{"id":1579,"date":"2024-06-19T12:00:58","date_gmt":"2024-06-19T19:00:58","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=1579"},"modified":"2020-04-09T20:22:46","modified_gmt":"2020-04-10T03:22:46","slug":"gmat-math-terminating-and-repeating-decimals","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-math-terminating-and-repeating-decimals\/","title":{"rendered":"GMAT Math: Terminating and Repeating Decimals"},"content":{"rendered":"<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2020\/03\/terminating-and-repeating-decimals.png\" alt=\"Endless colorful lanterns on string representing terminating and repeating decimals\" width=\"1200\" height=\"600\" class=\"aligncenter size-full wp-image-9322\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2020\/03\/terminating-and-repeating-decimals.png 1200w, https:\/\/magoosh.com\/gmat\/files\/2020\/03\/terminating-and-repeating-decimals-300x150.png 300w, https:\/\/magoosh.com\/gmat\/files\/2020\/03\/terminating-and-repeating-decimals-600x300.png 600w, https:\/\/magoosh.com\/gmat\/files\/2020\/03\/terminating-and-repeating-decimals-768x384.png 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>This post was updated in 2024 for the new GMAT.<\/strong><\/p>\n<p>You might not feel decimals are the most exciting thing in the world, but just look at our friend <strong><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_8edb2cf68079344a2edd739531259f6c.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"pi\" title=\"pi\"\/><\/strong>, a decimal with its own <a href=\"https:\/\/www.piday.org\/\" rel=\"noopener noreferrer\" target=\"_blank\">holiday<\/a>! In this edition of GMAT Math, we&#8217;re going over two specific types of decimals: terminating and repeating decimals.\u00a0 Learn about how to identify and solve questions with these decimals. Don&#8217;t forget to test your understanding with the practice questions at the end!<\/p>\n<h2>Rational Numbers<\/h2>\n<p>Integers are positive and negative whole numbers, including zero.\u00a0 Here are the integers:<\/p>\n<p align=\"center\">{ \u2026 -3, -2, -1, 0, 1, 2, 3, \u2026}<\/p>\n<p>When we take a <span style=\"text-decoration: underline\">ratio<\/span> of two integers, we get a <span style=\"text-decoration: underline\">ratio<\/span>nal number.\u00a0 <\/p>\n<ul class=\"no_bullet\">\n<li class=\"bulb\">A rational number is any number of the form a\/b, where a &amp; b are integers, and b \u2260 0.<\/li>\n<li class=\"bulb\">Rational numbers are the set of all <a href=\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\">fractions<\/a> made with integer ingredients.\u00a0\u00a0 Notice that all integers are included in the set of rational numbers, because, for example, 3\/1 = 3.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h2>Rational Numbers as Decimals<\/h2>\n<p>When we make a decimal out of a fraction, one of two things happens.\u00a0 It either terminates (comes to an end) or repeats (goes on forever in a pattern).\u00a0 Terminating rational numbers include:<\/p>\n<p>1\/2 = 0.5<\/p>\n<p>1\/8 = 0.125<\/p>\n<p>3\/20 = 0.15<\/p>\n<p>9\/160 = 0.05625<\/p>\n<p>&nbsp;<\/p>\n<p>Repeating rational numbers include:<\/p>\n<p>1\/3 = 0.333333333333333333333333333333333333\u2026<\/p>\n<p>1\/7 = 0.142857142857142857142857142857142857\u2026<\/p>\n<p>1\/11 = 0.090909090909090909090909090909090909\u2026<\/p>\n<p>1\/15 = 0.066666666666666666666666666666666666\u2026<\/p>\n<p>&nbsp;<\/p>\n<h2>When Do Rational Numbers Terminate?<\/h2>\n<p>The GMAT won&#8217;t give you a complicated fraction like 9\/160 and expect you to figure out what its decimal expression is.\u00a0 BUT, the GMAT could give you a fraction like 9\/160 and ask whether it terminates or not.