{"id":26845,"date":"2026-10-15T08:56:18","date_gmt":"2026-10-15T08:56:18","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=26845"},"modified":"2026-10-15T13:25:34","modified_gmt":"2026-10-15T13:25:34","slug":"decimal-representation-of-8-27","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/uncategorized\/decimal-representation-of-8-27\/","title":{"rendered":"Decimal Representation of 8\/27 (Step-by-Step Long Division + Why It Repeats)"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"26845\" class=\"elementor elementor-26845\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fe74926 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"fe74926\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-669c101\" data-id=\"669c101\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-61d85ae\" data-id=\"61d85ae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e72ca67 elementor-widget elementor-widget-heading\" data-id=\"e72ca67\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Decimal Representation of 8\/27 (Step-by-Step Long Division + Why It Repeats)<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d2f6a2e elementor-widget elementor-widget-text-editor\" data-id=\"d2f6a2e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The decimal representation of 8\/27 is <\/span><b>0.296296296&#8230;<\/b><span style=\"font-weight: 400;\">, and the block <\/span><b>296<\/b><span style=\"font-weight: 400;\"> repeats forever.<\/span><\/p><p><span style=\"font-weight: 400;\">You write it as <\/span><b>0.296<\/b><span style=\"font-weight: 400;\"> with a bar over all three digits, or as 0.(296) if a bar is awkward to type.<\/span><\/p><table><tbody><tr><td><p><b>Property<\/b><\/p><\/td><td><p><b>Value<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Fraction<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">8\/27<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Decimal form<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.296296&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Repeating block<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">296<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Length of the block<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3 digits<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Type of decimal<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Pure repeating (recurring)<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Rounded to 2 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.30<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Rounded to 3 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.296<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">As a percentage<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">about 29.63%<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">If that&#8217;s all you needed, you&#8217;re set. If you have to show your working for homework or an exam, keep reading. We&#8217;ll go through the long division, work out why the decimal repeats, and then turn it back into a fraction as a final check.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-22682ac elementor-align-center elementor-widget elementor-widget-button\" data-id=\"22682ac\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-258c03b elementor-widget elementor-widget-image\" data-id=\"258c03b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"647\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp\" class=\"attachment-large size-large wp-image-24739\" alt=\"Speech on value of time with a confident schoolboy, clock, and hourglass illustrating the importance of time management.\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-300x190.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-768x485.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1536x970.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value.webp 1578w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e54ffef\" data-id=\"e54ffef\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9b08855 elementor-widget elementor-widget-html\" data-id=\"9b08855\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; \r\n    z-index: 1 !important;\r\n    right: 70px;\r\n    width: 100%;\r\n    max-width: 340px;\r\n    font-family: 'Poppins', sans-serif;\r\n}\r\n\r\n\/* Image Styling *\/\r\n.toc-sticky-image {\r\n    width: 100%;\r\n    height: auto;\r\n    border-radius: 22px; \r\n    margin-bottom: 15px; \r\n    box-shadow: 0 4px 12px rgba(0,0,0,0.05);\r\n    display: block;\r\n}\r\n\r\n\/* TOC Container *\/\r\n.custom-toc {\r\n    background: #fff;\r\n    border: 1px solid #dfe3ea;\r\n    border-radius: 22px;\r\n    padding: 0px 28px;\r\n    width: 100%;\r\n    max-height: 24rem; \r\n    overflow-y: auto;\r\n    box-shadow: 0 2px 8px rgba(0,0,0,0.03);\r\n}\r\n\r\n\/* Title *\/\r\n.custom-toc .toc-title {\r\n    font-size: 25px;\r\n    font-weight: 700;\r\n    color: #17233c;\r\n    position: sticky;\r\n    padding: 0.6rem 0rem;\r\n    background-color: white;\r\n    top: 0px;\r\n    margin-bottom: 12px;\r\n    display: flex;\r\n    align-items: center;\r\n    gap: 10px;\r\n    z-index: 2;\r\n}\r\n\r\n\/* List *\/\r\n.custom-toc .toc-list {\r\n    list-style: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\n\/* Links *\/\r\n.custom-toc .toc-list li {\r\n    margin-bottom: 22px;\r\n}\r\n\r\n.custom-toc .toc-list a {\r\n    text-decoration: none;\r\n    color: #6b7a99;\r\n    font-size: 14px;\r\n    font-weight: 600;\r\n    transition: 0.3s ease;\r\n    display: inline-block;\r\n}\r\n\r\n\/* Hover Color *\/\r\n.custom-toc .toc-list a:hover {\r\n    color: #8B0000;\r\n}\r\n\r\n\/* H2 Indent *\/\r\n.custom-toc .toc-list .toc-h2 {\r\n    padding-left: 18px;\r\n    font-size: 14px;\r\n}\r\n\r\n\/* Mobile Responsive Adjustments *\/\r\n@media(max-width:768px) {\r\n    .toc-wrapper {\r\n        position: relative;\r\n        top: 0;\r\n        z-index: 1;\r\n        right: 0;\r\n        max-width: 100%;\r\n        margin-bottom: 30px;\r\n    }\r\n\r\n    .custom-toc {\r\n        max-height: none; \r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8459300 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"8459300\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-acd082c\" data-id=\"acd082c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1df8865\" data-id=\"1df8865\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4702dff elementor-widget elementor-widget-heading\" data-id=\"4702dff\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What Does \"Decimal Representation of a Fraction\" Mean?