{"id":233276,"date":"2025-06-12T21:33:04","date_gmt":"2025-06-12T21:33:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233276"},"modified":"2025-06-12T21:33:06","modified_gmt":"2025-06-12T21:33:06","slug":"express-the-repeating-decimal-as-the-ratio-of-two-integers-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/express-the-repeating-decimal-as-the-ratio-of-two-integers-2\/","title":{"rendered":"Express the repeating decimal as the ratio of two integers."},"content":{"rendered":"\n

Express the repeating decimal as the ratio of two integers. 0.\\overline{5} = 0.5555\u2026 The ratio of two integers is (Type an integer or a simplified fraction<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To express the repeating decimal 0.5\u203e=0.5555\u20260.\\overline{5} = 0.5555\\ldots0.5=0.5555\u2026 as a ratio of two integers<\/strong>, we follow a standard algebraic method:<\/p>\n\n\n\n

\n\n\n\n

Step-by-step solution:<\/strong><\/h3>\n\n\n\n

Letx=0.5\u203ex = 0.\\overline{5}x=0.5<\/p>\n\n\n\n

This means:x=0.5555\u2026x = 0.5555\\ldotsx=0.5555\u2026<\/p>\n\n\n\n

Now multiply both sides of the equation by 10 to move the decimal point one place to the right:10x=5.5555\u202610x = 5.5555\\ldots10x=5.5555\u2026<\/p>\n\n\n\n

Now subtract the original equation (x=0.5555\u2026x = 0.5555\\ldotsx=0.5555\u2026) from this new equation:10x\u2212x=5.5555\u2026\u22120.5555\u202610x – x = 5.5555\\ldots – 0.5555\\ldots10x\u2212x=5.5555\u2026\u22120.5555\u20269x=59x = 59x=5<\/p>\n\n\n\n

Now solve for xxx:x=59x = \\frac{5}{9}x=95\u200b<\/p>\n\n\n\n

\n\n\n\n

Final Answer:<\/strong><\/h3>\n\n\n\n

59\\boxed{\\frac{5}{9}}95\u200b\u200b<\/p>\n\n\n\n

\n\n\n\n

Textbook-style Explanation<\/strong><\/h3>\n\n\n\n

Repeating decimals are decimal numbers in which a digit or group of digits repeat infinitely. A common problem in algebra is converting a repeating decimal into a rational number<\/strong>, which means expressing it as a ratio of two integers<\/strong> (a fraction in simplest form). In this case, we are given the repeating decimal 0.5\u203e0.\\overline{5}0.5, which is shorthand notation for the infinite decimal 0.5555\u20260.5555\\ldots0.5555\u2026, where the digit 5 repeats forever.<\/p>\n\n\n\n

To convert this repeating decimal to a fraction, we use a simple algebraic trick. First, we assign a variable to the repeating decimal. Let x=0.5555\u2026x = 0.5555\\ldotsx=0.5555\u2026. To eliminate the repeating part, we multiply both sides of the equation by 10. This moves the decimal one place to the right and gives us 10x=5.5555\u202610x = 5.5555\\ldots10x=5.5555\u2026.<\/p>\n\n\n\n

Now, notice that both xxx and 10x10x10x have the same repeating decimal part 0.5555\u20260.5555\\ldots0.5555\u2026. If we subtract the original equation from the new one, the repeating parts cancel out:10x\u2212x=5.5555\u2026\u22120.5555\u2026\u21d29x=510x – x = 5.5555\\ldots – 0.5555\\ldots \\Rightarrow 9x = 510x\u2212x=5.5555\u2026\u22120.5555\u2026\u21d29x=5<\/p>\n\n\n\n

Solving this simple equation for xxx, we divide both sides by 9:x=59x = \\frac{5}{9}x=95\u200b<\/p>\n\n\n\n

Thus, 0.5\u203e=590.\\overline{5} = \\frac{5}{9}0.5=95\u200b. This shows that the repeating decimal is a rational number. All repeating decimals can be expressed in this way using algebra, and this method provides a reliable way to convert them to exact fractions.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Express the repeating decimal as the ratio of two integers. 0.\\overline{5} = 0.5555\u2026 The ratio of two integers is (Type an integer or a simplified fraction The Correct Answer and Explanation is: To express the repeating decimal 0.5\u203e=0.5555\u20260.\\overline{5} = 0.5555\\ldots0.5=0.5555\u2026 as a ratio of two integers, we follow a standard algebraic method: Step-by-step solution: Letx=0.5\u203ex […]<\/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-233276","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233276","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=233276"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233276\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}