{"id":8205,"date":"2023-09-25T17:26:17","date_gmt":"2023-09-25T17:26:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8205"},"modified":"2024-10-18T20:56:37","modified_gmt":"2024-10-18T20:56:37","slug":"classify-and-convert-rational-numbers-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/classify-and-convert-rational-numbers-learn-it-2\/","title":{"raw":"Classify and Convert Rational Numbers: Learn It 2","rendered":"Classify and Convert Rational Numbers: Learn It 2"},"content":{"raw":"<h2>Simplify Fractions<\/h2>\r\n<p>A fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>simplified fraction<\/h3>\r\n<p>A fraction is considered simplified, or reduced, if there are no common factors in the numerator and denominator.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>For example,<\/p>\r\n<ul id=\"fs-id1302300\">\r\n\t<li>[latex]\\Large\\frac{2}{3}[\/latex] is simplified because there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\r\n\t<li>[latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\r\n<\/ul>\r\n<p>The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can use the Equivalent Fractions Property in reverse to simplify fractions.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>equivalent fractions property<\/h3>\r\n<p>If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then <center>[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/center><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Simplify\/Reduce a Fraction<\/strong><\/p>\r\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\r\n\t<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\r\n\t<li>Simplify, using the equivalent fractions property, by removing common factors.<\/li>\r\n\t<li>Multiply any remaining factors.<\/li>\r\n<\/ol>\r\n<p><em>Note: To simplify a negative fraction, we use the same process as above. Remember to keep the negative sign<\/em><\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.<\/p>\r\n<\/section>\r\n<p>Let's simplify the fraction we saw earlier.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Simplify: [latex]\\Large\\frac{10}{15}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"160936\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160936\"]<\/p>\r\n\r\n\r\nTo simplify the fraction, we look for any common factors in the numerator and the denominator.\r\n\r\n\r\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{10}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the numerator and denominator.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factors.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{2}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]12634[\/ohm2_question]<\/p>\r\n<\/section>\r\n<p>Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>To identify the prime factors of a number using the factor tree method, start by dividing the given number into two factors. Continue breaking down each factor into smaller factors until you're left with only prime numbers. These prime numbers, found at the \"leaves\" of your factor tree, are the prime factors of the original number. Remember a prime number is a whole number greater than [latex]1[\/latex] that can only be divided by [latex]1[\/latex] and itself without a remainder.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{210}{385}[\/latex] [reveal-answer q=\"721590\"]Show Solution[\/reveal-answer] [hidden-answer a=\"721590\"]\r\n\r\n\r\n<table id=\"eip-id1168467251049\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The fraction 210 over 385 is shown. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Use factor trees to factor the numerator and denominator.<\/td>\r\n<td>[latex] \\Large\\frac{210}{385}[\/latex] <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220913\/CNX_BMath_Figure_04_02_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator as the product of the primes.<\/td>\r\n<td>[latex]{\\Large\\frac{210}{385}}={\\Large\\frac{2\\cdot 3\\cdot 5\\cdot 7}{5\\cdot 7\\cdot 11}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factors.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot 3\\cdot\\color{blue}{5}\\cdot\\color{red}{7}}{\\color{blue}{5}\\cdot\\color{red}{7}\\color{black}\\cdot 11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{2\\cdot 3}{11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply any remaining factors.<\/td>\r\n<td>[latex]\\Large\\frac{6}{11}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]12635[\/ohm2_question]<\/p>\r\n<\/section>\r\n<p>We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.<\/p>\r\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{5xy}{15x}[\/latex] [reveal-answer q=\"283612\"]Show Solution[\/reveal-answer] [hidden-answer a=\"283612\"]\r\n\r\n\r\n<table id=\"eip-id1168469401986\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td style=\"width:30%\">[latex]\\Large\\frac{5xy}{15x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite numerator and denominator showing common factors.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot x\\cdot y}{3\\cdot 5\\cdot x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{5}\\cdot \\color{red}{x}\\cdot \\color{black}{y}}{3\\cdot \\color{red}{5}\\cdot \\color{red}{x}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{y}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n[\/hidden-answer]<\/section>","rendered":"<h2>Simplify Fractions<\/h2>\n<p>A fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>simplified fraction<\/h3>\n<p>A fraction is considered simplified, or reduced, if there are no common factors in the numerator and denominator.