{"id":4647,"date":"2023-06-19T13:28:23","date_gmt":"2023-06-19T13:28:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4647"},"modified":"2024-10-18T20:52:43","modified_gmt":"2024-10-18T20:52:43","slug":"general-problem-solving-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/general-problem-solving-background-youll-need-1\/","title":{"raw":"General Problem Solving: Background You'll Need 1","rendered":"General Problem Solving: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Simplify Fractions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Simplify Fractions<\/h2>\r\n<p>There are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we\u2019ll use the fraction that is in <em>simplified<\/em> form.<\/p>\r\n<p>A fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. A common factor is a number that divides both the numerator and the denominator of a fraction evenly, without leaving a remainder. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing these 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 if there are no common factors in the numerator and denominator.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.<\/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<\/p>\r\n<p>&nbsp;<\/p>\r\n<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><\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">Notice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Simplify 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<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{10}{15}[\/latex][reveal-answer q=\"431362\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"431362\"]To simplify the fraction, we look for any common factors in the numerator and the denominator.\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\">[ohm2_question hide_question_numbers=1]8295[\/ohm2_question]<\/section>\r\n<p>To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.<\/p>\r\n<section class=\"textbox example\">Simplify: [latex]\\Large-\\frac{18}{24}[\/latex][reveal-answer q=\"242151\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"242151\"]\r\n\r\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\r\n<tbody>\r\n<tr>\r\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\r\n<td>[latex]\\Large-\\frac{18}{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large-\\frac{3}{4}[\/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\">[ohm2_question hide_question_numbers=1]8296[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify Fractions<\/li>\n<\/ul>\n<\/section>\n<h2>Simplify Fractions<\/h2>\n<p>There are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we\u2019ll use the fraction that is in <em>simplified<\/em> form.<\/p>\n<p>A fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. A common factor is a number that divides both the numerator and the denominator of a fraction evenly, without leaving a remainder. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing these common factors.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Simplified Fraction<\/h3>\n<p>A fraction is considered simplified if there are no common factors in the numerator and denominator.<\/p>\n<\/div>\n<\/section>\n<p>The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.<\/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<p>&nbsp;<\/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 proTip\">Notice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Simplify 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<\/section>\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{10}{15}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q431362\">Show Answer<\/button><\/p>\n<div id=\"q431362\" class=\"hidden-answer\" style=\"display: none\">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\"><iframe loading=\"lazy\" id=\"ohm8295\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8295&theme=lumen&iframe_resize_id=ohm8295&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.<\/p>\n<section class=\"textbox example\">Simplify: [latex]\\Large-\\frac{18}{24}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q242151\">Show Answer<\/button><\/p>\n<div id=\"q242151\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\n<tbody>\n<tr>\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\n<td>[latex]\\Large-\\frac{18}{24}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large-\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8296\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8296&theme=lumen&iframe_resize_id=ohm8296&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":23,"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4647"}],"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\/16"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4647\/revisions"}],"predecessor-version":[{"id":12013,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4647\/revisions\/12013"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4647\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4647"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4647"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4647"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}