\u00a0 How do you know?<\/p>\n<p>Well, first of all, <strong>any terminating decimal (like 0.0376) is, essentially, a fraction with a power of ten in the dominator<\/strong>; for example, 0.0376 = 376\/10000 = 47\/1250.\u00a0 Notice we simplified this fraction, by cancelling a factor of 8 in the numerator.\u00a0 Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator.\u00a0 Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.<\/p>\n<ul class=\"no_bullet\">\n<li class=\"bulb\">If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates.\u00a0 If there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats.<\/li>\n<\/ul>\n<p>Here&#8217;s some examples of this concept at work:<\/p>\n<ul>\n<li>1\/24 <em>repeats<\/em> (there&#8217;s a factor of 3)<\/li>\n<li>1\/25 <em>terminates<\/em> (just powers of 5)<\/li>\n<li>1\/28 <em>repeats<\/em> (there&#8217;s a factor of 7)<\/li>\n<li>1\/32 <em>terminates<\/em> (just powers of 2)<\/li>\n<li>1\/40 <em>terminates<\/em> (just powers of 2 and 5)<\/li>\n<\/ul>\n<p>Notice, as long as the fraction is in lowest terms, the numerator doesn&#8217;t matter at all. Since 1\/40 terminates, then 7\/40, 13\/40, or any other integer over 40 also terminates. Since 1\/28 repeats, then 5\/28 and 15\/28 and 25\/28 all repeat; notice, though that 7\/28 doesn&#8217;t repeat, because of the cancellation: 7\/28 = 1\/4 = 0.25.<\/p>\n<h2>Shortcut Decimals to Know<\/h2>\n<p>There are certain decimals that are good to know as shortcuts, both for fraction-to-decimal conversions and for fraction-to-percent conversions.\u00a0 These are:<\/p>\n<ul>\n<li>1\/2 = 0.5<\/li>\n<li>1\/3 = 0.33333333333333333333333333\u2026<\/li>\n<li>2\/3 = 0.66666666666666666666666666\u2026<\/li>\n<li>1\/4 = 0.25<\/li>\n<li>3\/4 = 0.75<\/li>\n<li>1\/5 = 0.2 (and times 2, 3, and 4 for other easy decimals)<\/li>\n<li>1\/6 = 0.166666666666666666666666666\u2026.<\/li>\n<li>5\/6 = 0.833333333333333333333333333\u2026<\/li>\n<li>1\/8 = 0.125<\/li>\n<li>1\/9 = 0.111111111111111111111111111\u2026 (and times other digits for other easy decimals)<\/li>\n<li>1\/11 = 0.09090909090909090909090909\u2026 (and times other digits for other easy decimals)<\/li>\n<\/ul>\n<h2>Irrational Numbers<\/h2>\n<p>There&#8217;s another category of decimals that don&#8217;t terminate (they go on forever) and they have no repeating pattern.\u00a0\u00a0 These numbers, the non-terminating non-repeating decimals, are called the <strong>irrational numbers<\/strong>.\u00a0 <\/p>\n<ul class=\"no_bullet\">\n<li class=\"bulb\"> It is impossible to write any irrational number as a ratio of two integers.<\/li>\n<p>\u00a0\n<\/ul>\n<p>Mr. Pythagoras (c. 570 \u2013 c. 495 bce) was the first to prove a number irrational: he proved that the square-root of <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_1b3047e87641ecfdcdbfc48989c0da9a.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"{2 \u2014 sqrt(2)}\" title=\"{2 \u2014 sqrt(2)}\"\/> is irrational.\u00a0 We now know: all square-roots of integers that don&#8217;t come out evenly are irrational.\u00a0 Another famous irrational number is <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_8edb2cf68079344a2edd739531259f6c.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"pi\" title=\"pi\"\/>, or pi, the ratio of a circle&#8217;s circumference to its diameter.\u00a0 For example,<\/p\n\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_8edb2cf68079344a2edd739531259f6c.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"pi\" title=\"pi\"\/>\u00a0= 3.141592653589793238462643383279502884197169399375105820974944592307816<\/p>\n<p>That&#8217;s the first 70 digits of pi, and the digits never repeat&mdash;they go on forever with no repeating pattern.