{"id":43180,"date":"2025-06-29T14:31:28","date_gmt":"2025-06-29T14:31:28","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=43180"},"modified":"2025-06-29T14:31:29","modified_gmt":"2025-06-29T14:31:29","slug":"write-the-repeating-decimal-first-as-a-geometric-series-and-then-as-a-fraction","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/write-the-repeating-decimal-first-as-a-geometric-series-and-then-as-a-fraction\/","title":{"rendered":"Write the repeating decimal first as a geometric series and then as a fraction"},"content":{"rendered":"\n

Write the repeating decimal first as a geometric series and then as a fraction (a ratio of two integers) 3.8549 = 3.85494949_ 0A 3.8549 = 0.0049(0.01)k k=0 B. 3.8549 = 3.85 + 2 0.00490.01)k k=0 0 C 3.8549 = 0.0049 + 2 3.85(0.001)k k=0 0 D: 3.8549 = 3.85(0.1)k k=0 Write the repeating decimal as a fraction (a ratio of two integers) 3.8549 (Type an integer or a simplified fraction:)<\/p>\n\n\n\n

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

To solve this problem, we first need to represent the repeating decimal 3.85494949…<\/strong> as a geometric series. The repeating part is “49”, so this part repeats every two decimal places. Here’s how we can break it down:<\/p>\n\n\n\n

Step 1: Express the repeating decimal as a geometric series<\/h3>\n\n\n\n

Let\u2019s first separate the non-repeating part from the repeating part.<\/p>\n\n\n\n

We have:3.85494949…=3.85+0.00494949…3.85494949… = 3.85 + 0.00494949…3.85494949…=3.85+0.00494949…<\/p>\n\n\n\n

Now, focus on the repeating part 0.00494949…0.00494949…0.00494949…. This is a repeating decimal, which can be written as a geometric series:0.00494949…=0.0049\u22c5(1+0.01+0.012+0.013+\u2026\u2009)0.00494949… = 0.0049 \\cdot (1 + 0.01 + 0.01^2 + 0.01^3 + \\dots)0.00494949…=0.0049\u22c5(1+0.01+0.012+0.013+\u2026)<\/p>\n\n\n\n

This is a geometric series with:<\/p>\n\n\n\n

    \n
  • First term a=0.0049a = 0.0049a=0.0049<\/li>\n\n\n\n
  • Common ratio r=0.01r = 0.01r=0.01<\/li>\n<\/ul>\n\n\n\n

    So, we can express it as:0.00494949…=0.0049\u2211k=0\u221e(0.01)k0.00494949… = 0.0049 \\sum_{k=0}^{\\infty} (0.01)^k0.00494949…=0.0049k=0\u2211\u221e\u200b(0.01)k<\/p>\n\n\n\n

    This series sums up to:S=a1\u2212r=0.00491\u22120.01=0.00490.99=499900S = \\frac{a}{1 – r} = \\frac{0.0049}{1 – 0.01} = \\frac{0.0049}{0.99} = \\frac{49}{9900}S=1\u2212ra\u200b=1\u22120.010.0049\u200b=0.990.0049\u200b=990049\u200b<\/p>\n\n\n\n

    Step 2: Combine the non-repeating part and the sum of the geometric series<\/h3>\n\n\n\n

    Now, we add the non-repeating part 3.853.853.85 and the sum we just calculated:3.85494949…=3.85+4999003.85494949… = 3.85 + \\frac{49}{9900}3.85494949…=3.85+990049\u200b<\/p>\n\n\n\n

    To combine these, express 3.853.853.85 as a fraction:3.85=3851003.85 = \\frac{385}{100}3.85=100385\u200b<\/p>\n\n\n\n

    Thus, the total expression becomes:3.85494949…=385100+4999003.85494949… = \\frac{385}{100} + \\frac{49}{9900}3.85494949…=100385\u200b+990049\u200b<\/p>\n\n\n\n

    Now, we need a common denominator to combine the fractions. The least common denominator (LCD) is 9900:385100=385\u00d799100\u00d799=381159900\\frac{385}{100} = \\frac{385 \\times 99}{100 \\times 99} = \\frac{38115}{9900}100385\u200b=100\u00d799385\u00d799\u200b=990038115\u200b<\/p>\n\n\n\n

    Now, add the fractions:381159900+499900=38115+499900=381649900\\frac{38115}{9900} + \\frac{49}{9900} = \\frac{38115 + 49}{9900} = \\frac{38164}{9900}990038115\u200b+990049\u200b=990038115+49\u200b=990038164\u200b<\/p>\n\n\n\n

    Thus, the repeating decimal 3.85494949…<\/strong> as a fraction is:381649900\\frac{38164}{9900}990038164\u200b<\/p>\n\n\n\n

    Final Answer:<\/h3>\n\n\n\n

    The correct geometric series representation is:3.85494949…=3.85+\u2211k=0\u221e0.0049\u22c5(0.01)k3.85494949… = 3.85 + \\sum_{k=0}^{\\infty} 0.0049 \\cdot (0.01)^k3.85494949…=3.85+k=0\u2211\u221e\u200b0.0049\u22c5(0.01)k<\/p>\n\n\n\n

    And as a fraction, the repeating decimal 3.8549<\/strong> is:381649900\\frac{38164}{9900}990038164\u200b<\/p>\n\n\n\n

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

    Write the repeating decimal first as a geometric series and then as a fraction (a ratio of two integers) 3.8549 = 3.85494949_ 0A 3.8549 = 0.0049(0.01)k k=0 B. 3.8549 = 3.85 + 2 0.00490.01)k k=0 0 C 3.8549 = 0.0049 + 2 3.85(0.001)k k=0 0 D: 3.8549 = 3.85(0.1)k k=0 Write the repeating decimal as […]<\/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-43180","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43180","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=43180"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43180\/revisions"}],"predecessor-version":[{"id":43182,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43180\/revisions\/43182"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=43180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=43180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=43180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}