{"id":2699,"date":"2025-08-13T18:28:21","date_gmt":"2025-08-13T18:28:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2699"},"modified":"2025-09-25T20:59:08","modified_gmt":"2025-09-25T20:59:08","slug":"probability-and-counting-theory-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-and-counting-theory-background-youll-need-3\/","title":{"raw":"Probability and Counting Theory: Background You'll Need 3","rendered":"Probability and Counting Theory: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Change numbers between fractions, decimals, and percentages<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Decimals to Percents<\/h2>\r\nConverting a decimal into a percent is a straightforward process once you understand that 'percent' means 'out of [latex]100[\/latex]'. This conversion is based on shifting the value of a decimal two places to the right to find its equivalent percent.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How To: Convert a Decimal to a Percent<\/strong>\r\n<ol id=\"eip-id1168469368569\" class=\"stepwise\">\r\n \t<li><strong>Express the Decimal as a Fraction<\/strong>: Start by writing the decimal as a fraction with 1 as the denominator.<\/li>\r\n \t<li><strong>Adjust to a Denominator of [latex]100[\/latex]<\/strong>: Since a percent is a fraction out of [latex]100[\/latex], scale the fraction so that the denominator is [latex]100[\/latex]. This may involve multiplying both the numerator and the denominator by an appropriate factor.<\/li>\r\n \t<li><strong>Convert to Percent<\/strong>: Once the denominator is [latex]100[\/latex], the numerator is the percent value. Write this number followed by the percent sign (%).<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Convert each decimal to a percent:\r\n<ol>\r\n \t<li>[latex]0.05[\/latex]<\/li>\r\n \t<li>[latex]0.83[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"872303\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"872303\"]\r\n<ol>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.05[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a fraction. The denominator is [latex]100[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{5}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write this ratio as a percent.<\/td>\r\n<td>[latex]5[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.83[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The denominator is [latex]100[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{83}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write this ratio as a percent.<\/td>\r\n<td>[latex]83[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">To convert a mixed number to a percent, we first write it as an improper fraction.<\/section><section class=\"textbox example\">Convert each decimal to a percent:\r\n<ol>\r\n \t<li>[latex]1.05[\/latex]<\/li>\r\n \t<li>[latex]0.075[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"515005\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"515005\"]\r\n<ol>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.05[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a fraction.<\/td>\r\n<td>[latex]1{\\Large\\frac{5}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as an improper fraction. The denominator is [latex]100[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{105}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write this ratio as a percent.<\/td>\r\n<td>[latex]105[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that since [latex]1.05&gt;1[\/latex], the result is more than [latex]\\text{100%.}[\/latex]<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.075[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a fraction. The denominator is [latex]1,000[\/latex].<\/td>\r\n<td style=\"width: 30%;\">[latex]{\\Large\\frac{75}{1,000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide the numerator and denominator by [latex]10[\/latex], so that the denominator is [latex]100[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{7.5}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write this ratio as a percent.<\/td>\r\n<td>[latex]7.5[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6638[\/ohm2_question]<\/section>\r\n<h2>Fraction to Percents<\/h2>\r\nThe process of converting fractions to percents involves two main steps: changing the fraction into a decimal, and then transforming that decimal into a percent. Understanding that 'percent' means 'per hundred' simplifies the conversion, as you're essentially finding out how many parts out of [latex]100[\/latex] the fraction represents.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How To: Convert a Fraction to a Percent<\/strong>\r\n<ol id=\"eip-id1168467345664\" class=\"stepwise\">\r\n \t<li><strong>Convert the Fraction to a Decimal<\/strong>: Divide the numerator (the top number of the fraction) by the denominator (the bottom number of the fraction) to get a decimal.<\/li>\r\n \t<li><strong>Multiply by [latex]100[\/latex] to Find the Percent<\/strong>: Take the decimal result and multiply it by [latex]100[\/latex]. Then, add the percent sign ([latex]\\%[\/latex]) to the product to express the answer as a percentage.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Convert each fraction or mixed number to a percent:\r\n<ol>\r\n \t<li>[latex]{\\Large\\frac{3}{4}}[\/latex]<\/li>\r\n \t<li>[latex]{\\Large\\frac{11}{8}}[\/latex]<\/li>\r\n \t<li>[latex]2{\\Large\\frac{1}{5}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"589499\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"589499\"]To convert a fraction to a decimal, divide the numerator by the denominator.\r\n<ol>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Change to a decimal.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221904\/CNX_BMath_Figure_06_01_004_img-01.png\" alt=\"0.75 showing the decimal place moves to the right to spaces.\" width=\"41\" height=\"24\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a percent by moving the decimal two places.<\/td>\r\n<td>[latex]75[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Change to a decimal.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221905\/CNX_BMath_Figure_06_01_005_img-01.png\" alt=\"1.375 showing the decimal place moves to the right to spaces.\" width=\"43\" height=\"26\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a percent by moving the decimal two places.<\/td>\r\n<td>[latex]137.5[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Write as an improper fraction.<\/td>\r\n<td>[latex]{\\Large\\frac{11}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change to a decimal.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221905\/CNX_BMath_Figure_06_01_006_img-01.png\" alt=\"2.20 showing the decimal place moves to the right to spaces.\" width=\"45\" height=\"26\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a percent.<\/td>\r\n<td>[latex]220[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.