can compute the equivalent percentage: 3200\/215000 = 0.01488, or about 1.49% effective rate.\r\n\r\n[\/hidden-answer]\r\n\r\nHere is another example:\r\n\r\nIn Washington DC, the tax on a property assessed at $690,000 is $9,660. If tax rates are proportional in this city, how much would the tax be on a property assessed at $610,000?\r\n\r\nWe can use proportions to solve this problem.\r\n\r\n<img class=\"wp-image-935 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-300x194.png\" alt=\"\" width=\"567\" height=\"367\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nTaxes are often referred to as progressive, regressive, or flat.\r\n<ul>\r\n \t<li>A <strong>flat tax<\/strong>, or proportional tax, charges a constant percentage rate.<\/li>\r\n \t<li>A <strong>progressive tax<\/strong> increases the percent rate as the base amount increases.<\/li>\r\n \t<li>A <strong>regressive tax <\/strong>decreases the percent rate as the base amount increases.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Federal Income Tax<\/h3>\r\n<strong>Income Tax<\/strong>\r\n\r\nThe United States federal government collects income tax and most (but not all) states collect a state income tax.\u00a0\u00a0The United States federal income tax on earned wages is an example of a progressive tax. People with a higher wage income pay a higher percent tax on their income.\r\n\r\nIncome tax is based on gross income which is your total earned income for the year (this includes not just your wages but also interest earnings and even lottery winnings).\u00a0 However gross income is not directly used to calculate the amount of income tax that you owe for the year.\u00a0 You are allowed to adjust your gross income by subtracting deposits made to a retirement account (along with other things).\u00a0 After making these adjustments, you have an adjusted gross income.\u00a0 Wait!\u00a0 You are allowed deductions (and in some years we were also allowed personal exemptions).\u00a0 Taxable income is what is used in the calculations for your income taxes.\r\n\r\nAdjusted Gross Income = Gross Income \u2013 Adjustments\r\n\r\nTaxable Income = Adjusted Gross Income \u2013 Deductions \u2013 Any Allowed Exemptions\r\n\r\nBelow are a couple of basic examples to show calculations of income taxes based on the taxable income and the federal tax table for the year 2011.\u00a0 These tax tables are updated each year so it is important to make sure that you are using the correct table.\r\n\r\n<strong>Examples:<\/strong>\r\n\r\nFor a single person in 2011, taxable income (subtract deductions and any personal exemptions from adjusted gross income) up to $8,500 was taxed at 10%. Income over $8,500 but up to $34,500 was taxed at 15%.\r\n<h4>Taxable Income of $10,000<\/h4>\r\nStephen's taxable income was $10,000 in 2011. He would pay 10% on the portion of his\u00a0income up to $8,500, and 15% on the income over $8,500.\r\n\r\n8500(0.10) = 850 \u00a0 \u00a0 10% of $8500\r\n1500(0.15) = 225 \u00a0 \u00a0 \u00a015% of the remaining $1500 of income\r\nTotal tax:\u00a0\u00a0 = $1075\r\n\r\nWhat was Stephen's effective tax rate?\r\n[reveal-answer q=\"696772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"696772\"]\r\n\r\nThe effective tax rate paid is 1075\/10000 = 10.75%\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n<h4>Taxable Income of $30,000<\/h4>\r\nD'Andrea's taxable income was $30,000 in 2011. She would also pay 10% on the portion of her\u00a0income up to $8,500, and 15% on the income over $8,500.\r\n\r\n8500(0.10) = 850\u00a0\u00a0\u00a0\u00a0\u00a0 10% of $8500\r\n21500(0.15) = 3225\u00a0\u00a0\u00a0 15% of the remaining $21500 of income\r\nTotal tax:\u00a0\u00a0 = $4075\r\n\r\nWhat was D'Andrea's effective tax rate?\r\n[reveal-answer q=\"648795\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"648795\"]\r\n\r\nThe effective tax rate paid is 4075\/30000 = 13.58%.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nNotice that the effective rate has increased with income, showing this is a progressive tax.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Gasoline Tax<\/h3>\r\nA gasoline tax is a flat tax when considered in terms of consumption. A tax of, say, $0.30 per gallon is proportional to the amount of gasoline purchased. Someone buying 10 gallons of gas at $4 a gallon would pay $3 in tax, which is $3\/$40 = 7.5%. Someone buying 30 gallons of gas at $4 a gallon would pay $9 in tax, which is $9\/$120 = 7.5%, the same effective rate.\r\n\r\nHowever, in terms of income, a gasoline tax is often considered a regressive tax. It is likely that someone earning $30,000 a year and someone earning $60,000 a year will drive about the same amount. If both pay $60 in gasoline taxes over a year, the person earning $30,000 has paid 0.2% of their income, while the person earning $60,000 has paid 0.1% of their income in gas taxes.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA sales tax is a fixed percentage tax on a person\u2019s purchases. Is this a flat, progressive, or regressive tax?\r\n\r\n[practice-area rows=\"4\"][\/practice-area]\r\n[reveal-answer q=\"807267\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807267\"]\r\n\r\nWhile sales tax is a flat percentage rate, it is often considered a regressive tax for the same reasons as the gasoline tax.