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-547a717 elementor-widget elementor-widget-text-editor\" data-id=\"547a717\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A fraction is just a division that hasn&#8217;t been done yet. The line in 8\/27 means &#8220;8 divided by 27.&#8221; So the decimal representation of 8\/27 is whatever you get when you actually carry out that division.<\/span><\/p><p><span style=\"font-weight: 400;\">Fractions turn into one of two kinds of decimals:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Terminating decimals<\/b><span style=\"font-weight: 400;\"> stop after a few digits, like 1\/4 = 0.25.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Repeating decimals<\/b><span style=\"font-weight: 400;\"> carry on forever in a pattern, like 1\/3 = 0.333&#8230;<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">8\/27 is the second kind. And because 8 is smaller than 27, the answer is below 1, so it starts with 0 before the decimal point.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0b323b2\" data-id=\"0b323b2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-80eb588 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"80eb588\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e5ed52b\" data-id=\"e5ed52b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8b0aa40\" data-id=\"8b0aa40\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8e45cb3 elementor-widget elementor-widget-heading\" data-id=\"8e45cb3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Step-by-Step Long Division of 8 \u00f7 27\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0364032 elementor-widget elementor-widget-text-editor\" data-id=\"0364032\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Let&#8217;s do it slowly. Once you see the pattern, it&#8217;s actually quite satisfying.<\/span><\/p><p><b>Setup:<\/b><span style=\"font-weight: 400;\"> Write 8 as 8.000000 so you have plenty of zeros to bring down.<\/span><\/p><p><b>Step 1: Start with 8 \u00f7 27.<\/b><span style=\"font-weight: 400;\"> 27 doesn&#8217;t go into 8, so write 0 in the answer, then a decimal point. Add a zero to 8 to make <\/span><b>80<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Step 2: Divide 80 by 27.<\/b><span style=\"font-weight: 400;\"> 27 \u00d7 2 = 54, and 27 \u00d7 3 = 81 is just too big. So the first digit after the decimal point is <\/span><b>2<\/b><span style=\"font-weight: 400;\">. Subtract: 80 \u2212 54 = <\/span><b>26<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Step 3: Bring down a zero to make 260.<\/b><span style=\"font-weight: 400;\"> 27 \u00d7 9 = 243, and 27 \u00d7 10 = 270 is too big. The next digit is <\/span><b>9<\/b><span style=\"font-weight: 400;\">. Subtract: 260 \u2212 243 = <\/span><b>17<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Step 4: Bring down a zero to make 170.<\/b><span style=\"font-weight: 400;\"> 27 \u00d7 6 = 162, and 27 \u00d7 7 = 189 is too big. The next digit is <\/span><b>6<\/b><span style=\"font-weight: 400;\">. Subtract: 170 \u2212 162 = <\/span><b>8<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Step 5: Spot the repeat.<\/b><span style=\"font-weight: 400;\"> The remainder is <\/span><b>8<\/b><span style=\"font-weight: 400;\">, which is exactly where we began. So the next step is 80 \u00f7 27 all over again, and the whole cycle starts again: 2, 9, 6, 2, 9, 6, and on it goes.<\/span><\/p><p><span style=\"font-weight: 400;\">Here&#8217;s the working on paper:<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a00 . 2 9 6 2 9 6 &#8230;<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0______________________<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a027\u00a0 | 8 . 0 0 0 0 0 0<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a08 0<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05 4<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0&#8212;-<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02 6 0<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02 4 3<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0&#8212;&#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 7 0<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 6 2<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0&#8212;&#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a08 0 \u00a0 &lt;- back where we began<\/span><\/p><p>\u00a0<\/p><p><b>8 \u00f7 27 = 0.296296&#8230;<\/b><span style=\"font-weight: 400;\"> \u2713<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-db3f69c\" data-id=\"db3f69c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4a249f1 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"4a249f1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-38fd719\" data-id=\"38fd719\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6854634\" data-id=\"6854634\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ffcb746 elementor-widget