<\/p>\n<\/div>\n<\/section>\n<p>For example,<\/p>\n<ul id=\"fs-id1302300\">\n<li>[latex]\\Large\\frac{2}{3}[\/latex] is simplified because there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\n<li>[latex]\\Large\\frac{10}{15}[\/latex] is not simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\n<\/ul>\n<p>The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can use the Equivalent Fractions Property in reverse to simplify fractions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>equivalent fractions property<\/h3>\n<p>If [latex]a,b,c[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then <\/p>\n<div style=\"text-align: center;\">[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Simplify\/Reduce a Fraction<\/strong><\/p>\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\n<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\n<li>Simplify, using the equivalent fractions property, by removing common factors.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<p><em>Note: To simplify a negative fraction, we use the same process as above. Remember to keep the negative sign<\/em><\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.<\/p>\n<\/section>\n<p>Let&#8217;s simplify the fraction we saw earlier.<\/p>\n<section class=\"textbox example\">\n<p>Simplify: [latex]\\Large\\frac{10}{15}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160936\">Show Solution<\/button> <\/p>\n<div id=\"q160936\" class=\"hidden-answer\" style=\"display: none\">\n<p>To simplify the fraction, we look for any common factors in the numerator and the denominator.<\/p>\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\n<td>[latex]\\Large\\frac{10}{15}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the numerator and denominator.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factors.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm12634\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12634&theme=lumen&iframe_resize_id=ohm12634&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.<\/p>\n<section class=\"textbox recall\">\n<p>To identify the prime factors of a number using the factor tree method, start by dividing the given number into two factors. Continue breaking down each factor into smaller factors until you&#8217;re left with only prime numbers. These prime numbers, found at the &#8220;leaves&#8221; of your factor tree, are the prime factors of the original number. Remember a prime number is a whole number greater than [latex]1[\/latex] that can only be divided by [latex]1[\/latex] and itself without a remainder.<\/p>\n<\/section>\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{210}{385}[\/latex] <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q721590\">Show Solution<\/button> <\/p>\n<div id=\"q721590\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467251049\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The fraction 210 over 385 is shown. The next line says,\">\n<tbody>\n<tr>\n<td>Use factor trees to factor the numerator and denominator.<\/td>\n<td>[latex]\\Large\\frac{210}{385}[\/latex] <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220913\/CNX_BMath_Figure_04_02_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator as the product of the primes.<\/td>\n<td>[latex]{\\Large\\frac{210}{385}}={\\Large\\frac{2\\cdot 3\\cdot 5\\cdot 7}{5\\cdot 7\\cdot 11}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factors.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot 3\\cdot\\color{blue}{5}\\cdot\\color{red}{7}}{\\color{blue}{5}\\cdot\\color{red}{7}\\color{black}\\cdot 11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{2\\cdot 3}{11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply any remaining factors.<\/td>\n<td>[latex]\\Large\\frac{6}{11}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm12635\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12635&theme=lumen&iframe_resize_id=ohm12635&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.<\/p>\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{5xy}{15x}[\/latex] <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q283612\">Show Solution<\/button> <\/p>\n<div id=\"q283612\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469401986\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td style=\"width:30%\">[latex]\\Large\\frac{5xy}{15x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite numerator and denominator showing common factors.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot x\\cdot y}{3\\cdot 5\\cdot x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{5}\\cdot \\color{red}{x}\\cdot \\color{black}{y}}{3\\cdot \\color{red}{5}\\cdot \\color{red}{x}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{y}{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":53,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction"}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8205"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":24,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8205\/revisions"}],"predecessor-version":[{"id":14132,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8205\/revisions\/14132"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/53"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8205\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8205"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8205"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8205"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}