\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]65941[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate sales and income taxes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Calculate sales and income taxes<\/span><\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/27192407\/5120304358_f64b93a2ec_o.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/27192407\/5120304358_f64b93a2ec_o-300x212.jpg\" alt=\"magnifying lens over the word &quot;tax&quot;\" width=\"442\" height=\"312\" \/><\/a><\/p>\n<p>Governments collect taxes to pay for the services they provide. In the United States, federal income taxes help fund the military, the environmental protection agency, and thousands of other programs. Property taxes help fund schools. Gasoline taxes help pay for road improvements. While very few people enjoy paying taxes, they are necessary to pay for the services we all depend upon.<\/p>\n<p>Taxes can be computed in a variety of ways, but are typically computed as a percentage of a sale, of one\u2019s income, or of one\u2019s assets.<\/p>\n<div class=\"textbox examples\">\n<h3>recall calculations with percents<\/h3>\n<p>To make calculations using percents, first convert the percent to a decimal by dropping the % symbol and moving the decimal two places to the left (adding more zeros to the left as needed).\u00a0 When you move the decimal two places to the left, this is the same as dividing by 100.\u00a0 Recall the &#8220;percent&#8221; means &#8220;per 100&#8221; and the word &#8220;per&#8221; in math is often used to refer to division.<\/p>\n<p>Ex. Write 2.7% as a decimal<\/p>\n<p>Write 2.7 then move the decimal two places left to divide by 100 (adding more zeros to the left as needed).<\/p>\n<p>0.027 is the decimal representation of 2.7%.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Sales Tax<\/h3>\n<p>The sales tax rate in a city is 9.3%. How much sales tax will you pay on a $140 purchase?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q620729\">Show Solution<\/span><\/p>\n<div id=\"q620729\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sales tax will be 9.3% of $140. To compute this, we multiply $140 by the percent written as a decimal: $140(0.093) = $13.02.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm80109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=80109&theme=oea&iframe_resize_id=ohm80109&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When taxes are not given as a fixed percentage rate, sometimes it is necessary to calculate the <strong>effective tax rate<\/strong>:<strong>\u00a0<\/strong>the equivalent percent rate of the tax paid out of the dollar amount the tax is based on.<\/p>\n<div class=\"textbox examples\">\n<h3>Write a ratio as a percent<\/h3>\n<p>To obtain a percent from a ratio, divide the part by the whole and multiply by 100.<\/p>\n<p>Ex. If 87 out of 932 people polled claim to dislike pizza, the percent of people polled who stated they don&#8217;t like pizza is:<\/p>\n<p>[latex]\\dfrac{\\text{ part }}{\\text{ whole }}=\\dfrac{87}{932} \\approx .0934[\/latex]<\/p>\n<p>[latex].0934 \\cdot 100 = 9.34 \\%[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Property Tax<\/h3>\n<p>Jaquim\u00a0paid $3,200 in property taxes on his\u00a0house valued at $215,000 last year. What is the effective tax rate?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q696377\">Show Solution<\/span><\/p>\n<div id=\"q696377\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can compute the equivalent percentage: 3200\/215000 = 0.01488, or about 1.49% effective rate.<\/p>\n<\/div>\n<\/div>\n<p>Here is another example:<\/p>\n<p>In Washington DC, the tax on a property assessed at $690,000 is $9,660. If tax rates are proportional in this city, how much would the tax be on a property assessed at $610,000?<\/p>\n<p>We can use proportions to solve this problem.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-935 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-300x194.png\" alt=\"\" width=\"567\" height=\"367\" srcset=\"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-300x194.png 300w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-768x496.png 768w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-1024x661.png 1024w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-65x42.png 65w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-225x145.png 225w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions-350x226.png 350w, https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-content\/uploads\/sites\/5857\/2023\/06\/Finding-property-taxes-using-proportions.png 1033w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Taxes are often referred to as progressive, regressive, or flat.<\/p>\n<ul>\n<li>A <strong>flat tax<\/strong>, or proportional tax, charges a constant percentage rate.<\/li>\n<li>A <strong>progressive tax<\/strong> increases the percent rate as the base amount increases.<\/li>\n<li>A <strong>regressive tax <\/strong>decreases the percent rate as the base amount increases.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example: Federal Income Tax<\/h3>\n<p><strong>Income Tax<\/strong><\/p>\n<p>The United States federal government collects income tax and most (but not all) states collect a state income tax.