How do you write $0.\\overline{09}$ as a fraction?<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Sure! Let’s convert the repeating decimal 0.09\u203e0.\\overline{09} into a fraction step-by-step.<\/p>\n\n\n\n Let x=0.09090909\u2026x = 0.09090909\\ldots<\/p>\n\n\n\n where “09” repeats indefinitely.<\/p>\n\n\n\n Since the repeating block has 2 digits (“09”), multiply both sides by 100100: 100x=9.09090909\u2026100x = 9.09090909\\ldots<\/p>\n\n\n\n Now subtract the original xx from this new equation: 100x\u2212x=9.09090909\u2026\u22120.09090909\u2026100x – x = 9.09090909\\ldots – 0.09090909\\ldots 99x=999x = 9<\/p>\n\n\n\n x=999=111x = \\frac{9}{99} = \\frac{1}{11}<\/p>\n\n\n\n 0.09\u203e=1110.\\overline{09} = \\frac{1}{11}<\/p>\n\n\n\n Repeating decimals are decimals in which a digit or group of digits repeats infinitely. Here, the decimal 0.09\u203e0.\\overline{09} means the two digits “09” repeat forever after the decimal point.<\/p>\n\n\n\n To convert a repeating decimal into a fraction, we use algebraic manipulation to isolate the repeating part.<\/p>\n\n\n\n First, we assign the decimal to a variable xx. In this case, x=0.090909\u2026x = 0.090909\\ldots<\/p>\n\n\n\n Next, because the repeating part has two digits (“09”), we multiply xx by 100100 (which shifts the decimal point two places to the right): 100x=9.090909\u2026100x = 9.090909\\ldots<\/p>\n\n\n\n The key idea is to subtract the original decimal from this equation to eliminate the repeating part: 100x\u2212x=9.090909\u2026\u22120.090909\u2026100x – x = 9.090909\\ldots – 0.090909\\ldots<\/p>\n\n\n\n On the right side, the infinite repeating parts cancel out, leaving: 99x=999x = 9<\/p>\n\n\n\n This equation is simpler to solve for xx: x=999x = \\frac{9}{99}<\/p>\n\n\n\n By dividing numerator and denominator by their greatest common divisor (9), we simplify this to: x=111x = \\frac{1}{11}<\/p>\n\n\n\n Thus, the repeating decimal 0.09\u203e0.\\overline{09} is exactly equal to the fraction 111\\frac{1}{11}.<\/p>\n\n\n\n This method works generally for any repeating decimal. The number of digits in the repeating block tells us what power of 10 to multiply by in order to shift the decimal point correctly and eliminate the repetition via subtraction. This algebraic technique turns a seemingly complex infinite decimal into a neat fraction!<\/p>\n","protected":false},"excerpt":{"rendered":" How do you write $0.\\overline{09}$ as a fraction? The correct answer and explanation is: Sure! Let’s convert the repeating decimal 0.09\u203e0.\\overline{09} into a fraction step-by-step. Step 1: Define the repeating decimal as a variable Let x=0.09090909\u2026x = 0.09090909\\ldots where “09” repeats indefinitely. Step 2: Multiply xx to shift the decimal point to the right, past […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-20453","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20453","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=20453"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20453\/revisions"}],"predecessor-version":[{"id":20454,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20453\/revisions\/20454"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=20453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=20453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=20453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Define the repeating decimal as a variable<\/h3>\n\n\n\n
\n\n\n\nStep 2: Multiply xx to shift the decimal point to the right, past one full repeating block<\/h3>\n\n\n\n
\n\n\n\nStep 3: Set up an equation to eliminate the repeating decimal<\/h3>\n\n\n\n
\n\n\n\nStep 4: Solve for xx<\/h3>\n\n\n\n
\n\n\n\nFinal answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation (300 words):<\/h3>\n\n\n\n