{"id":28790,"date":"2023-05-08T07:10:26","date_gmt":"2023-05-08T07:10:26","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=28790"},"modified":"2023-05-11T08:05:13","modified_gmt":"2023-05-11T08:05:13","slug":"operations-on-rational-numbers-methods-steps-facts-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers","title":{"rendered":"Operations on Rational Numbers &#8211; Methods, Steps, Facts, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-6c0c3cfb-aaea-414d-8b93-cd5ace50ec51\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#0-what-are-the-operations-on-rational-numbers>What Are the Operations on Rational Numbers?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#6-operations-on-rational-numbers-with-negative-signs>Operations on Rational Numbers with Negative Signs<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#7-properties-of-operations-on-rational-numbers>Properties of Operations on Rational Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#17-solved-examples-on-operations-on-rational-numbers>Solved Examples on Operations on Rational Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#18-practice-problems-on-operations-on-rational-numbers>Practice Problems on Operations on Rational Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers#19-frequently-asked-questions-on-operations-on-rational-numbers>Frequently Asked Questions on Operations on Rational Numbers<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-are-the-operations-on-rational-numbers\">What Are the Operations on Rational Numbers?<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Operations on rational numbers refer to the arithmetic operations given by addition, subtraction, multiplication, and division. Rational numbers are numbers that can be written in the form $\\frac{p}{q}$, where p and q are integers and $q \\neq 0$. The set of rational numbers is represented by the symbol \u211a.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Arithmetic operations on rational numbers refer to the mathematical operations carried out on two or more rational numbers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The basic arithmetic operations performed on rational numbers are:<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Addition of Rational Numbers (with same denominators and with different denominators)<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Subtraction of Rational Numbers (with same denominators and with different denominators)<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Multiplication of Rational Numbers<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Division of Rational Numbers<\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">Let&#8217;s study each of them in detail with the steps and examples.&nbsp;<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_make_10_pt.png\" alt=\"Add 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-and-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_2d_1d_match_pt.png\" alt=\"Add 2-Digit and 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit and 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-numbers-by-regrouping\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_regrp_2d_2d_vertical_pt.png\" alt=\"Add 2-Digit Numbers By Regrouping Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit Numbers By Regrouping Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_pt.png\" alt=\"Add 3 Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-in-any-order\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_shifty_bridges_4_gm.png\" alt=\"Add 3 Numbers in Any Order Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers in Any Order Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-groups-of-objects\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_using_model_pt.png\" alt=\"Add 3 Numbers Using Groups of Objects Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Using Groups of Objects Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-model\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_using_models_pt.png\" alt=\"Add 3 Numbers using Model Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers using Model Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers-and-match\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_3d_1d_match_g3_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers and Match Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers and Match Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_3d_1d_vertical_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers-with-regrouping\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_regrp_3d_1d_vertical_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers with Regrouping Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers with Regrouping Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = 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});\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-addition-of-rational-numbers\">Addition of Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">There are two cases possible when adding two or more rational numbers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Case 1: When the denominators of the given rational numbers are equal.