{"id":179389,"date":"2024-12-30T19:25:26","date_gmt":"2024-12-30T19:25:26","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179389"},"modified":"2024-12-30T19:25:28","modified_gmt":"2024-12-30T19:25:28","slug":"alan-rewrites-a-fraction-less-than-1-as-a-decimal-the-numerator-is-a-whole-the-number-greater-than-0-for-which-denominator-will-the-fraction-always-convert-to-the-terminating-decimal","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/12\/30\/alan-rewrites-a-fraction-less-than-1-as-a-decimal-the-numerator-is-a-whole-the-number-greater-than-0-for-which-denominator-will-the-fraction-always-convert-to-the-terminating-decimal\/","title":{"rendered":"Alan rewrites a fraction less than 1 as a decimal. The numerator is a whole the number greater than 0 . For which denominator will the fraction always convert to the terminating decimal"},"content":{"rendered":"\n<p>Alan rewrites a fraction less than 1 as a decimal. The numerator is a whole the number greater than 0 . For which denominator will the fraction always convert to the terminating decimal?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine for which denominator a fraction will always convert to a terminating decimal, we must examine the relationship between the numerator and denominator of a fraction.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Definition of Terminating Decimal:<\/strong><\/h3>\n\n\n\n<p>A <strong>terminating decimal<\/strong> is a decimal number that has a finite number of digits after the decimal point. For example, 12=0.5\\frac{1}{2} = 0.5, 34=0.75\\frac{3}{4} = 0.75, and 78=0.875\\frac{7}{8} = 0.875 are terminating decimals.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Fraction Conversion to Decimal:<\/strong><\/h3>\n\n\n\n<p>To convert a fraction to a decimal, we divide the numerator by the denominator. Some fractions, when divided, result in decimals that either repeat (like 13=0.3333\u2026\\frac{1}{3} = 0.3333\\ldots) or terminate after a finite number of decimal places (like 12=0.5\\frac{1}{2} = 0.5).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Criteria for Terminating Decimal:<\/strong><\/h3>\n\n\n\n<p>A fraction ab\\frac{a}{b} will always convert to a terminating decimal if and only if the denominator bb (after simplifying the fraction) has no prime factors other than 2 or 5. In other words, the denominator must be of the form 2m\u00d75n2^m \\times 5^n, where mm and nn are non-negative integers.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Why does this work?<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The long division process works by repeatedly dividing the numerator by the denominator. For a fraction to have a terminating decimal, the division must end after a finite number of steps.<\/li>\n\n\n\n<li>If the denominator has prime factors other than 2 or 5, the division will continue indefinitely, leading to a repeating decimal. For example, 13\\frac{1}{3} gives a repeating decimal because 3 is not divisible by 2 or 5.<\/li>\n\n\n\n<li>If the denominator is a power of 2, a power of 5, or a combination of both (like 2, 4, 5, 8, 10, 25, 50, etc.), the division process will terminate.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Examples of Denominators that Work:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Denominators like 2, 4, 5, 8, 10, 20, 25, 50, 100, etc., will always convert to terminating decimals.\n<ul class=\"wp-block-list\">\n<li>For example, 18=0.125\\frac{1}{8} = 0.125 and 110=0.1\\frac{1}{10} = 0.1.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion:<\/strong><\/h3>\n\n\n\n<p>The denominator of a fraction must be a product of powers of 2 and 5 (i.e., 2m\u00d75n2^m \\times 5^n) for the fraction to convert to a terminating decimal. Thus, denominators like 2, 4, 5, 8, 10, 25, 50, 100, etc., will always result in a terminating decimal.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Alan rewrites a fraction less than 1 as a decimal. The numerator is a whole the number greater than 0 . For which denominator will the fraction always convert to the terminating decimal? The correct answer and explanation is: To determine for which denominator a fraction will always convert to a terminating decimal, we must [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-179389","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=179389"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179389\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}