elementor-widget-heading\" data-id=\"ffcb746\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The Remainder Table (This Is the Key to Everything)\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ddb18d8 elementor-widget elementor-widget-text-editor\" data-id=\"ddb18d8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Long division for any fraction is really a story about remainders. Here&#8217;s the story for 8\/27:<\/span><\/p><table><tbody><tr><td><p><b>Step<\/b><\/p><\/td><td><p><b>Dividend<\/b><\/p><\/td><td><p><b>Digit<\/b><\/p><\/td><td><p><b>Remainder<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">80<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">26<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">260<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">9<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">17<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">3<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">170<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">6<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">8<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">4<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">80<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">26<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">5<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">260<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">9<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">17<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">&#8230;<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">&#8230;<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">&#8230;<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">&#8230;<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">The remainders go 26, 17, 8, then 26, 17, 8 again. The moment a remainder comes back, the digits repeat too, because the next step starts from the exact same place as before. That&#8217;s the whole reason repeating decimals exist.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b2776bc\" data-id=\"b2776bc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e3ccb2 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"0e3ccb2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-edf2995\" data-id=\"edf2995\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-866e993\" data-id=\"866e993\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dced56a elementor-widget elementor-widget-heading\" data-id=\"dced56a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Why Does 8\/27 Repeat Forever?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fcd9385 elementor-widget elementor-widget-text-editor\" data-id=\"fcd9385\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">There are two ways to explain it: a simple one and a more precise one.<\/span><\/p><p><b>The simple reason: there are only so many remainders<\/b><\/p><p><span style=\"font-weight: 400;\">When you divide by 27, every remainder has to be somewhere between 0 and 26. That&#8217;s only 27 possibilities. So sooner or later you&#8217;re going to land on a remainder you&#8217;ve already seen, and from that point the pattern loops. The only way out is hitting a remainder of 0, which ends the decimal. With 8\/27, we never get there.<\/span><\/p><p><b>The precise reason: 27 has a factor of 3<\/b><\/p><p><span style=\"font-weight: 400;\">A fraction in lowest terms gives a terminating decimal only if its denominator has no prime factors other than 2 and 5. That&#8217;s because we count in base 10, and 10 = 2 \u00d7 5.<\/span><\/p><p><span style=\"font-weight: 400;\">Let&#8217;s test 8\/27:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Is it in lowest terms?<\/b><span style=\"font-weight: 400;\"> 8 = 2\u00b3 and 27 = 3\u00b3. They share no factors, so yes.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>What are the prime factors of the denominator?<\/b><span style=\"font-weight: 400;\"> 27 = 3 \u00d7 3 \u00d7 3.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Are they only 2s and 5s?<\/b><span style=\"font-weight: 400;\"> No, they&#8217;re 3s.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">So 8\/27 can&#8217;t terminate. It has to repeat.<\/span><\/p><p><b>Why a block of exactly 3 digits?<\/b><\/p><p><span style=\"font-weight: 400;\">Here&#8217;s a neat bit of number magic. 10\u00b3 \u2212 1 = 999, and 999 = 27 \u00d7 37. Because 27 divides evenly into 999, the decimal repeats every <\/span><b>3<\/b><span style=\"font-weight: 400;\"> digits. (For a denominator like 7, you&#8217;d need 999999 before it divides evenly, which is why 1\/7 has a 6-digit block.)<\/span><\/p><p><span style=\"font-weight: 400;\">This gives us a handy shortcut too: 1\/27 = 37\/999 = 0.037037&#8230;, and we&#8217;ll use that below.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c418477\" data-id=\"c418477\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-86e0c5a elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"86e0c5a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2399b5f\" data-id=\"2399b5f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7f23af1\" data-id=\"7f23af1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5d465e2 elementor-widget elementor-widget-heading\" data-id=\"5d465e2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">How to Turn 0.296296... Back Into a Fraction\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-57ce934 elementor-widget elementor-widget-text-editor\" data-id=\"57ce934\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This is a great way to check your answer. Let x = 0.296296&#8230;<\/span><\/p><p><b>Step 1:<\/b><span style=\"font-weight: 400;\"> The repeating block has 3 digits, so multiply by 10\u00b3 = 1000:<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> 1000x = 296.296296&#8230;<\/span><\/p><p><b>Step 2:<\/b><span style=\"font-weight: 400;\"> Subtract the original x from it:<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> 1000x \u2212 x = 296.296296&#8230; \u2212 0.296296&#8230;<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> 999x = 296<\/span><\/p><p><b>Step 3:<\/b><span style=\"font-weight: 400;\"> Solve for x:<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> x = 296\/999<\/span><\/p><p><b>Step 4:<\/b><span style=\"font-weight: 400;\"> Simplify. Both numbers divide by 37, since 296 = 8 \u00d7 37 and 999 = 27 \u00d7 37:<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> x = 8\/27 \u2713<\/span><\/p><p><span style=\"font-weight: 400;\">We&#8217;re right back where we started, so the decimal is correct.