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">When denominators are equal, add the numerators and keep the same denominator.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Addition of Rational Numbers (When denominators are equal.)<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Add the numerators. Keep the common denominator.<br>            <br>$\\frac{a}{c} + \\frac{b}{c} = \\frac{a + b}{c}$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:<\/strong><br>$\\bullet\\;\\frac{2}{8} + \\frac{3}{8} = \\frac{2 + 3}{8} = \\frac{5}{8}$<br><br>$\\bullet\\;\\frac{4}{6} + \\frac{1}{6} = \\frac{4 + 1}{6} = \\frac{5}{6}$<br><br>$\\bullet\\;\\frac{7}{16} + \\frac{9}{16} = \\frac{7 + 9}{16} = \\frac{16}{16} = 1$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Case 2: When the denominators of given rational numbers are different<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">When the denominators are not equal, we first need to find a common denominator. Let\u2019s understand this with an example.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example: Add the rational numbers <\/strong>$\\frac{2}{5}$<strong> and <\/strong>$\\frac{3}{4}$<strong>.<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 1:<\/strong> Find the LCM of the denominators of the given rational numbers.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, the LCM of 4 and 5 is 20.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 2: <\/strong>Change the denominator of each rational number to 20 by multiplying both numerator and denominator by an appropriate factor.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{2 \\times 4}{5 \\times 4} = \\frac{8}{20}$ and $\\frac{3 \\times 5}{4 \\times 5} = \\frac{15}{20}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 3:<\/strong> For these new rational numbers (having a common denominator), add the numerators and keep the common denominator.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{2}{5} + \\frac{3}{4} = \\frac{23}{20}$<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Addition of Rational Numbers (When denominators are different)<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Find the LCM of denominators and solve to find the common denominator.<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:&nbsp;<\/strong><br><br>$\\bullet\\;\\frac{2}{8} + \\frac{1}{3} = \\frac{6}{24} + \\frac{8}{24} = \\frac{6 + 8}{24} = \\frac{14}{24} = \\frac{7}{12}$<br><br>$\\bullet\\;\\frac{1}{6} + \\frac{5}{2} = \\frac{1}{6} + \\frac{15}{6} = \\frac{15 + 1}{6} = \\frac{16}{6} = \\frac{8}{3}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/1-and-2-more-within-10-horizontal-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/1-and-2-more-within-10-horizontal-addition.jpeg\" alt=\"1 and 2 more within 10: Horizontal Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/1-and-2-more-within-10-vertical-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/1-and-2-more-within-10-vertical-addition.jpeg\" alt=\"1 and 2 more within 10: Vertical Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/1-and-2-more-within-20-horizontal-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/1-and-2-more-within-20-horizontal-addition.jpeg\" alt=\"1 and 2 more within 20: Horizontal Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/1-and-2-more-within-20-vertical-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/1-and-2-more-within-20-vertical-addition.jpeg\" alt=\"1 and 2 more within 20: Vertical Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers.jpeg\" alt=\"Add & Subtract Ones & 2-Digit Numbers Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Find the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Match the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Addition Tables\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-wheel\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-wheel.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Addition Wheel\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-number-charts\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-number-charts.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Number Charts\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = 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slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-subtraction-of-rational-numbers\">Subtraction of Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">We will discuss the same cases for subtraction of rational numbers as well.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Case 1: When the denominators of the given rational numbers are equal:<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Subtract the numerators and keep the denominator the same.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Subtraction of Rational Numbers (When denominators are equal.)<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Subtract the numerators. Keep the common denominator.