{"id":7187,"date":"2017-05-09T20:23:15","date_gmt":"2017-05-09T20:23:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=7187"},"modified":"2020-09-11T00:27:34","modified_gmt":"2020-09-11T00:27:34","slug":"understand-percent","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/understand-percent\/","title":{"raw":"4.1.a - Defining Percents","rendered":"4.1.a &#8211; Defining Percents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the definition of percent<\/li>\r\n \t<li>Identify the amount, the base, and the percent in a percent problem<\/li>\r\n \t<li>Find the unknown in a percent problem<\/li>\r\n<\/ul>\r\n<\/div>\r\nPercents are used in many different applications. Percents are used widely to describe how something changed. For example, you may have heard that the amount of rainfall this month had decreased by [latex]12\\%[\/latex] from last year, or that the number of jobless claims has increase by [latex]5\\%[\/latex] this quarter over last quarter.\r\n\r\n[caption id=\"attachment_3014\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-3014 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/20194605\/Screen-Shot-2016-04-20-at-12.44.43-PM-300x209.png\" alt=\"A graph showing the unemployment rate, with the y-axis representing percent and the x-axis representing time.\" width=\"300\" height=\"209\" \/> Unemployment rate as percent by year between 2004 and 2014.[\/caption]\r\n\r\nWe regularly use this kind of language to quickly describe how much something increased or decreased over time or between significant events.\r\n\r\nHow many cents are in one dollar? There are [latex]100[\/latex] cents in a dollar. How many years are in a century? There are [latex]100[\/latex] years in a century. Does this give you a clue about what the word \"percent\" means? It is really two words, \"per cent,\" and means per one hundred. A percent is a ratio whose denominator is [latex]100[\/latex]. We use the percent symbol [latex]\\text{%,}[\/latex] to show percent.\r\n<div class=\"textbox shaded\">\r\n<h3>Percent<\/h3>\r\nA percent is a ratio whose denominator is [latex]100[\/latex].\r\n\r\n<\/div>\r\nAccording to data from the American Association of Community Colleges [latex]\\left(2015\\right)[\/latex], about [latex]\\text{57%}[\/latex] of community college students are female. This means [latex]57[\/latex] out of every [latex]100[\/latex] community college students are female, as the image below\u00a0<span style=\"font-size: 16px\">shows. Out of the [latex]100[\/latex] squares on the grid, [latex]57[\/latex] are shaded, which we write as the ratio [latex]\\Large\\frac{57}{100}[\/latex].<\/span>\r\n\r\nAmong every [latex]100[\/latex] community college students, [latex]57[\/latex] are female.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221858\/CNX_BMath_Figure_06_01_001.png\" alt=\"The figure shows a hundred flat with 57 units shaded.\" \/>\r\nSimilarly, [latex]\\text{25%}[\/latex] means a ratio of [latex]\\Large\\frac{25}{100}\\normalsize ,\\text{3%}[\/latex] means a ratio of [latex]\\Large\\frac{3}{100}[\/latex] and [latex]\\text{100%}[\/latex] means a ratio of [latex]\\Large\\frac{100}{100}[\/latex]. In words, \"one hundred percent\" means the total [latex]\\text{100%}[\/latex] is [latex]\\Large\\frac{100}{100}[\/latex], and since [latex]\\Large\\frac{100}{100}\\normalsize =1[\/latex], we see that [latex]\\text{100%}[\/latex] means [latex]1[\/latex] whole.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nAccording to the Public Policy Institute of California [latex]\\left(2010\\right)\\text{, }\\text{44%}[\/latex] of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.\r\n\r\nSolution\r\n<table id=\"eip-id1168468695064\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>The amount we want to convert is [latex]44%[\/latex].<\/td>\r\n<td>[latex]44%[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the percent as a ratio. Remember that <em>percent<\/em> means per [latex]100[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{44}{100}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question]146655[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIn [latex]2007[\/latex], according to a U.S. Department of Education report, [latex]21[\/latex] out of every [latex]100[\/latex] first-time freshmen college students at [latex]\\text{4-year}[\/latex] public institutions took at least one remedial course. Write this as a ratio and then as a percent.