{"id":26197,"date":"2023-06-04T22:44:23","date_gmt":"2023-06-04T21:44:23","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=26197"},"modified":"2023-06-04T23:24:16","modified_gmt":"2023-06-04T22:24:16","slug":"um-problema-sobre-os-termos-de-uma-fracao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=26197","title":{"rendered":"Um problema sobre os termos de uma fra\u00e7\u00e3o"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_26197' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_26197' class='GTTabs_curr'><a  id=\"26197_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_26197' ><a  id=\"26197_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_26197'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Se\u00a0\\(\\frac{x}{y}\\) \u00e9 uma fra\u00e7\u00e3o equivalente a \\(\\frac{2}{7}\\) e a soma dos seus termos \u00e9 igual a 72, ent\u00e3o \\(x &#8211; y\\) \u00e9 igual a:<\/p>\n<p><strong>[A]<\/strong> \u221256\u00a0 \u00a0 \u00a0 \u00a0 <strong>[B]<\/strong> 40\u00a0 \u00a0 \u00a0 \u00a0 <strong>[C]<\/strong> \u221240\u00a0 \u00a0 \u00a0 \u00a0 <strong>[D]<\/strong> 56<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_26197' onClick='GTTabs_show(1,26197)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_26197'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Se\u00a0\\(\\frac{x}{y}\\) \u00e9 uma fra\u00e7\u00e3o equivalente a \\(\\frac{2}{7}\\) e a soma dos seus termos \u00e9 igual a 72, ent\u00e3o \\(x &#8211; y\\) \u00e9 igual a:<\/p>\n<p><strong>[A]<\/strong> \u221256\u00a0 \u00a0 \u00a0 \u00a0 <strong>[B]<\/strong> 40\u00a0 \u00a0 \u00a0 \u00a0 <strong>[C]<\/strong> \u221240\u00a0 \u00a0 \u00a0 \u00a0 <strong>[D]<\/strong> 56<\/p>\n<\/blockquote>\n<p>\u00a0<\/p>\n<p>Ora, se \\(\\frac{2}{7}\\) \u00e9 uma fra\u00e7\u00e3o irredut\u00edvel equivalente a \\(\\frac{x}{y}\\), ent\u00e3o essa fra\u00e7\u00e3o irredut\u00edvel pode ter sido obtida dividindo os termos da fra\u00e7\u00e3o inicial por um n\u00famero natural <em>n<\/em>. Isto \u00e9, a fra\u00e7\u00e3o inicial \u00e9 da forma \\(\\frac{{2n}}{{7n}}\\).<\/p>\n<p>Como a soma dos termos dessa fra\u00e7\u00e3o \u00e9 igual a 72, ent\u00e3o ser\u00e1 \\(\\begin{array}{*{20}{c}}{2n + 7n = 72}&amp; \\Leftrightarrow &amp;{n = 8}\\end{array}\\).<\/p>\n<p>Assim, a fra\u00e7\u00e3o inicial (\\(\\frac{x}{y}\\)) \u00e9 \\(\\frac{{16}}{{56}}\\).<\/p>\n<p>Logo, \\(x &#8211; y\\) \u00e9 igual a \\(16 &#8211; 56 = &#8211; 40\\).<\/p>\n<p>Portanto, a op\u00e7\u00e3o correta \u00e9 <strong>[C]<\/strong> \u221240.<\/p>\n<p>\u00a0<\/p>\n<h6>Uma alternativa com um sistema de equa\u00e7\u00f5es<\/h6>\n<p>Equacionando o problema e resolvendo o sistema de equa\u00e7\u00f5es, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\left\\{ {\\begin{array}{*{20}{l}}{\\frac{x}{y} = \\frac{2}{7}}\\\\{x + y = 72}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{7x = 2y}\\\\{x = 72 &#8211; y}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{504 &#8211; 7y = 2y}\\\\{x = 72 &#8211; y}\\end{array}} \\right.}&amp; \\Leftrightarrow \\\\{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{9y = 504}\\\\{x = 72 &#8211; y}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = 56}\\\\{x = 16}\\end{array}} \\right.}&amp;{}\\end{array}\\]<\/p>\n<p>Logo, \\(x &#8211; y\\) \u00e9 igual a \\(16 &#8211; 56 = &#8211; 40\\).<\/p>\n<p>Portanto, a op\u00e7\u00e3o correta \u00e9 <strong>[C]<\/strong> \u221240.<\/p>\n<p>\u00a0<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_26197' onClick='GTTabs_show(0,26197)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Se\u00a0\\(\\frac{x}{y}\\) \u00e9 uma fra\u00e7\u00e3o equivalente a \\(\\frac{2}{7}\\) e a soma dos seus termos \u00e9 igual a 72, ent\u00e3o \\(x &#8211; y\\) \u00e9 igual a: [A] \u221256\u00a0 \u00a0 \u00a0 \u00a0 [B] 40\u00a0&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19267,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,714],"tags":[424,345,239],"series":[],"class_list":["post-26197","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-equacoes-literais-e-sistemas","tag-8-o-ano","tag-funcao-afim","tag-sistema-de-equacoes"],"views":72,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat88.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/26197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26197"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/26197\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19267"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=26197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=26197"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=26197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}