<br><br>$\\frac{a}{c} \\;-\\; \\frac{b}{c} = \\frac{a \\;-\\; b}{c}$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:&nbsp;<\/strong><br><br>$\\bullet\\;\\frac{5}{8} \\;-\\; \\frac{2}{8} = \\frac{3}{8}$<br><br>$\\bullet\\;\\frac{2}{8} \\;-\\; \\frac{3}{8} = \\frac{2 \\;-\\; 3}{8} = \\frac{-1}{8}$<br><br>$\\bullet\\;\\frac{4}{6} \\;-\\; \\frac{1}{6} = \\frac{4 \\;-\\; 1}{6} = \\frac{3}{6} = \\frac{1}{2}$<br><br>$\\bullet\\;\\frac{7}{16} \\;-\\; \\frac{9}{16} = \\frac{7 \\;-\\; 9}{16} = \\frac{-2}{16} = \\frac{-1}{8}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Case 2: When the denominators of the given numbers are unequal:<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, we first make the denominators equal using the LCM method.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Subtraction of Rational Numbers (When denominators are different)<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Find the LCM of denominators and solve to find the common denominator.<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:<\/strong><br><br>$\\bullet\\;\\frac{2}{8} \\;-\\; \\frac{1}{4} = \\frac{2}{8} \\;-\\; \\frac{2}{8} = 0$<br><br>$\\bullet\\;\\frac{1}{5} \\;-\\; \\frac{5}{3} = \\frac{3}{15} \\;-\\; \\frac{25}{15} = \\frac{-22}{15}$&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example: subtract <\/strong>$\\frac{1}{3}$<strong> from <\/strong>$\\frac{1}{2}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 1: <\/strong>Find the LCM of the denominators of the given rational numbers.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">In this case, the LCM of 2 and 3 is 6.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 2:<\/strong> Convert each rational number into an equivalent rational number with the LCM as the new denominator.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{1 \\times 3}{2 \\times 3} = \\frac{3}{6}$ and $1 \\times 2}{3 \\times 2 = \\frac{2}{6}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 3:<\/strong> Subtract the numerators. Keep the common denominator.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{3}{6}\\;-\\; \\frac{2}{6} = \\frac{1}{6}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Therefore, $\\frac{1}{2} \\;-\\; \\frac{1}{3} =&nbsp;\\frac{1}{6}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-multiplication-of-rational-numbers\">Multiplication of Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">It is very easy to multiply rational numbers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 1:<\/strong> Multiply the numerators. Write the product as the numerator of the answer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 2:<\/strong> Multiply the denominators. Write the product as the denominator of the answer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 3:<\/strong> Reduce the final answer to its lowest form.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Multiplication of Rational Numbers<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Multiply the numerators. Multiply the denominators.<br><br>$\\frac{a}{c} \\times \\frac{b}{d} = \\frac{a \\times b}{c \\times d}$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:&nbsp;<\/strong><br><br>$\\bullet\\;\\frac{2}{5} \\times \\frac{3}{4} = \\frac{2 \\times 3}{5 \\times 4} = \\frac{6}{20}$<br><br>$\\bullet\\;\\frac{2}{8}\\frac{\\;-\\;3}{8} = \\frac{2 (\\;-\\;3)}{8 \\times 8} = \\frac{-6}{64} = \\frac{-3}{32}$<br><br>$\\bullet\\;\\frac{(\\;-\\;4)}{6} \\times \\frac{1}{4} = \\frac{(\\;-\\;4) \\times 1}{6\\times 4} = \\frac{\\;-\\;4}{24} = \\frac{\\;-\\;1}{6}$<br><br>$\\bullet\\;\\frac{7}{16} \\times \\frac{9}{10} = \\frac{7 \\times 9}{16 \\times 10} = \\frac{63}{160}$&nbsp;&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-division-of-rational-numbers\">Division of Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">To divide a rational number by another rational number, we multiply the first rational number (dividend) by the reciprocal of the second rational number (divisor).<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Division of Rational Numbers<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>RULE: <\/strong>Multiply the dividend with the reciprocal of the divisor.<br><br>$\\frac{a}{c} \\div \\frac{b}{d} = \\frac{a}{c}\\times\\frac{d}{b}$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Examples:<\/strong><br><br>$\\bullet\\;\\frac{2}{5} \\div \\frac{3}{4} = \\frac{2}{5} \\times \\frac{4}{3} = \\frac{8}{15}$<br><br>$\\bullet\\;\\frac{7}{4} \\div \\frac{2}{7} = \\frac{7}{4} \\times \\frac{7}{2} = \\frac{49}{8}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example: Find <\/strong>$\\frac{2}{3} \\div \\frac{1}{5}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 1: <\/strong>Find the reciprocal of the divisor.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Reciprocal of $\\frac{1}{5} = \\frac{5}{1}$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Step 2:<\/strong> Multiply the dividend with the reciprocal of the divisor.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{2}{3} \\times \\frac{5}{1} = = \\frac{2 \\times 5}{3 \\times 1} = \\frac{10}{3}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Multiplication or division of integers with the same signs produces a positive result.