\u00a0\u00a0The United States federal income tax on earned wages is an example of a progressive tax. People with a higher wage income pay a higher percent tax on their income.<\/p>\n<p>Income tax is based on gross income which is your total earned income for the year (this includes not just your wages but also interest earnings and even lottery winnings).\u00a0 However gross income is not directly used to calculate the amount of income tax that you owe for the year.\u00a0 You are allowed to adjust your gross income by subtracting deposits made to a retirement account (along with other things).\u00a0 After making these adjustments, you have an adjusted gross income.\u00a0 Wait!\u00a0 You are allowed deductions (and in some years we were also allowed personal exemptions).\u00a0 Taxable income is what is used in the calculations for your income taxes.<\/p>\n<p>Adjusted Gross Income = Gross Income \u2013 Adjustments<\/p>\n<p>Taxable Income = Adjusted Gross Income \u2013 Deductions \u2013 Any Allowed Exemptions<\/p>\n<p>Below are a couple of basic examples to show calculations of income taxes based on the taxable income and the federal tax table for the year 2011.\u00a0 These tax tables are updated each year so it is important to make sure that you are using the correct table.<\/p>\n<p><strong>Examples:<\/strong><\/p>\n<p>For a single person in 2011, taxable income (subtract deductions and any personal exemptions from adjusted gross income) up to $8,500 was taxed at 10%. Income over $8,500 but up to $34,500 was taxed at 15%.<\/p>\n<h4>Taxable Income of $10,000<\/h4>\n<p>Stephen&#8217;s taxable income was $10,000 in 2011. He would pay 10% on the portion of his\u00a0income up to $8,500, and 15% on the income over $8,500.<\/p>\n<p>8500(0.10) = 850 \u00a0 \u00a0 10% of $8500<br \/>\n1500(0.15) = 225 \u00a0 \u00a0 \u00a015% of the remaining $1500 of income<br \/>\nTotal tax:\u00a0\u00a0 = $1075<\/p>\n<p>What was Stephen&#8217;s effective tax rate?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q696772\">Show Solution<\/span><\/p>\n<div id=\"q696772\" class=\"hidden-answer\" style=\"display: none\">\n<p>The effective tax rate paid is 1075\/10000 = 10.75%<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h4>Taxable Income of $30,000<\/h4>\n<p>D&#8217;Andrea&#8217;s taxable income was $30,000 in 2011. She would also pay 10% on the portion of her\u00a0income up to $8,500, and 15% on the income over $8,500.<\/p>\n<p>8500(0.10) = 850\u00a0\u00a0\u00a0\u00a0\u00a0 10% of $8500<br \/>\n21500(0.15) = 3225\u00a0\u00a0\u00a0 15% of the remaining $21500 of income<br \/>\nTotal tax:\u00a0\u00a0 = $4075<\/p>\n<p>What was D&#8217;Andrea&#8217;s effective tax rate?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q648795\">Show Solution<\/span><\/p>\n<div id=\"q648795\" class=\"hidden-answer\" style=\"display: none\">\n<p>The effective tax rate paid is 4075\/30000 = 13.58%.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that the effective rate has increased with income, showing this is a progressive tax.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Gasoline Tax<\/h3>\n<p>A gasoline tax is a flat tax when considered in terms of consumption. A tax of, say, $0.30 per gallon is proportional to the amount of gasoline purchased. Someone buying 10 gallons of gas at $4 a gallon would pay $3 in tax, which is $3\/$40 = 7.5%. Someone buying 30 gallons of gas at $4 a gallon would pay $9 in tax, which is $9\/$120 = 7.5%, the same effective rate.<\/p>\n<p>However, in terms of income, a gasoline tax is often considered a regressive tax. It is likely that someone earning $30,000 a year and someone earning $60,000 a year will drive about the same amount. If both pay $60 in gasoline taxes over a year, the person earning $30,000 has paid 0.2% of their income, while the person earning $60,000 has paid 0.1% of their income in gas taxes.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A sales tax is a fixed percentage tax on a person\u2019s purchases. Is this a flat, progressive, or regressive tax?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"4\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q807267\">Show Solution<\/span><\/p>\n<div id=\"q807267\" class=\"hidden-answer\" style=\"display: none\">\n<p>While sales tax is a flat percentage rate, it is often considered a regressive tax for the same reasons as the gasoline tax.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm65941\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=65941&theme=oea&iframe_resize_id=ohm65941&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-83\">\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>Revision and Adaptation. <strong>Provided 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>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>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><strong>Authored by<\/strong>: Calita Kabir. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/flic.kr\/p\/8NsU6C\">https:\/\/flic.kr\/p\/8NsU6C<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 65951. <strong>Authored by<\/strong>: Parker,Gary. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 80109. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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