<\/section>Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6639[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Change numbers between fractions, decimals, and percentages<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Decimals to Percents<\/h2>\n<p>Converting a decimal into a percent is a straightforward process once you understand that &#8216;percent&#8217; means &#8216;out of [latex]100[\/latex]&#8216;. This conversion is based on shifting the value of a decimal two places to the right to find its equivalent percent.<\/p>\n<section class=\"textbox questionHelp\"><strong>How To: Convert a Decimal to a Percent<\/strong><\/p>\n<ol id=\"eip-id1168469368569\" class=\"stepwise\">\n<li><strong>Express the Decimal as a Fraction<\/strong>: Start by writing the decimal as a fraction with 1 as the denominator.<\/li>\n<li><strong>Adjust to a Denominator of [latex]100[\/latex]<\/strong>: Since a percent is a fraction out of [latex]100[\/latex], scale the fraction so that the denominator is [latex]100[\/latex]. This may involve multiplying both the numerator and the denominator by an appropriate factor.<\/li>\n<li><strong>Convert to Percent<\/strong>: Once the denominator is [latex]100[\/latex], the numerator is the percent value. Write this number followed by the percent sign (%).<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Convert each decimal to a percent:<\/p>\n<ol>\n<li>[latex]0.05[\/latex]<\/li>\n<li>[latex]0.83[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q872303\">Show Answer<\/button><\/p>\n<div id=\"q872303\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a fraction. The denominator is [latex]100[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{5}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write this ratio as a percent.<\/td>\n<td>[latex]5[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.83[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The denominator is [latex]100[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{83}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write this ratio as a percent.<\/td>\n<td>[latex]83[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">To convert a mixed number to a percent, we first write it as an improper fraction.<\/section>\n<section class=\"textbox example\">Convert each decimal to a percent:<\/p>\n<ol>\n<li>[latex]1.05[\/latex]<\/li>\n<li>[latex]0.075[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q515005\">Show Answer<\/button><\/p>\n<div id=\"q515005\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a fraction.<\/td>\n<td>[latex]1{\\Large\\frac{5}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as an improper fraction. The denominator is [latex]100[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{105}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write this ratio as a percent.<\/td>\n<td>[latex]105[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that since [latex]1.05>1[\/latex], the result is more than [latex]\\text{100%.}[\/latex]<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.075[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a fraction. The denominator is [latex]1,000[\/latex].<\/td>\n<td style=\"width: 30%;\">[latex]{\\Large\\frac{75}{1,000}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator and denominator by [latex]10[\/latex], so that the denominator is [latex]100[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{7.5}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write this ratio as a percent.<\/td>\n<td>[latex]7.5[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6638\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6638&theme=lumen&iframe_resize_id=ohm6638&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Fraction to Percents<\/h2>\n<p>The process of converting fractions to percents involves two main steps: changing the fraction into a decimal, and then transforming that decimal into a percent. Understanding that &#8216;percent&#8217; means &#8216;per hundred&#8217; simplifies the conversion, as you&#8217;re essentially finding out how many parts out of [latex]100[\/latex] the fraction represents.<\/p>\n<section class=\"textbox questionHelp\"><strong>How To: Convert a Fraction to a Percent<\/strong><\/p>\n<ol id=\"eip-id1168467345664\" class=\"stepwise\">\n<li><strong>Convert the Fraction to a Decimal<\/strong>: Divide the numerator (the top number of the fraction) by the denominator (the bottom number of the fraction) to get a decimal.<\/li>\n<li><strong>Multiply by [latex]100[\/latex] to Find the Percent<\/strong>: Take the decimal result and multiply it by [latex]100[\/latex]. Then, add the percent sign ([latex]\\%[\/latex]) to the product to express the answer as a percentage.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Convert each fraction or mixed number to a percent:<\/p>\n<ol>\n<li>[latex]{\\Large\\frac{3}{4}}[\/latex]<\/li>\n<li>[latex]{\\Large\\frac{11}{8}}[\/latex]<\/li>\n<li>[latex]2{\\Large\\frac{1}{5}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q589499\">Show Answer<\/button><\/p>\n<div id=\"q589499\" class=\"hidden-answer\" style=\"display: none\">To convert a fraction to a decimal, divide the numerator by the denominator.<\/p>\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Change to a decimal.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221904\/CNX_BMath_Figure_06_01_004_img-01.png\" alt=\"0.75 showing the decimal place moves to the right to spaces.\" width=\"41\" height=\"24\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write as a percent by moving the decimal two places.<\/td>\n<td>[latex]75[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Change to a decimal.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221905\/CNX_BMath_Figure_06_01_005_img-01.png\" alt=\"1.375 showing the decimal place moves to the right to spaces.\" width=\"43\" height=\"26\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write as a percent by moving the decimal two places.<\/td>\n<td>[latex]137.5[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Write as an improper fraction.<\/td>\n<td>[latex]{\\Large\\frac{11}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change to a decimal.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221905\/CNX_BMath_Figure_06_01_006_img-01.png\" alt=\"2.20 showing the decimal place moves to the right to spaces.\" width=\"45\" height=\"26\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write as a percent.<\/td>\n<td>[latex]220[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.<\/section>\n<p>Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6639\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6639&theme=lumen&iframe_resize_id=ohm6639&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2699"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":2,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2699\/revisions"}],"predecessor-version":[{"id":4441,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2699\/revisions\/4441"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2699\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2699"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2699"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2699"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2699"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}