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Multiplication or division of two integers with unlike signs results in a negative answer.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">These rules can be applied to the multiplication and division of rational numbers as well.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-order-of-operations-with-rational-numbers\">Order of Operations with Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Order of operations with rational numbers is no different from the order of operations you have used so far for. Order of operations tells us the correct sequence in which a mathematical expression should be evaluated. We use the PEMDAS rule to remember the order.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>P<\/strong> &#8211; Parentheses<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>E<\/strong> &#8211; Exponents<strong>&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>M <\/strong>&#8211; Multiplication<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>D<\/strong> &#8211; Division<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>A<\/strong> &#8211; Addition<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>S<\/strong> &#8211; Subtraction<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-operations-on-rational-numbers-with-negative-signs\">Operations on Rational Numbers with Negative Signs<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">A rational number is said to be negative if the numerator and denominator have opposite signs. Rational number operations with negatives follow the same rules as integers.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Operation<\/strong><\/th><th><strong>Rule<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\" rowspan=\"2\"><strong>Addition&nbsp;<\/strong><strong>&amp; Subtraction&nbsp;<\/strong><\/td><td>If the numbers have SAME SIGN, then ADD the numbers and KEEP THE SIGN.<br><strong>Examples:<\/strong><br>$9 + 7 = 16$<br><br>$(\\;-\\;9) + (\\;-\\;7) = (\\;-\\;16)$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">If the numbers have DIFFERENT SIGNS, then SUBTRACT the numbers and take the SIGN OF THE GREATER NUMBER.<br><strong>Examples:<\/strong><br><br>$(-9) + 7 = \\;-\\;2$<br><br>$9 + (\\;-\\;7) = 2$<br><br><strong>Special case:<\/strong> $a \\;-\\; (\\;-\\;b) = a + b$<br><br>$9 \\;-\\;(\\;-\\;7) = 9 + 7 = 16$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <\/strong><br><br><br><strong>Multiplication&nbsp;&amp;Division<\/strong><\/td><td><strong>SAME SIGN: The answer will be POSITIVE.<\/strong><br><br>$(+) (+) = (+)$<br><br>$(-) (-) = (+)$<br>Examples:  $5 \\times 4 = 20$<br><br>$(\\;-\\;2) \\times (\\;-\\;1) = 2$<br><br>$(\\;-\\;20) \\div (\\;-\\;5) = 4$<br><br><strong>DIFFERENT SIGNS: The answer will be NEGATIVE.<\/strong><br><br>$(\\;-\\;) (+) = (\\;-\\;)$<br><br>Examples: $(\\;-\\;5) \\times (\\;-\\;4) = 20$<br><br>$(2) \\times (\\;-\\;1) = \\;-\\;2$<br><br>$(\\;-\\;20) \\div 5 = \\;-\\;4$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-properties-of-operations-on-rational-numbers\">Properties of Operations on Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Properties of rational numbers make it easy to perform different mathematical operations on them. These properties come handy when simplifying expressions or solving equations.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"8-closure-property-of-rational-numbers\">Closure Property of Rational Numbers<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">For two rational numbers, the addition, subtraction, and multiplication always results in a rational number. Thus, rational numbers are closed under addition, subtraction, and multiplication. The closure property isn\u2019t applicable for the division of rational numbers as division by zero isn\u2019t defined.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"9-associative-property-of-rational-numbers\">Associative Property of Rational Numbers<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">Rational numbers obey the associative property for addition and multiplication.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, for any three rational numbers x, y, and z, we have<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x + (y + z) = (x + y) + z$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x \\times (y \\times z) = (x \\times y) \\times z$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Examples:&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">$\\frac{1}{3} + (\\frac{1}{4} + \\frac{3}{3}) = (\\frac{1}{3} + \\frac{1}{4}) + \\frac{3}{3}$<\/li>\n\n\n\n<li class=\" eplus-wrapper\">$\\frac{1}{3} \\times (\\frac{1}{4}\\times \\frac{3}{3}) = (\\frac{1}{3} \\times \\frac{1}{4}) \\times \\frac{3}{3}$<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"10-commutative-property-of-rational-numbers\">Commutative Property of Rational Numbers<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">The addition and multiplication of rational numbers is always commutative. Subtraction of rational numbers doesn\u2019t obey commutative property.