\r\n[reveal-answer q=\"469361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"469361\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467356132\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>The amount we want to convert is [latex]21[\/latex] out of [latex]100[\/latex] .<\/td>\r\n<td>[latex]21[\/latex] out of [latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a ratio.<\/td>\r\n<td>[latex]\\Large\\frac{21}{100}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert the [latex]21[\/latex] per [latex]100[\/latex] to percent.<\/td>\r\n<td>[latex]21[\/latex]%<\/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\u00a0it<\/h3>\r\n[ohm_question]146656[\/ohm_question]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\nFrom the previous example, we can identify\u00a0three important parts to finding the percent of a whole:\r\n<ul>\r\n \t<li>the <b>percent<\/b>,\u00a0<b> <\/b>has the percent symbol (%) or the word \u201cpercent\u201d<\/li>\r\n \t<li>the <b>amount<\/b>, the amount is\u00a0part of the whole<\/li>\r\n \t<li>and the <b>base, <\/b>the base is the whole amount<\/li>\r\n<\/ul>\r\nThe following examples show how to identify the three parts: the percent, the base, and the amount.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the percent, amount, and base in this problem.\r\n\r\n[latex]30[\/latex] is [latex]20\\%[\/latex] of what number?\r\n\r\n[reveal-answer q=\"204160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"204160\"]\r\n\r\n<b>Percent:<\/b> The percent is the number with the % symbol: [latex]20\\%[\/latex].\r\n\r\n<b>Base:<\/b> The base is the whole amount, which in this case is unknown.\r\n\r\n<b>Amount:<\/b> The amount based on the percent is [latex]30[\/latex].\r\n<h4>Answer<\/h4>\r\nPercent = [latex]20\\%[\/latex]\r\nAmount = [latex]30[\/latex]\r\nBase = unknown or [latex]x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe previous problem states that [latex]30[\/latex] is a portion of another number. That means \u00a0[latex]30[\/latex] is the amount. Note that this problem could be rewritten: [latex]20\\%[\/latex] of what number is [latex]30[\/latex]?\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the percent, amount, and base in this problem.\r\n\r\nWhat percent of [latex]30[\/latex] is [latex]3[\/latex]?\r\n\r\n[reveal-answer q=\"318375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"318375\"]\r\n\r\n<b>Percent:<\/b> The percent is unknown, because the problem states \u201cwhat percent?\u201d.\r\n\r\n<b>Base:<\/b> The base is the whole amount, so the base is [latex]30[\/latex].\r\n\r\n<b>Amount:<\/b> The amount is a portion of the whole, which is [latex]3[\/latex] in this case.\r\n<h4>Answer<\/h4>\r\nPercent = unknown or [latex]x[\/latex]\r\nAmount = \u00a0[latex]3[\/latex]\r\nBase = [latex]30[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the percent, amount, and base in this problem.\r\n\r\nWhat is \u00a0[latex]60\\%[\/latex] of [latex]45[\/latex]?\r\n\r\n[reveal-answer q=\"102763\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"102763\"]\r\n\r\n<b>Percent:<\/b> The percent is known\r\n\r\n<b>Base:\u00a0<\/b>The base is the whole amount, so the base is \u00a0[latex]45[\/latex].\r\n\r\n<b>Amount:<\/b> The amount is a portion of the whole, which is what we want to identify.\r\n<h4>Answer<\/h4>\r\nPercent = \u00a0[latex]60\\%[\/latex]\r\nAmount = unknown or [latex]x[\/latex]\r\nBase = \u00a0[latex]45[\/latex]\r\n\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]80094[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides more examples that describe how to identify the percent, amount, and base in a percent problem.\r\n\r\nhttps:\/\/youtu.be\/zwT58-LJCvs\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the definition of percent<\/li>\n<li>Identify the amount, the base, and the percent in a percent problem<\/li>\n<li>Find the unknown in a percent problem<\/li>\n<\/ul>\n<\/div>\n<p>Percents are used in many different applications. Percents are used widely to describe how something changed. For example, you may have heard that the amount of rainfall this month had decreased by [latex]12\\%[\/latex] from last year, or that the number of jobless claims has increase by [latex]5\\%[\/latex] this quarter over last quarter.