{"id":11719,"date":"2014-01-29T14:33:43","date_gmt":"2014-01-29T14:33:43","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11719"},"modified":"2022-01-22T00:58:19","modified_gmt":"2022-01-22T00:58:19","slug":"quantos-alunos-foram-almocar","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11719","title":{"rendered":"Quantos alunos foram almo\u00e7ar?"},"content":{"rendered":"<p><ul id='GTTabs_ul_11719' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11719' class='GTTabs_curr'><a  id=\"11719_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11719' ><a  id=\"11719_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_11719'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Um grupo de alunos de uma turma resolveu ir almo\u00e7ar no \u00faltimo dia de aulas. No final, a conta paga foi de $60$ \u20ac.<\/p>\n<p>Como dois desses alunos n\u00e3o tinham dinheiro, os outros resolveram a quest\u00e3o dando cada um mais $8$ \u20ac.<\/p>\n<p>Quantos alunos foram almo\u00e7ar?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11719' onClick='GTTabs_show(1,11719)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11719'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Seja $n \\in \\mathbb{N}$ o n\u00famero de alunos que foram almo\u00e7ar. Assim, temos:<\/p>\n<ul>\n<li>Quantia (em euros) que cabe a cada um dos alunos para pagamento da conta: $\\frac{{60}}{n}$<\/li>\n<li>Quantia (em euros) em falta para pagamento da conta: $2 \\times \\frac{{60}}{n}$<\/li>\n<li>Quantia (em euros) em falta, obtida pela alternativa escolhida: $8 \\times \\left( {n &#8211; 2} \\right)$<\/li>\n<\/ul>\n<p>Equacionando o problema e resolvendo a equa\u00e7\u00e3o, vem:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{\\begin{array}{*{20}{c}}<br \/>\n{2 \\times \\frac{{60}}{n} = 8 \\times \\left( {n &#8211; 2} \\right)}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{120}}{n} &#8211; 8n + 16 = 0}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{120 &#8211; 8{n^2} + 16n}}{n} = 0}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{{n^2} &#8211; 2n &#8211; 15}}{n} = 0}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{\\left( {n + 3} \\right)\\left( {n &#8211; 5} \\right)}}{n} = 0}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\left( {\\begin{array}{*{20}{c}}<br \/>\n{\\left( {n + 3} \\right)\\left( {n &#8211; 5} \\right) = 0}&amp; \\wedge &amp;{n \\ne 0}<br \/>\n\\end{array}} \\right)}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\left( {\\begin{array}{*{20}{c}}<br \/>\n{\\left( {\\begin{array}{*{20}{c}}<br \/>\n{n =\u00a0 &#8211; 3}&amp; \\vee &amp;{n = 5}<br \/>\n\\end{array}} \\right)}&amp; \\wedge &amp;{n \\ne 0}<br \/>\n\\end{array}} \\right)}&amp; \\wedge &amp;{n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n = 5}<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, foram almo\u00e7ar cinco alunos da turma.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11719' onClick='GTTabs_show(0,11719)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um grupo de alunos de uma turma resolveu ir almo\u00e7ar no \u00faltimo dia de aulas. No final, a conta paga foi de $60$ \u20ac. Como dois desses alunos n\u00e3o tinham dinheiro,&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20862,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,425,126],"series":[],"class_list":["post-11719","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-11-o-ano","tag-equacoes","tag-funcao-racional"],"views":3019,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11V2Pag36-12_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11719","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=11719"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11719\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20862"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11719"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11719"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11719"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}