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Commutative Law of Addition:&nbsp; $x + y = y + x$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Example: $\\frac{1}{3} + \\frac{2}{3} = \\frac{2}{3} + \\frac{1}{3} = \\frac{3}{3}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Commutative Law of Multiplication: $x \\times y = y \\times x$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Example: $\\frac{1}{2} \\times \\frac{2}{3} = \\frac{2}{3} \\times \\frac{1}{2} = \\frac{2}{6}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"11-distributive-property-of-rational-numbers\">Distributive Property of Rational Numbers<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">Rational numbers follow distributive property over addition and subtraction.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">For rational numbers A, B, and C, we have<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">&nbsp;$A \\times (B \\;-\\; C) = (A \\times B) \\;-\\; (A \\times C)$<\/li>\n\n\n\n<li class=\" eplus-wrapper\">&nbsp;$A \\times (B + C) = (A \\times B) + (A \\times C)$<\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example: <\/strong>$\\frac{1}{3} \\times (\\frac{1}{4} + \\frac{2}{5}) = (\\frac{1}{3} \\times \\frac{1}{4})$ + (\\frac{1}{3} \\times \\frac{2}{5})$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">L.H.S. $= \\frac{1}{3} \\times (\\frac{1}{4} + \\frac{2}{5}) = \\frac{1}{3} \\times (\\frac{17}{20}) = \\frac{17}{60}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">R.H.S. $= (\\frac{1}{3} \\times \\frac{1}{4}) + (\\frac{1}{3} \\times \\frac{2}{5}) = \\frac{1}{12} + \\frac{2}{10} = \\frac{17}{60}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"12-additive-identity-and-multiplicative-identity\">Additive Identity and Multiplicative Identity<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">0 is the additive identity of any rational number. When we add 0 to any rational number, the resultant is the number itself.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x + 0 = x$ \u2026for any rational number x&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">1 is the multiplicative identity of any rational number. When we multiply 1 with any rational number, the result is the number itself.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x \\times 1 = x$ \u2026for any rational number x&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"13-additive-inverse-and-multiplicative-inverse\">Additive Inverse and Multiplicative Inverse<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">For any rational number $\\frac{x}{y}$, the additive inverse is given by $(\\;-\\;\\frac{x}{y})$.The addition of a rational number and its additive inverse is always 0.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">For any rational number $\\frac{x}{y}$, the multiplicative inverse or reciprocal is given by $\\frac{y}{x}$ . The product of a rational number and its reciprocal is always 1.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"14-rational-numbers-operations-anchor-chart\">Rational Numbers Operations Anchor Chart<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">An anchor chart displays the important information from a given lesson. Let\u2019s recall and summarize a few important points.<\/p>\n\n\n\n<figure class=\"wj-custom-table wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Addition of Rational Numbers<\/strong><\/th><th><strong>Subtraction of Rational Numbers<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">$\\bullet\\;$Same denominators: $\\frac{a}{c} + \\frac{b}{c} = \\frac{a + b}{c}$<br>$\\bullet\\;$Different denominators: Use the LCM method.<\/td><td>$\\bullet\\;$Same denominators: $\\frac{a}{c} \\;-\\; \\frac{b}{c} = \\frac{a \\;-\\; b}{c}$<br>$\\bullet\\;$Different denominators: Use the LCM method.<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\"><strong>Multiplication of Rational Numbers<\/strong><br>$\\frac{a}{c} \\times \\frac{b}{d} = \\frac{a \\times b}{c \\times d}$<\/td><td><strong>Division of Rational Numbers<\/strong><br>$\\frac{a}{c}\\div \\frac{b}{d} = \\frac{a}{ c}\\times \\frac{d}{b}$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\" colspan=\"2\"><strong>Adding and Subtracting Rational Numbers<\/strong><br>$\\bullet\\;$with SAME signs: ADD and KEEP&nbsp;<br>$\\bullet\\;$with DIFFERENT signs: SUBTRACT and take the SIGN OF THE LARGER NUMBER.<br>$\\bullet\\;$Special case: $a \\;-\\; (\\;-\\;b) = a + b$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\" colspan=\"2\"><strong>Multiplying and Dividing Rational Numbers<\/strong><br>$\\bullet\\;$with SAME signs: Answer is POSITIVE<br>$\\bullet\\;$with DIFFERENT signs: Answer is NEGATIVE<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"15-facts-about-operations-on-rational-numbers\">Facts about Operations on Rational Numbers<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Rational numbers are denoted by the letter \u211a.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Pythagoras and his disciples believed that all numbers were rational. It is said that the discovery of irrational numbers shocked them!