{"id":223638,"date":"2025-06-02T06:48:22","date_gmt":"2025-06-02T06:48:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=223638"},"modified":"2025-06-02T06:48:24","modified_gmt":"2025-06-02T06:48:24","slug":"is-7-x-1-then-find-the-decimal-expansion-of-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/02\/is-7-x-1-then-find-the-decimal-expansion-of-x\/","title":{"rendered":"Is 7 x = 1 then find the decimal expansion of x"},"content":{"rendered":"\n

Is 7 x = 1 then find the decimal expansion of x<\/p>\n\n\n\n

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

To solve the equation:<\/p>\n\n\n\n

7x = 1<\/strong><\/p>\n\n\n\n

We divide both sides by 7: x=17x = \\frac{1}{7}<\/p>\n\n\n\n

Now, let\u2019s find the decimal expansion<\/strong> of 17\\frac{1}{7}: 17=0.142857\u203e\\frac{1}{7} = 0.\\overline{142857}<\/p>\n\n\n\n

This means the decimal repeats every 6 digits: 0.142857142857142857\u2026<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation<\/h3>\n\n\n\n

To understand why 17=0.142857\u203e\\frac{1}{7} = 0.\\overline{142857}, we can use long division<\/strong> and explore the nature of repeating decimals<\/strong>.<\/p>\n\n\n\n

First, consider what dividing 1 by 7 means: you are distributing 1 equally among 7 parts. Since 1 is smaller than 7, the result will be less than 1<\/strong>, i.e., a decimal number.<\/p>\n\n\n\n

Using long division:<\/p>\n\n\n\n

    \n
  • 7 goes into 10 once (1 \u00d7 7 = 7), remainder 3 \u2192 write down 1<\/strong><\/li>\n\n\n\n
  • Bring down a 0 \u2192 7 into 30 = 4 times (4 \u00d7 7 = 28), remainder 2 \u2192 write 4<\/strong><\/li>\n\n\n\n
  • Bring down a 0 \u2192 7 into 20 = 2 times (2 \u00d7 7 = 14), remainder 6 \u2192 write 2<\/strong><\/li>\n\n\n\n
  • 7 into 60 = 8 (8 \u00d7 7 = 56), remainder 4 \u2192 write 8<\/strong><\/li>\n\n\n\n
  • 7 into 40 = 5 (5 \u00d7 7 = 35), remainder 5 \u2192 write 5<\/strong><\/li>\n\n\n\n
  • 7 into 50 = 7 (7 \u00d7 7 = 49), remainder 1 \u2192 write 7<\/strong><\/li>\n<\/ul>\n\n\n\n

    Now we\u2019re back to the original situation: remainder 1, meaning the process repeats.<\/p>\n\n\n\n

    So, 17\\frac{1}{7} has a repeating cycle of six digits<\/strong>: 142857<\/strong>.<\/p>\n\n\n\n

    This is an example of a repeating decimal<\/strong>, which occurs whenever the denominator of a fraction (in lowest terms) has prime factors other than 2 or 5. Since 7 is a prime other than 2 or 5, 17\\frac{1}{7} cannot terminate and must repeat.<\/p>\n\n\n\n

    Interestingly, 142857<\/strong> is a cyclic number. Multiplying it by numbers 1 to 6 produces cyclic permutations of itself:<\/p>\n\n\n\n

      \n
    • 1 \u00d7 142857 = 142857<\/li>\n\n\n\n
    • 2 \u00d7 142857 = 285714<\/li>\n\n\n\n
    • 3 \u00d7 142857 = 428571<\/li>\n\n\n\n
    • \u2026and so on.<\/li>\n<\/ul>\n\n\n\n

      Thus, the decimal expansion of xx in the equation 7x=17x = 1 is: x=0.142857\u203e\\boxed{x = 0.\\overline{142857}}<\/p>\n\n\n\n

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

      Is 7 x = 1 then find the decimal expansion of x The Correct Answer and Explanation is: To solve the equation: 7x = 1 We divide both sides by 7: x=17x = \\frac{1}{7} Now, let\u2019s find the decimal expansion of 17\\frac{1}{7}: 17=0.142857\u203e\\frac{1}{7} = 0.\\overline{142857} This means the decimal repeats every 6 digits: 0.142857142857142857\u2026 Explanation […]<\/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-223638","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223638","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=223638"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223638\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=223638"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=223638"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=223638"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}