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e4728c4\" data-id=\"e4728c4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e76d351 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"e76d351\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-831033e\" data-id=\"831033e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0a68cde\" data-id=\"0a68cde\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-81aae5f elementor-widget elementor-widget-heading\" data-id=\"81aae5f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">A Handy Shortcut: The \"27 Pattern\"\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-82188fd elementor-widget elementor-widget-text-editor\" data-id=\"82188fd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">27 is a friendly denominator, because 1\/27 = 0.037037&#8230; and every other fraction over 27 is just a multiple of that. Take a look:<\/span><\/p><table><tbody><tr><td><p><b>Fraction<\/b><\/p><\/td><td><p><b>Decimal<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">1\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.037037&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">2\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.074074&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">4\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.148148&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">5\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.185185&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">7\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.259259&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><b>8\/27<\/b><\/p><\/td><td><p><b>0.296296&#8230;<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">10\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.370370&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">13\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.481481&#8230;<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">26\/27<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.962962&#8230;<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">Spot the trick? The repeating block is just the numerator times 37. For 8\/27, that&#8217;s 8 \u00d7 37 = 296. It&#8217;s a quick way to check your long division in a few seconds (as long as the numerator is below 27, so the block stays within 3 digits).<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-07d15f8\" data-id=\"07d15f8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9ec18e1 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"9ec18e1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-848daae\" data-id=\"848daae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ee41aa8\" data-id=\"ee41aa8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d6f5e1c elementor-widget elementor-widget-heading\" data-id=\"d6f5e1c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Rounding 8\/27 to Different Decimal Places\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-221e09c elementor-widget elementor-widget-text-editor\" data-id=\"221e09c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Since the decimal never ends, you&#8217;ll often be asked to round it:<\/span><\/p><table><tbody><tr><td><p><b>Rounded to<\/b><\/p><\/td><td><p><b>Answer<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">1 decimal place<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.3<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">2 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.30<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">3 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.296<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">4 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.2963<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">5 decimal places<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.29630<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">Watch the 4 decimal place case. The digits run 0.2962 96&#8230;, and the next digit is 9, so the last 2 rounds up to a 3. This is a spot where lots of people slip.<\/span><\/p><p><span style=\"font-weight: 400;\">As a percentage, 8\/27 is about 29.63%.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-30037bf\" data-id=\"30037bf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-77b0bdc elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"77b0bdc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-19f2bc2\" data-id=\"19f2bc2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-68dc36c\" data-id=\"68dc36c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2769a6e elementor-widget elementor-widget-heading\" data-id=\"2769a6e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Is 8\/27 a Rational Number?