<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Identity property holds true for addition and multiplication. It does not hold true for subtraction and division of rational numbers.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Subtraction of rational numbers doesn\u2019t obey commutative property.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The inverse property does not apply to the division and subtraction of rational numbers.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"16-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">In this article, we have learned about operations on rational numbers. We learned that the techniques followed in the arithmetic operations are quite similar to operations on rational numbers as well. Now, let&#8217;s apply rational number operations and properties to solve a few examples and practice problems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"17-solved-examples-on-operations-on-rational-numbers\">Solved Examples on Operations on Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><strong>1. Find the additive inverse of <\/strong>$\\frac{5}{12}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Additive inverse of rational number $\\frac{x}{y}$ is $(\\frac{-x}{y})$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, additive inverse of $\\frac{5}{12}$ is $\\frac{-5}{12}$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>2. Solve the following equation using the distributive property: <\/strong>$6 \\times (20 + 5)$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">According to the distributive property over addition,<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;$A (B + C)&nbsp; = (A \\times B) + (A \\times C)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$6 \\times (20 + 5) = (6 \\times 20) + (6 \\times 5)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 120 + 30$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 150$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, $6 \\times (20 + 5) = 150$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>3. From a rope 252 ft long, a part from end measuring <\/strong>$\\frac{62}{8}$<strong> ft is cut off. Find the length of the remaining rope.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Total length of the rope $= \\frac{25}{2}$ ft<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Length of the rope cut off $=&nbsp; \\frac{62}{8}$ ft&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The length of the remaining rope $= \\frac{25}{2} \\;-\\; \\frac{62}{8}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The LCM of 8 and 2 is 8.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{25}{2} \\;-\\; \\frac{62}{8} = \\frac{25\\times 4}{2 \\times 4} \\;-\\; \\frac{62 \\times 1}{8 \\times 1}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;$= \\frac{100}{8} \\;-\\; \\frac{62}{8}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp; $= \\frac{38}{8}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;$= \\frac{19}{4}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The length of the remaining rope is $\\frac{19}{4}$ ft<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>4. Divide the rational numbers <\/strong>$\\frac{2}{5} \\div \\frac{3}{6}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Reciprocal of the divisor $\\frac{3}{6}$ is $\\frac{6}{3}$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Multiply the dividend by the reciprocal of the divisor.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{2}{5} \\div \\frac{3}{6} =&nbsp;\\frac{2}{5} \\times \\frac{6}{3}$&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= \\frac{2 \\times 6}{5 \\times 3}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$=&nbsp;\\frac{12}{15}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= \\frac{4}{5}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, $\\frac{2}{5} \\div \\frac{3}{6} = \\frac{4}{5}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>5. Find the multiplicative inverse of <\/strong>$\\frac{2}{3} + \\frac{3}{2}$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">To find the reciprocal, we need to simplify the expression first.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;$\\frac{2}{3} + \\frac{3}{2} = \\frac{2\\times 2}{3\\times 2} + \\frac{3 \\times 3}{2\\times 3}$&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;$&nbsp;= \\frac{4}{6} +\\frac{9}{6}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"> $&nbsp;= \\frac{13}{6}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, the multiplicative inverse of $\\frac{13}{6}$ is $\\frac{6}{13}$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"18-practice-problems-on-operations-on-rational-numbers\">Practice Problems on Operations on Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Operations on Rational Numbers - Methods, Steps, Facts, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Rational numbers are closed under the operations of _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">addition and multiplication<\/div><div class=\"spq_answer_block\" data-value=\"1\">subtraction