{"id":8318,"date":"2023-09-29T14:37:48","date_gmt":"2023-09-29T14:37:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8318"},"modified":"2024-10-18T21:00:17","modified_gmt":"2024-10-18T21:00:17","slug":"math-in-arts-common-scenarios-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/math-in-arts-common-scenarios-background-youll-need-2\/","title":{"raw":"Math in Arts - Common Scenarios: Background You'll Need 2","rendered":"Math in Arts &#8211; Common Scenarios: Background You&#8217;ll Need 2"},"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>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 [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].<\/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\nTo 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] [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\r\n[\/hidden-answer]<\/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>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 [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].<\/p>\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\"><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":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":8095,"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\/8318"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8318\/revisions"}],"predecessor-version":[{"id":14898,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8318\/revisions\/14898"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/8095"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8318\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8318"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8318"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8318"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}