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5d614f5 elementor-widget elementor-widget-text-editor\" data-id=\"5d614f5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Yes. A rational number is any number that can be written as a fraction of two whole numbers, and 8\/27 is exactly that. Every repeating decimal is rational. Irrational numbers like \u03c0 or \u221a2 are the ones whose digits go on forever with no repeating pattern at all.<\/span><\/p><p><span style=\"font-weight: 400;\">So although 0.296296&#8230; never ends, there&#8217;s nothing mysterious about it. It&#8217;s predictable, and it always converts back to a neat fraction.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-810373c\" data-id=\"810373c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-125bc8e elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"125bc8e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d260f81\" data-id=\"d260f81\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3a0f26a\" data-id=\"3a0f26a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6bc9e2e elementor-widget elementor-widget-heading\" data-id=\"6bc9e2e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Mistakes Almost Everyone Makes\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e1f011c elementor-widget elementor-widget-text-editor\" data-id=\"e1f011c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Writing 0.296 and stopping.<\/b><span style=\"font-weight: 400;\"> That&#8217;s only an approximation. The exact value is 0.296296&#8230;, with the bar over the whole block.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Putting the bar over the wrong digits.<\/b><span style=\"font-weight: 400;\"> The bar goes over all of 296, not just the 6.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Forgetting the starting zero.<\/b><span style=\"font-weight: 400;\"> Because 8 is smaller than 27, the answer begins 0. and your first step is dividing 80 by 27.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Waiting for a zero remainder.<\/b><span style=\"font-weight: 400;\"> You&#8217;ll be waiting forever. Instead, watch for a remainder you&#8217;ve seen before.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Mixing up 1\/27 and 8\/27.<\/b><span style=\"font-weight: 400;\"> 1\/27 = 0.037037&#8230; and 8\/27 = 0.296296&#8230; Always check the numerator.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Rounding the 4th decimal place wrongly.<\/b><span style=\"font-weight: 400;\"> 0.2962963&#8230; becomes 0.2963, not 0.2962.<\/span><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ca04277\" data-id=\"ca04277\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3fcf753 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"3fcf753\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c3900b5\" data-id=\"c3900b5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e492a83\" data-id=\"e492a83\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6f6fc9a elementor-widget elementor-widget-heading\" data-id=\"6f6fc9a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Try It Yourself\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2fb3903 elementor-widget elementor-widget-text-editor\" data-id=\"2fb3903\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Want to see if it&#8217;s sunk in? Try these with long division:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">5\/27 (answer: 0.185185&#8230;)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">4\/27 (answer: 0.148148&#8230;)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">2\/9 (answer: 0.222&#8230;)<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Find the remainder that comes back around, and you&#8217;ll find the repeating block every time.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a8bfdf0\" data-id=\"a8bfdf0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7455093 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"7455093\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-aa89a1d\" data-id=\"aa89a1d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-51532dd\" data-id=\"51532dd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-97993a8 elementor-widget elementor-widget-heading\" data-id=\"97993a8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5a646bf elementor-widget elementor-widget-text-editor\" data-id=\"5a646bf\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<ul><li style=\"list-style-type: none;\"><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>8\/27 = 0.296296&#8230;<\/b><span style=\"font-weight: 400;\">, with a bar over 296<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The repeating block is <\/span><b>296<\/b><span style=\"font-weight: 400;\">, and it has 3 digits<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Long division gives remainders 26, 17 and 8, and when 8 comes back, the digits repeat<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">It repeats because 27 = 3\u00b3 contains a factor other than 2 or 5<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Shortcut: 1\/27 = 0.037037&#8230;, and the block for k\/27 is k \u00d7 37<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Convert it back with 999x = 296, which gives 296\/999 = 8\/27<\/span><\/li><\/ul><\/li><\/ul><p><span style=\"font-weight: 400;\">Once you&#8217;ve worked through this one by hand, repeating decimals stop feeling scary. They&#8217;re just a reminder coming back around.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-28ad701\" data-id=\"28ad701\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Decimal Representation of 8\/27 (Step-by-Step Long Division + Why It Repeats) The decimal representation of 8\/27 is 0.296296296&#8230;, and the [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[1],"tags":[],"class_list":["post-26845","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - 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