<\/p>\n<div id=\"attachment_3014\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3014\" class=\"wp-image-3014 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/20194605\/Screen-Shot-2016-04-20-at-12.44.43-PM-300x209.png\" alt=\"A graph showing the unemployment rate, with the y-axis representing percent and the x-axis representing time.\" width=\"300\" height=\"209\" \/><\/p>\n<p id=\"caption-attachment-3014\" class=\"wp-caption-text\">Unemployment rate as percent by year between 2004 and 2014.<\/p>\n<\/div>\n<p>We regularly use this kind of language to quickly describe how much something increased or decreased over time or between significant events.<\/p>\n<p>How many cents are in one dollar? There are [latex]100[\/latex] cents in a dollar. How many years are in a century? There are [latex]100[\/latex] years in a century. Does this give you a clue about what the word &#8220;percent&#8221; means? It is really two words, &#8220;per cent,&#8221; and means per one hundred. A percent is a ratio whose denominator is [latex]100[\/latex]. We use the percent symbol [latex]\\text{%,}[\/latex] to show percent.<\/p>\n<div class=\"textbox shaded\">\n<h3>Percent<\/h3>\n<p>A percent is a ratio whose denominator is [latex]100[\/latex].<\/p>\n<\/div>\n<p>According to data from the American Association of Community Colleges [latex]\\left(2015\\right)[\/latex], about [latex]\\text{57%}[\/latex] of community college students are female. This means [latex]57[\/latex] out of every [latex]100[\/latex] community college students are female, as the image below\u00a0<span style=\"font-size: 16px\">shows. Out of the [latex]100[\/latex] squares on the grid, [latex]57[\/latex] are shaded, which we write as the ratio [latex]\\Large\\frac{57}{100}[\/latex].<\/span><\/p>\n<p>Among every [latex]100[\/latex] community college students, [latex]57[\/latex] are female.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221858\/CNX_BMath_Figure_06_01_001.png\" alt=\"The figure shows a hundred flat with 57 units shaded.\" \/><br \/>\nSimilarly, [latex]\\text{25%}[\/latex] means a ratio of [latex]\\Large\\frac{25}{100}\\normalsize ,\\text{3%}[\/latex] means a ratio of [latex]\\Large\\frac{3}{100}[\/latex] and [latex]\\text{100%}[\/latex] means a ratio of [latex]\\Large\\frac{100}{100}[\/latex]. In words, &#8220;one hundred percent&#8221; means the total [latex]\\text{100%}[\/latex] is [latex]\\Large\\frac{100}{100}[\/latex], and since [latex]\\Large\\frac{100}{100}\\normalsize =1[\/latex], we see that [latex]\\text{100%}[\/latex] means [latex]1[\/latex] whole.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>According to the Public Policy Institute of California [latex]\\left(2010\\right)\\text{, }\\text{44%}[\/latex] of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468695064\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>The amount we want to convert is [latex]44%[\/latex].<\/td>\n<td>[latex]44%[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the percent as a ratio. Remember that <em>percent<\/em> means per [latex]100[\/latex].<\/td>\n<td>[latex]\\Large\\frac{44}{100}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146655\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146655&theme=oea&iframe_resize_id=ohm146655&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>In [latex]2007[\/latex], according to a U.S. Department of Education report, [latex]21[\/latex] out of every [latex]100[\/latex] first-time freshmen college students at [latex]\\text{4-year}[\/latex] public institutions took at least one remedial course. Write this as a ratio and then as a percent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q469361\">Show Solution<\/span><\/p>\n<div id=\"q469361\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467356132\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>The amount we want to convert is [latex]21[\/latex] out of [latex]100[\/latex] .<\/td>\n<td>[latex]21[\/latex] out of [latex]100[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a ratio.