To express a repeating decimal as the ratio of two integers, we can follow a systematic approach using algebra.<\/p>\n\n\n\n
Let’s use the example of the repeating decimal 0.ab\u203e0.\\overline{ab}0.ab, where “ab” represents the repeating part (two digits). For the sake of explanation, let\u2019s consider a concrete example: 0.15\u203e0.\\overline{15}0.15, which means 0.151515…0.151515…0.151515….<\/p>\n\n\n\n
Step 1: Set up the equation<\/h3>\n\n\n\n
Let x=0.15\u203ex = 0.\\overline{15}x=0.15. This means x=0.151515…x = 0.151515…x=0.151515….<\/p>\n\n\n\n
Step 2: Eliminate the repeating decimal<\/h3>\n\n\n\n
To eliminate the repeating part, multiply both sides of the equation by 100 (because there are two digits in the repeating part). This gives:100x=15.151515…100x = 15.151515…100x=15.151515…<\/p>\n\n\n\n
Now you have the original repeating decimal part (0.151515…) on both sides of the equation.<\/p>\n\n\n\n
Step 3: Subtract the original equation from the new equation<\/h3>\n\n\n\n
Subtract the original equation x=0.151515…x = 0.151515…x=0.151515… from 100x=15.151515…100x = 15.151515…100x=15.151515…. This gives:100x\u2212x=15.151515…\u22120.151515…100x – x = 15.151515… – 0.151515…100x\u2212x=15.151515…\u22120.151515…<\/p>\n\n\n\n
Simplifying both sides:99x=1599x = 1599x=15<\/p>\n\n\n\n
Step 4: Solve for xxx<\/h3>\n\n\n\n
Now, solve for xxx:x=1599x = \\frac{15}{99}x=9915\u200b<\/p>\n\n\n\n
Step 5: Simplify the fraction<\/h3>\n\n\n\n
Simplify the fraction 1599\\frac{15}{99}9915\u200b. The greatest common divisor (GCD) of 15 and 99 is 3, so divide both the numerator and the denominator by 3:x=15\u00f7399\u00f73=533x = \\frac{15 \\div 3}{99 \\div 3} = \\frac{5}{33}x=99\u00f7315\u00f73\u200b=335\u200b<\/p>\n\n\n\n
Final Answer<\/h3>\n\n\n\n
Thus, the repeating decimal 0.15\u203e0.\\overline{15}0.15 can be expressed as the fraction 533\\frac{5}{33}335\u200b.<\/p>\n\n\n\n
This method works for any repeating decimal. The key is to multiply both sides of the equation by a power of 10 that matches the number of repeating digits, subtract the two equations, and then simplify the resulting fraction.<\/p>\n\n\n\n Express the repeating decimal as the ratio of two integers.The ratio of two integers is. (Type an integer or a simplified fraction. The Correct Answer and Explanation is: To express a repeating decimal as the ratio of two integers, we can follow a systematic approach using algebra. Let’s use the example of the repeating decimal […]<\/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-45387","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45387","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=45387"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45387\/revisions"}],"predecessor-version":[{"id":45389,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45387\/revisions\/45389"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=45387"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=45387"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=45387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-25.jpeg\")
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