\u00a0 There are infinitely many irrational numbers: in fact, the infinity of irrational numbers is infinitely bigger than the infinity of the rational numbers, but that gets into some math that is much more advanced than what you need to know for the GMAT.<\/p>\n<h2>Terminating and Repeating Decimals: Practice Questions<\/h2>\n<p>Now here&#8217;s your chance to test your understanding! Try to answer these practice questions and then check the answer and explanation.<\/p>\n<ol>\n<li><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_62e27c937c445d71f3f3e68df3578098.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"{(0.16666...)\/(0.44444...)} =\" title=\"{(0.16666...)\/(0.44444...)} =\"\/>\n<ol type=\"A\">\n<li>2\/27<\/li>\n<li>3\/2\t<\/li>\n<li>3\/4<\/li>\n<li>3\/8<\/li>\n<li>9\/16<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<details>\n<summary style=\"color:#005bc2;\">Show answer and explanation<\/summary>\n<p>From our shortcuts, we know 0.166666666666\u2026 = 1\/6, and 0.444444444444\u2026 = 4\/9.\u00a0 Therefore\u00a0(1\/6)*(9\/4) = 3\/8.\u00a0 <\/p>\n<p>Answer = <strong>D<\/strong><br \/>\n<\/details>\n<p>&nbsp;<\/p>\n<p>And now for a sample question from our <a href=\"https:\/\/gmat.magoosh.com?utm_source=gmatblog&#038;utm_medium=blog&#038;utm_campaign=gmatquestions&#038;utm_term=inline&#038;utm_content=gmat-math-terminating-and-repeating-decimals\" rel=\"noopener noreferrer\" target=\"_blank\">GMAT product<\/a>:<\/p>\n<blockquote><p>2. Given that <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_163649480838189dca87be31c32bce7b.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"{0.overline{k}}\" title=\"{0.overline{k}}\"\/> represents a decimal in which the digit k repeats without end, then what is the value of <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_969_31e0bdd42f6fc6d56d0a7fc6a9ba5661.png\" style=\"vertical-align:-31px; display: inline-block ;\" alt=\"1\/{(0.5)-(0.overline{6})^2}\" title=\"1\/{(0.5)-(0.overline{6})^2}\"\/>?<\/p><\/blockquote>\n<\/ol>\n<p><input type=\"radio\" name=\"repeatdecimal\" value=\"0.1\">0.1 <br \/>\n<input type=\"radio\" name=\"repeatdecimal\" value=\"1\">1 <br \/>\n<input type=\"radio\" name=\"repeatdecimal\" value=\"4.5\">4.5 <br \/>\n<input type=\"radio\" name=\"repeatdecimal\" value=\"6\">6 <br \/>\n<input type=\"radio\" name=\"repeatdecimal\" value=\"18\">18 <\/p>\n<p><a href=\"https:\/\/gmat.magoosh.com\/questions\/12628?utm_source=gmatblog&#038;utm_medium=blog&#038;utm_campaign=gmatquestions&#038;utm_term=inline&#038;utm_content=gmat-math-terminating-and-repeating-decimals\" target=\"_blank\" rel=\"noopener noreferrer\">Click here for the answer and video explanation!<\/a><\/p>\n<p>If you&#8217;d like to practice more with decimals, check out our <a href=\"https:\/\/magoosh.com\/gmat\/gmat-practice-questions-with-fractions-and-decimals\/\" rel=\"noopener noreferrer\" target=\"_blank\">GMAT practice questions with fractions and decimals<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post was updated in 2024 for the new GMAT. You might not feel decimals are the most exciting thing in the world, but just look at our friend , a decimal with its own holiday! In this edition of GMAT Math, we&#8217;re going over two specific types of decimals: terminating and repeating decimals.\u00a0 Learn [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-1579","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Math: Terminating and Repeating Decimals - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"What&#039;s the difference between terminating and repeating decimals? 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