Rewrite as a simplified fraction. 0.67 where 7 repeats forever<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To convert the repeating decimal (0.67\\overline{7}) into a simplified fraction, we can follow a systematic approach. The notation (0.67\\overline{7}) means that the digits “67” are followed by an infinite series of the digit “7” (i.e., 0.677777\u2026).<\/p>\n\n\n\n Let (x = 0.67\\overline{7}).<\/p>\n\n\n\n To eliminate the repeating part, we can multiply (x) by a power of 10 that moves the decimal point to the right, covering one full cycle of the repeating digits. Since the “7” repeats after two digits (“67”), we multiply by (1000) (two for “67” and one for the repeat):<\/p>\n\n\n\n [ Next, we can create a second equation without the repeating part. Multiply (x) by (10) to cover just the “67”:<\/p>\n\n\n\n [ Now, we subtract the second equation from the first to eliminate the repeating part:<\/p>\n\n\n\n [ This simplifies to:<\/p>\n\n\n\n [ Next, divide both sides by (990):<\/p>\n\n\n\n [ To simplify (\\frac{670}{990}), we can find the greatest common divisor (GCD) of (670) and (990). The prime factorization shows:<\/p>\n\n\n\n The common factors are (2) and (5). The GCD is (10). Now, we divide both the numerator and the denominator by (10):<\/p>\n\n\n\n [ Thus, the simplified fraction of the repeating decimal (0.67\\overline{7}) is:<\/p>\n\n\n\n [ This fraction is now in its simplest form, representing the original repeating decimal accurately.<\/p>\n","protected":false},"excerpt":{"rendered":" Rewrite as a simplified fraction. 0.67 where 7 repeats forever The Correct Answer and Explanation is: To convert the repeating decimal (0.67\\overline{7}) into a simplified fraction, we can follow a systematic approach. The notation (0.67\\overline{7}) means that the digits “67” are followed by an infinite series of the digit “7” (i.e., 0.677777\u2026). Step 1: Define […]<\/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-155803","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/155803","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=155803"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/155803\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=155803"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=155803"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=155803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Define the Decimal as a Variable<\/h3>\n\n\n\n
Step 2: Eliminate the Repeating Part<\/h3>\n\n\n\n
1000x = 677.7777\\overline{7}
]<\/p>\n\n\n\nStep 3: Set Up a System of Equations<\/h3>\n\n\n\n
10x = 6.7777\\overline{7}
]<\/p>\n\n\n\nStep 4: Subtract the Two Equations<\/h3>\n\n\n\n
1000x – 10x = 677.7777\\overline{7} – 6.7777\\overline{7}
]<\/p>\n\n\n\n
990x = 670
]<\/p>\n\n\n\nStep 5: Solve for (x)<\/h3>\n\n\n\n
x = \\frac{670}{990}
]<\/p>\n\n\n\nStep 6: Simplify the Fraction<\/h3>\n\n\n\n
\n
\\frac{670 \\div 10}{990 \\div 10} = \\frac{67}{99}
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
\\boxed{\\frac{67}{99}}
]<\/p>\n\n\n\n