{"id":4633,"date":"2020-04-21T00:19:12","date_gmt":"2020-04-21T00:19:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/percents\/"},"modified":"2021-03-23T19:32:54","modified_gmt":"2021-03-23T19:32:54","slug":"percents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-mathforliberalartscorequisite\/chapter\/percents\/","title":{"raw":"Percents","rendered":"Percents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given the part and the whole, write a percent<\/li>\r\n \t<li>Evaluate changes in amounts with percent calculations<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>A percent is a fraction<\/h3>\r\nRecall that a fraction is written [latex]\\dfrac{a}{b},[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b \\neq 0[\/latex]. In a fraction, [latex]a[\/latex] is called the numerator and [latex]b[\/latex] is called the denominator.\r\n\r\nA\u00a0<strong>percent<\/strong> can be expressed as a fraction, that is a\u00a0<strong>ratio,\u00a0<\/strong>of some part of a quantity out of the whole quantity,\u00a0 [latex]\\dfrac{\\text{part}}{\\text{whole}}[\/latex].\r\n\r\nEx. Suppose you take an informal poll of your classmates to find out how many of them like pizza. You find that, out of 25 classmates, 20 of them like pizza. You can represent your findings as a\u00a0<strong>ratio<\/strong> of how many like pizza out of how many classmates you asked.\r\n<p style=\"padding-left: 30px\">[latex]\\dfrac{20}{25}[\/latex] represents the 20 out of 25 classmates who like pizza.<\/p>\r\nTo find out what\u00a0<strong>percent<\/strong> of the 25 asked said they like pizza, divide the numerator by the denominator, then multiply by 100.\r\n<p style=\"padding-left: 30px\">[latex]\\dfrac{20}{25} = 20 \\div 25 = 0.80 = 80 \\%[\/latex]<\/p>\r\n\r\n<\/div>\r\n<strong>Percent <\/strong>literally means \u201cper 100,\u201d or \u201cparts per hundred.\u201d When we write 40%, this is equivalent to the fraction [latex]\\displaystyle\\frac{40}{100}[\/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\\displaystyle\\frac{80}{200}=\\frac{10}{25}=\\frac{40}{100}[\/latex].\r\n\r\n[caption id=\"attachment_494\" align=\"aligncenter\" width=\"500\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\"><img class=\"wp-image-494\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\" alt=\"Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty.\" width=\"500\" height=\"282\" \/><\/a> A visual depiction of 40%[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>convert a percent to a\u00a0 decimal or fraction<\/h3>\r\nTo do mathematical calculations with a given percent, we must first write it in numerical form. A percent may be represented as a percent, a fraction, or a decimal.\r\n\r\n<strong>Convert a percent to a fraction<\/strong>\r\n<ol>\r\n \t<li>Write the percent over a denominator of [latex]100[\/latex] and drop the percent symbol %.<\/li>\r\n \t<li>Reduce the resulting fraction as needed.<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 30px\">Ex. [latex]80 \\% =\\dfrac{80}{100}=\\dfrac{4\\cdot 20}{5\\cdot 20}=\\dfrac{4}{5}[\/latex]<\/p>\r\n<strong>Convert a percent to a decimal<\/strong>\r\n\r\nThere are three methods for writing a percent as a decimal.\r\n<ul>\r\n \t<li>You can write the percent as a fraction, simply fully, then divide the numerator by the denominator.\r\n<ul>\r\n \t<li>[latex]80 \\% =\\dfrac{80}{100}=\\dfrac{4\\cdot 20}{5\\cdot 20}=\\dfrac{4}{5}=0.8[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>You can write the percent as a fraction, simplify to a denominator of 10, 100, 1000, etc., then rewrite as a decimal.\r\n<ul>\r\n \t<li>[latex]80 \\% =\\dfrac{80}{100}=\\dfrac{8\\cdot 10}{10\\cdot 10}=\\dfrac{8}{10}=\\text{ eight tenths }=0.8[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Write the percent without the percent symbol %, then place a decimal after the ones place and move it two places to the left.\r\n<ul>\r\n \t<li>[latex]80 \\% =80.0=0.80=0.8[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Percent<\/h3>\r\nIf we have a <em>part<\/em> that is some <em>percent<\/em> of a <em>whole<\/em>, then\u00a0[latex]\\displaystyle\\text{percent}=\\frac{\\text{part}}{\\text{whole}}[\/latex], or equivalently, [latex]\\text{percent}\\cdot\\text{whole}=\\text{part}[\/latex].