<\/td>\n<td>[latex]\\Large\\frac{21}{100}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert the [latex]21[\/latex] per [latex]100[\/latex] to percent.<\/td>\n<td>[latex]21[\/latex]%<\/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\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146656\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146656&theme=oea&iframe_resize_id=ohm146656&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<p>From the previous example, we can identify\u00a0three important parts to finding the percent of a whole:<\/p>\n<ul>\n<li>the <b>percent<\/b>,\u00a0<b> <\/b>has the percent symbol (%) or the word \u201cpercent\u201d<\/li>\n<li>the <b>amount<\/b>, the amount is\u00a0part of the whole<\/li>\n<li>and the <b>base, <\/b>the base is the whole amount<\/li>\n<\/ul>\n<p>The following examples show how to identify the three parts: the percent, the base, and the amount.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the percent, amount, and base in this problem.<\/p>\n<p>[latex]30[\/latex] is [latex]20\\%[\/latex] of what number?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q204160\">Show Solution<\/span><\/p>\n<div id=\"q204160\" class=\"hidden-answer\" style=\"display: none\">\n<p><b>Percent:<\/b> The percent is the number with the % symbol: [latex]20\\%[\/latex].<\/p>\n<p><b>Base:<\/b> The base is the whole amount, which in this case is unknown.<\/p>\n<p><b>Amount:<\/b> The amount based on the percent is [latex]30[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>Percent = [latex]20\\%[\/latex]<br \/>\nAmount = [latex]30[\/latex]<br \/>\nBase = unknown or [latex]x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The previous problem states that [latex]30[\/latex] is a portion of another number. That means \u00a0[latex]30[\/latex] is the amount. Note that this problem could be rewritten: [latex]20\\%[\/latex] of what number is [latex]30[\/latex]?<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the percent, amount, and base in this problem.<\/p>\n<p>What percent of [latex]30[\/latex] is [latex]3[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q318375\">Show Solution<\/span><\/p>\n<div id=\"q318375\" class=\"hidden-answer\" style=\"display: none\">\n<p><b>Percent:<\/b> The percent is unknown, because the problem states \u201cwhat percent?\u201d.<\/p>\n<p><b>Base:<\/b> The base is the whole amount, so the base is [latex]30[\/latex].<\/p>\n<p><b>Amount:<\/b> The amount is a portion of the whole, which is [latex]3[\/latex] in this case.<\/p>\n<h4>Answer<\/h4>\n<p>Percent = unknown or [latex]x[\/latex]<br \/>\nAmount = \u00a0[latex]3[\/latex]<br \/>\nBase = [latex]30[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the percent, amount, and base in this problem.<\/p>\n<p>What is \u00a0[latex]60\\%[\/latex] of [latex]45[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q102763\">Show Solution<\/span><\/p>\n<div id=\"q102763\" class=\"hidden-answer\" style=\"display: none\">\n<p><b>Percent:<\/b> The percent is known<\/p>\n<p><b>Base:\u00a0<\/b>The base is the whole amount, so the base is \u00a0[latex]45[\/latex].<\/p>\n<p><b>Amount:<\/b> The amount is a portion of the whole, which is what we want to identify.<\/p>\n<h4>Answer<\/h4>\n<p>Percent = \u00a0[latex]60\\%[\/latex]<br \/>\nAmount = unknown or [latex]x[\/latex]<br \/>\nBase = \u00a0[latex]45[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm80094\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=80094&theme=oea&iframe_resize_id=ohm80094&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides more examples that describe how to identify the percent, amount, and base in a percent problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Identify the Percent, Base, and Amount of a Percent Question\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zwT58-LJCvs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/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-7187\">\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: 146656, 146655. <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 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 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