and multiplication<\/div><div class=\"spq_answer_block\" data-value=\"2\">subtraction and division<\/div><div class=\"spq_answer_block\" data-value=\"3\">addition and division<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: addition and multiplication<br\/>Rational numbers are numbers that can be written in the form $\\frac{p}{q}$, where $p$ and $q$ are integers and $q \\neq 0$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">$\\frac{1}{2} + \\frac{2}{4} =$ _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\frac{3}{6}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\frac{2}{8}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\frac{1}{2}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$1$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $1$<br\/>$\\frac{1}{2} + \\frac{2}{4} = \\frac{1 \\times 2}{2 \\times 2} + \\frac{2}{4} = \\frac{2}{4} + \\frac{2}{4} = \\frac{4}{4} = 1$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">The value of $\\frac{1}{4} \\div \\frac{2}{5}$ is _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\frac{2}{20}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\frac{8}{5}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\frac{5}{8}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\frac{2}{9}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\\frac{5}{8}$<br\/>$\\frac{1}{4} \\div 25= \\frac{1}{4} \\times \\frac{5}{2} = \\frac{5}{8}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The commutative property of rational numbers is not applicable for the _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">Addition operation<\/div><div class=\"spq_answer_block\" data-value=\"1\">Subtraction operation<\/div><div class=\"spq_answer_block\" data-value=\"2\">Division operation<\/div><div class=\"spq_answer_block\" data-value=\"3\">Both b and c<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: Both b and c<br\/>Subtraction and division of rational numbers is not commutative.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">According to distributive property, $\\frac{1}{3} \\times (\\frac{1}{4} + \\frac{2}{5}) =$  _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\frac{1}{3} + \\frac{1}{4} + \\frac{2}{5}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$(\\frac{1}{3} \\times \\frac{1}{4}) + (\\frac{1}{3} \\times \\frac{2}{5})$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$(\\frac{1}{3} + \\frac{1}{4}) \\times ( \\frac{1}{3} + \\frac{2}{5} )$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\frac{1}{3} \\times \\frac{1}{4} \\times \\frac{2}{5}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $(\\frac{1}{3} \\times \\frac{1}{4}) + (\\frac{1}{3} \\times \\frac{2}{5})$<br\/>According to the distributive property, <br>$A \\times (B + C) = (A \\times B) + (A \\times C)$<br>\r\nThus, $\\frac{1}{3} \\times (\\frac{1}{4} + \\frac{2}{5}) = (\\frac{1}{3} \\times \\frac{1}{4}) + (\\frac{1}{3} \\times \\frac{2}{5})$<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Operations on Rational Numbers - Methods, Steps, Facts, Examples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Operations on Rational Numbers\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Rational numbers are closed under the operations of _______.\",\n                    \"text\": \"Rational numbers are closed under the operations of _______.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"subtraction and multiplication\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"subtraction and division\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"addition and division\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"addition and multiplication\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Rational numbers are numbers that can be written in the form $$\\\\frac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q \\\\neq 0$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} =$$ _______.\",\n                    \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} =$$ _______.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{3}{6}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{2}{8}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{1}{2}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$1$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\frac{1}{2} + \\\\frac{2}{4} = \\\\frac{1 \\\\times 2}{2 \\\\times 2} + \\\\frac{2}{4} = \\\\frac{2}{4} + \\\\frac{2}{4} = \\\\frac{4}{4} = 1$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The value of $$\\\\frac{1}{4} \\\\div \\\\frac{2}{5}$$ is _______.\",\n                    \"text\": \"The value of $$\\\\frac{1}{4} \\\\div \\\\frac{2}{5}$$ is _______.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{2}{20}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{8}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{2}{9}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\frac{5}{8}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\frac{1}{4} \\\\div 25= \\\\frac{1}{4} \\\\times \\\\frac{5}{2} = \\\\frac{5}{8}$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The commutative property of rational numbers is not applicable for the _______.