\r\n\r\nTo do calculations using percents, we write the percent as a decimal or fraction.\r\n\r\nThe video and following few examples demonstrate how to convert between percent, fraction, and decimal representations.\r\n\r\nhttps:\/\/youtu.be\/Z229RysttR8\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn a survey, 243 out of 400 people state that they like dogs. What percent is this?\r\n[reveal-answer q=\"987171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"987171\"]\r\n\r\n[latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex] This is 60.75%.\r\n\r\nNotice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite each as a percent:\r\n<ol>\r\n \t<li>[latex]\\displaystyle\\frac{1}{4}[\/latex]<\/li>\r\n \t<li>0.02<\/li>\r\n \t<li>2.35<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"660805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660805\"]\r\n<ol>\r\n \t<li>[latex]\\displaystyle\\frac{1}{4}=0.25[\/latex] = 25%<\/li>\r\n \t<li>0.02 = 2%<\/li>\r\n \t<li>2.35 = 235%<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]17441[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Given the part and the whole, write a percent<\/li>\n<li>Evaluate changes in amounts with percent calculations<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>A percent is a fraction<\/h3>\n<p>Recall that a fraction is written [latex]\\dfrac{a}{b},[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b \\neq 0[\/latex]. In a fraction, [latex]a[\/latex] is called the numerator and [latex]b[\/latex] is called the denominator.<\/p>\n<p>A\u00a0<strong>percent<\/strong> can be expressed as a fraction, that is a\u00a0<strong>ratio,\u00a0<\/strong>of some part of a quantity out of the whole quantity,\u00a0 [latex]\\dfrac{\\text{part}}{\\text{whole}}[\/latex].<\/p>\n<p>Ex. Suppose you take an informal poll of your classmates to find out how many of them like pizza. You find that, out of 25 classmates, 20 of them like pizza. You can represent your findings as a\u00a0<strong>ratio<\/strong> of how many like pizza out of how many classmates you asked.<\/p>\n<p style=\"padding-left: 30px\">[latex]\\dfrac{20}{25}[\/latex] represents the 20 out of 25 classmates who like pizza.<\/p>\n<p>To find out what\u00a0<strong>percent<\/strong> of the 25 asked said they like pizza, divide the numerator by the denominator, then multiply by 100.<\/p>\n<p style=\"padding-left: 30px\">[latex]\\dfrac{20}{25} = 20 \\div 25 = 0.80 = 80 \\%[\/latex]<\/p>\n<\/div>\n<p><strong>Percent <\/strong>literally means \u201cper 100,\u201d or \u201cparts per hundred.\u201d When we write 40%, this is equivalent to the fraction [latex]\\displaystyle\\frac{40}{100}[\/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\\displaystyle\\frac{80}{200}=\\frac{10}{25}=\\frac{40}{100}[\/latex].<\/p>\n<div id=\"attachment_494\" style=\"width: 510px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-494\" class=\"wp-image-494\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/14203900\/percent-40844_1280.png\" alt=\"Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty.\" width=\"500\" height=\"282\" \/><\/a><\/p>\n<p id=\"caption-attachment-494\" class=\"wp-caption-text\">A visual depiction of 40%<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>convert a percent to a\u00a0 decimal or fraction<\/h3>\n<p>To do mathematical calculations with a given percent, we must first write it in numerical form. A percent may be represented as a percent, a fraction, or a decimal.<\/p>\n<p><strong>Convert a percent to a fraction<\/strong><\/p>\n<ol>\n<li>Write the percent over a denominator of [latex]100[\/latex] and drop the percent symbol %.<\/li>\n<li>Reduce the resulting fraction as needed.<\/li>\n<\/ol>\n<p style=\"padding-left: 30px\">Ex. [latex]80 \\% =\\dfrac{80}{100}=\\dfrac{4\\cdot 20}{5\\cdot 20}=\\dfrac{4}{5}[\/latex]<\/p>\n<p><strong>Convert a percent to a decimal<\/strong><\/p>\n<p>There are three methods for writing a percent as a decimal.<\/p>\n<ul>\n<li>You can write the percent as a fraction, simply fully, then divide the numerator by the denominator.