Can the fraction 84\/250 be written as a finite decimal? Explain why or why not without dividing.<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n The fraction ( \\frac{84}{250} ) can indeed be written as a finite decimal. To understand why, we need to examine the denominator’s prime factorization and apply a rule regarding fractions that have finite decimal representations.<\/p>\n\n\n\n To check if a fraction can be written as a finite decimal, we start by factoring the denominator into its prime factors. Let’s break down 250:<\/p>\n\n\n\n [ The key here is that a fraction will have a finite decimal representation if and only if the denominator, when fully simplified, consists only of the prime factors 2 and\/or 5. This is because the decimal system is based on powers of 10 (which is ( 2 \\times 5 )), so if the denominator has only these factors, the fraction can be written as a terminating (finite) decimal.<\/p>\n\n\n\n Next, we can look at the numerator, 84, to see if it simplifies with the denominator:<\/p>\n\n\n\n [ Now we simplify ( \\frac{84}{250} ) by canceling out any common factors in the numerator and the denominator. We observe that both 84 and 250 share a factor of 2:<\/p>\n\n\n\n [ Now we check the denominator of 125:<\/p>\n\n\n\n [ Since 125 is composed entirely of the prime factor 5, the fraction ( \\frac{42}{125} ) has a denominator with only the prime factor 5, which means it can be written as a finite decimal.<\/p>\n\n\n\n Since the denominator of the simplified fraction ( \\frac{42}{125} ) consists only of the prime factor 5, the fraction can indeed be written as a finite decimal.<\/p>\n\n\n\n [ The fraction ( \\frac{84}{250} ) can be written as the finite decimal 0.168 because its denominator, when simplified, only contains the prime factor 5, satisfying the condition for finite decimals.<\/p>\n","protected":false},"excerpt":{"rendered":" Can the fraction 84\/250 be written as a finite decimal? Explain why or why not without dividing. The correct answer and explanation is : The fraction ( \\frac{84}{250} ) can indeed be written as a finite decimal. To understand why, we need to examine the denominator’s prime factorization and apply a rule regarding fractions that […]<\/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-209420","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209420","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=209420"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209420\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209420"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209420"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Prime Factorization of the Denominator<\/h3>\n\n\n\n
250 = 2 \\times 5^3
]<\/p>\n\n\n\nStep 2: Prime Factorization of the Numerator<\/h3>\n\n\n\n
84 = 2^2 \\times 3 \\times 7
]<\/p>\n\n\n\nStep 3: Simplifying the Fraction<\/h3>\n\n\n\n
\\frac{84}{250} = \\frac{84 \\div 2}{250 \\div 2} = \\frac{42}{125}
]<\/p>\n\n\n\n
125 = 5^3
]<\/p>\n\n\n\nStep 4: Conclusion<\/h3>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
\\frac{84}{250} = 0.168
]<\/p>\n\n\n\n