{"id":229494,"date":"2025-06-08T08:43:41","date_gmt":"2025-06-08T08:43:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229494"},"modified":"2025-06-08T08:43:43","modified_gmt":"2025-06-08T08:43:43","slug":"a-student-wrote-a-proof-about-the-product-of-two-rational-numbers-1-let-xab-and-let-ycd-where-a-and-c-are-defined-to-be-integers-and-b-and-d","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/a-student-wrote-a-proof-about-the-product-of-two-rational-numbers-1-let-xab-and-let-ycd-where-a-and-c-are-defined-to-be-integers-and-b-and-d\/","title":{"rendered":"A student wrote a proof about the product of two rational numbers: 1. Let\u00a0x=ab\u00a0and let\u00a0y=cd, where\u00a0a\u00a0and\u00a0c\u00a0are defined to be integers, and\u00a0b\u00a0and\u00a0d\u00a0are nonzero integers."},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">A student wrote a proof about the product of two rational numbers: 1. Let&nbsp;x=ab&nbsp;and let&nbsp;y=cd, where&nbsp;a&nbsp;and&nbsp;c&nbsp;are defined to be integers, and&nbsp;b&nbsp;and&nbsp;d&nbsp;are nonzero integers. 2. By substitution,&nbsp;xy=acbd. 3. By applying the closure property of integers and nonzero integers on multiplication,&nbsp;ac&nbsp;is an integer and&nbsp;bd&nbsp;is a nonzero integer. What conclusion can the student now make about the product&nbsp;xy? The product&nbsp;xy&nbsp;may be either rational or irrational because the values of&nbsp;a,&nbsp;b,&nbsp;c, and&nbsp;d&nbsp;are unknown. The product&nbsp;xy&nbsp;is rational because it can be written as the quotient of an integer and a nonzero integer. The product&nbsp;xy&nbsp;is a nonzero integer because nonzero integers are closed on division. The product&nbsp;xy&nbsp;cannot be an integer because&nbsp;bd&nbsp;is a nonzero integer.<\/pre>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><br><strong>The product&nbsp;<em>xy<\/em> is rational because it can be written as the quotient of an integer and a nonzero integer.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation<\/strong><br>The product of two rational numbers is always <strong>rational<\/strong>. Let&#8217;s break down the reasoning behind this.<\/p>\n\n\n\n<p>We start with the assumption that the student lets<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x=abx = \\frac{a}{b}x=ba\u200b,<\/li>\n\n\n\n<li>y=cdy = \\frac{c}{d}y=dc\u200b,<br>where aaa and ccc are integers, and bbb and ddd are <strong>nonzero<\/strong> integers. These definitions satisfy the condition of rational numbers, which are defined as numbers that can be written as the <strong>quotient of two integers<\/strong>, where the <strong>denominator is nonzero<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>The student then multiplies the two rational numbers:xy=(ab)(cd)=acbdxy = \\left(\\frac{a}{b}\\right) \\left(\\frac{c}{d}\\right) = \\frac{ac}{bd}xy=(ba\u200b)(dc\u200b)=bdac\u200b<\/p>\n\n\n\n<p>Now, consider the numerator and denominator of the resulting expression:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since aaa and ccc are integers, and integers are closed under multiplication, acacac is an integer.<\/li>\n\n\n\n<li>Since bbb and ddd are nonzero integers, and nonzero integers are also closed under multiplication, bdbdbd is a <strong>nonzero integer<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the result acbd\\frac{ac}{bd}bdac\u200b is a <strong>quotient of an integer over a nonzero integer<\/strong>, which fits the definition of a <strong>rational number<\/strong>.<\/p>\n\n\n\n<p>Now, let\u2019s examine the incorrect answer choices:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Saying that the product may be irrational is incorrect because rational \u00d7 rational = rational.<\/li>\n\n\n\n<li>Saying that the product is a <strong>nonzero integer<\/strong> is too specific and not always true (e.g., 12\u00d723=13\\frac{1}{2} \u00d7 \\frac{2}{3} = \\frac{1}{3}21\u200b\u00d732\u200b=31\u200b, which is not an integer).<\/li>\n\n\n\n<li>The statement that \u201cthe product cannot be an integer because bd is nonzero\u201d is flawed reasoning. The denominator being nonzero doesn\u2019t prevent the product from being an integer\u2014it depends on the numerator and denominator values.<\/li>\n<\/ul>\n\n\n\n<p><strong>Therefore, the best conclusion is that the product <em>xy<\/em> is rational.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-134.jpeg\" alt=\"\" class=\"wp-image-229495\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A student wrote a proof about the product of two rational numbers: 1. Let&nbsp;x=ab&nbsp;and let&nbsp;y=cd, where&nbsp;a&nbsp;and&nbsp;c&nbsp;are defined to be integers, and&nbsp;b&nbsp;and&nbsp;d&nbsp;are nonzero integers. 2. By substitution,&nbsp;xy=acbd. 3. By applying the closure property of integers and nonzero integers on multiplication,&nbsp;ac&nbsp;is an integer and&nbsp;bd&nbsp;is a nonzero integer. What conclusion can the student now make about the product&nbsp;xy? [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":"","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-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":"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":""},"mobile":{"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":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-229494","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229494","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=229494"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229494\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}