\",\n                    \"text\": \"The commutative property of rational numbers is not applicable for the _______.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Addition operation\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Subtraction operation\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Division operation\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"Both b and c\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Subtraction and division of rational numbers is not commutative.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"According to distributive property, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) =$$  _______.\",\n                    \"text\": \"According to distributive property, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) =$$  _______.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{1}{3} + \\\\frac{1}{4} + \\\\frac{2}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(\\\\frac{1}{3} + \\\\frac{1}{4}) \\\\times ( \\\\frac{1}{3} + \\\\frac{2}{5} )$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{1}{3} \\\\times \\\\frac{1}{4} \\\\times \\\\frac{2}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$(\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"According to the distributive property, <br>$$A \\\\times (B + C) = (A \\\\times B) + (A \\\\times C)$$<br>\r\nThus, $$\\\\frac{1}{3} \\\\times (\\\\frac{1}{4} + \\\\frac{2}{5}) = (\\\\frac{1}{3} \\\\times \\\\frac{1}{4}) + (\\\\frac{1}{3} \\\\times \\\\frac{2}{5})$$\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"19-frequently-asked-questions-on-operations-on-rational-numbers\">Frequently Asked Questions on Operations on Rational Numbers<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-90d383f5-f078-4e0e-9f76-dbdf6f29765f\"><strong>How to find the decimal form of a rational number?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-90d383f5-f078-4e0e-9f76-dbdf6f29765f\">\n\n<p class=\" eplus-wrapper\">Divide the numerator by the denominator using the long division method.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$\\frac{5}{8} = 5\\div8 = 0.625$<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-90d383f5-f078-4e0e-9f76-dbdf6f29765f\"><strong>What are irrational numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-90d383f5-f078-4e0e-9f76-dbdf6f29765f\">\n\n<p class=\" eplus-wrapper\">Irrational numbers are the type of real numbers that cannot be expressed in the rational form&nbsp;$\\frac{p}{q}$, where p, q are integers and $q \\neq 0$. In simple words, all the real numbers that are not rational numbers are irrational.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-90d383f5-f078-4e0e-9f76-dbdf6f29765f\"><strong>How many irrational numbers are between two rational numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-90d383f5-f078-4e0e-9f76-dbdf6f29765f\">\n\n<p class=\" eplus-wrapper\">There are infinite irrational numbers between any two rational numbers.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-90d383f5-f078-4e0e-9f76-dbdf6f29765f\"><strong>Can the sum of two irrational numbers be a rational number?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-90d383f5-f078-4e0e-9f76-dbdf6f29765f\">\n\n<p class=\" eplus-wrapper\">Whenever you add or subtract two irrational numbers, you may get a rational number or an irrational number.Example: $(2 + \\sqrt{5}) + (3\\;-\\;\\sqrt{5}) = 5$<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-90d383f5-f078-4e0e-9f76-dbdf6f29765f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-90d383f5-f078-4e0e-9f76-dbdf6f29765f\"><strong>What is the sum of a rational number and an irrational number?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-90d383f5-f078-4e0e-9f76-dbdf6f29765f\">\n\n<p class=\" eplus-wrapper\">Adding a rational number and an irrational number always results in an irrational number.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are the Operations on Rational Numbers? Operations on rational numbers refer to the arithmetic operations given by addition, subtraction, multiplication, and division. Rational numbers are numbers that can be written in the form $\\frac{p}{q}$, where p and q are integers and $q \\neq 0$. The set of rational numbers is represented by the symbol &#8230; <a title=\"Operations on Rational Numbers &#8211; Methods, Steps, Facts, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/operations-on-rational-numbers\" aria-label=\"More on Operations on Rational Numbers &#8211; Methods, Steps, Facts, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-28790","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=28790"}],"version-history":[{"count":30,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28790\/revisions"}],"predecessor-version":[{"id":29224,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28790\/revisions\/29224"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=28790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=28790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=28790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}