{"id":47654,"date":"2025-07-02T14:37:36","date_gmt":"2025-07-02T14:37:36","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=47654"},"modified":"2025-07-02T14:37:38","modified_gmt":"2025-07-02T14:37:38","slug":"in-this-problem-we-shall-see-how-to-write-numbers-whose-decimal-is-repeating-as-fraction","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/in-this-problem-we-shall-see-how-to-write-numbers-whose-decimal-is-repeating-as-fraction\/","title":{"rendered":"In this problem, we shall see how to write numbers whose decimal is repeating as fraction"},"content":{"rendered":"\n

In this problem, we shall see how to write numbers whose decimal is repeating as fraction_ Suppose we have the number 0.123 0.123123123 We can write this number as 123 103 123 123 106 109 Express this sum using series notation_ Calculate the sum of the series and express the answer as simplified fraction:<\/p>\n\n\n\n

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

To express a repeating decimal like 0.123\u203e0.\\overline{123}0.123 (where the “123” repeats) as a fraction, we can use series notation. Here\u2019s the step-by-step breakdown:<\/p>\n\n\n\n

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

We can rewrite the decimal 0.123\u203e0.\\overline{123}0.123 as: 0.123123123\u2026=0.123+0.000123+0.000000123+\u20260.123123123\\ldots = 0.123 + 0.000123 + 0.000000123 + \\ldots0.123123123\u2026=0.123+0.000123+0.000000123+\u2026<\/p>\n\n\n\n

This is an infinite geometric series with the first term a=0.123a = 0.123a=0.123 and the common ratio r=10\u22123r = 10^{-3}r=10\u22123 (since each new term is 100010001000 times smaller than the previous term).<\/p>\n\n\n\n

So, the sum of this infinite series is given by the formula for the sum of a geometric series: S=a1\u2212rS = \\frac{a}{1 – r}S=1\u2212ra\u200b<\/p>\n\n\n\n

where a=0.123a = 0.123a=0.123 and r=10\u22123r = 10^{-3}r=10\u22123.<\/p>\n\n\n\n

Step 2: Calculate the sum of the series<\/h3>\n\n\n\n

Substitute a=0.123a = 0.123a=0.123 and r=10\u22123r = 10^{-3}r=10\u22123 into the formula: S=0.1231\u221210\u22123=0.1230.999S = \\frac{0.123}{1 – 10^{-3}} = \\frac{0.123}{0.999}S=1\u221210\u221230.123\u200b=0.9990.123\u200b<\/p>\n\n\n\n

Now simplify: S=123999S = \\frac{123}{999}S=999123\u200b<\/p>\n\n\n\n

Step 3: Simplify the fraction<\/h3>\n\n\n\n

To simplify 123999\\frac{123}{999}999123\u200b, we can find the greatest common divisor (GCD) of 123 and 999. Using the Euclidean algorithm:<\/p>\n\n\n\n

    \n
  • 999\u00f7123=8999 \\div 123 = 8999\u00f7123=8 with a remainder of 999\u22128\u00d7123=999\u2212984=15999 – 8 \\times 123 = 999 – 984 = 15999\u22128\u00d7123=999\u2212984=15<\/li>\n\n\n\n
  • 123\u00f715=8123 \\div 15 = 8123\u00f715=8 with a remainder of 123\u22128\u00d715=123\u2212120=3123 – 8 \\times 15 = 123 – 120 = 3123\u22128\u00d715=123\u2212120=3<\/li>\n\n\n\n
  • 15\u00f73=515 \\div 3 = 515\u00f73=5 with a remainder of 000<\/li>\n<\/ul>\n\n\n\n

    So, the GCD of 123 and 999 is 3. Now, divide both the numerator and the denominator of 123999\\frac{123}{999}999123\u200b by 3: 123999=123\u00f73999\u00f73=41333\\frac{123}{999} = \\frac{123 \\div 3}{999 \\div 3} = \\frac{41}{333}999123\u200b=999\u00f73123\u00f73\u200b=33341\u200b<\/p>\n\n\n\n

    Thus, the simplified fraction for 0.123\u203e0.\\overline{123}0.123 is 41333\\frac{41}{333}33341\u200b.<\/p>\n\n\n\n

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

    The repeating decimal 0.123\u203e0.\\overline{123}0.123 can be expressed as the simplified fraction: 41333\\frac{41}{333}33341\u200b<\/p>\n\n\n\n

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

    In this problem, we shall see how to write numbers whose decimal is repeating as fraction_ Suppose we have the number 0.123 0.123123123 We can write this number as 123 103 123 123 106 109 Express this sum using series notation_ Calculate the sum of the series and express the answer as simplified fraction: The […]<\/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-47654","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47654","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=47654"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47654\/revisions"}],"predecessor-version":[{"id":47656,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47654\/revisions\/47656"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=47654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=47654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=47654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}