\n<ul>\n<li>[latex]80 \\% =\\dfrac{80}{100}=\\dfrac{4\\cdot 20}{5\\cdot 20}=\\dfrac{4}{5}=0.8[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>You can write the percent as a fraction, simplify to a denominator of 10, 100, 1000, etc., then rewrite as a decimal.\n<ul>\n<li>[latex]80 \\% =\\dfrac{80}{100}=\\dfrac{8\\cdot 10}{10\\cdot 10}=\\dfrac{8}{10}=\\text{ eight tenths }=0.8[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Write the percent without the percent symbol %, then place a decimal after the ones place and move it two places to the left.\n<ul>\n<li>[latex]80 \\% =80.0=0.80=0.8[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Percent<\/h3>\n<p>If we have a <em>part<\/em> that is some <em>percent<\/em> of a <em>whole<\/em>, then\u00a0[latex]\\displaystyle\\text{percent}=\\frac{\\text{part}}{\\text{whole}}[\/latex], or equivalently, [latex]\\text{percent}\\cdot\\text{whole}=\\text{part}[\/latex].<\/p>\n<p>To do calculations using percents, we write the percent as a decimal or fraction.<\/p>\n<p>The video and following few examples demonstrate how to convert between percent, fraction, and decimal representations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Review of basic percents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Z229RysttR8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In a survey, 243 out of 400 people state that they like dogs. What percent is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987171\">Show Solution<\/span><\/p>\n<div id=\"q987171\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\displaystyle\\frac{243}{400}=0.6075=\\frac{60.75}{100}[\/latex] This is 60.75%.<\/p>\n<p>Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write each as a percent:<\/p>\n<ol>\n<li>[latex]\\displaystyle\\frac{1}{4}[\/latex]<\/li>\n<li>0.02<\/li>\n<li>2.35<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660805\">Show Solution<\/span><\/p>\n<div id=\"q660805\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\displaystyle\\frac{1}{4}=0.25[\/latex] = 25%<\/li>\n<li>0.02 = 2%<\/li>\n<li>2.35 = 235%<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm17441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=17441&theme=oea&iframe_resize_id=ohm17441&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-4633\">\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>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>Caution sign. <strong>Authored by<\/strong>: JDDesign. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/\">https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>40% shaded rectangle. <strong>Authored by<\/strong>: Clker-Free-Vector-Images. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/percent-40-bar-progress-meter-40844\/\">https:\/\/pixabay.com\/en\/percent-40-bar-progress-meter-40844\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Review of basic percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Z229RysttR8\">https:\/\/youtu.be\/Z229RysttR8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Absolute and Relative Differences. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QjVeurkg8CQ\">https:\/\/youtu.be\/QjVeurkg8CQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Importance of base in percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nygw69JqwoQ\">https:\/\/youtu.be\/nygw69JqwoQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Combining percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/4HNxwYMTNl8\">https:\/\/youtu.be\/4HNxwYMTNl8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluating claims involving percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Svlu2Lurmsc\">https:\/\/youtu.be\/Svlu2Lurmsc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Percentage points and averaging percents. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vtgEkQUB5F8\">https:\/\/youtu.be\/vtgEkQUB5F8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 17441, 17447, 17443. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Problem Solving\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Caution sign\",\"author\":\"JDDesign\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/industrial-safety-signal-symbol-1492046\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"40% shaded 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